diff --git a/lean/OAI/Probability/SLE/Loewner/ForwardHeightSquareIncrement.lean b/lean/OAI/Probability/SLE/Loewner/ForwardHeightSquareIncrement.lean index 2be6838383aafa7198c39812d90271b6cda26487..8fcad00b621a1361dedcb248759d13d89f919d35 100644 --- a/lean/OAI/Probability/SLE/Loewner/ForwardHeightSquareIncrement.lean +++ b/lean/OAI/Probability/SLE/Loewner/ForwardHeightSquareIncrement.lean @@ -1,55 +1,55 @@ -import OAI.Probability.SLE.Loewner.CapacityPointwiseLoewner -import OAI.Probability.SLE.Geometry.EndpointMonotonicity - -namespace OAI - -/-! The squared height can decrease by at most twice elapsed capacity. -The proof uses pointwise right derivatives and includes both endpoints. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma height_kernel_square_bound (z : ℂ) (u : ℝ) : - 2*z.im^2/Complex.normSq (z-(u:ℂ))-2≤0 := by - have hsq : z.im^2≤Complex.normSq (z-(u:ℂ)) := by - simp only [Complex.normSq_apply,sub_im,ofReal_im,sub_zero] - nlinarith [sq_nonneg (z-(u:ℂ)).re] - by_cases hn : Complex.normSq (z-(u:ℂ))=0 - · simp only [hn,div_zero,zero_sub] - norm_num - · have hpos := lt_of_le_of_ne (Complex.normSq_nonneg (z-(u:ℂ))) (Ne.symm hn) - have hb : 2*z.im^2/Complex.normSq (z-(u:ℂ))≤2 := - (div_le_iff₀ hpos).mpr (by nlinarith) - linarith - -theorem height_square_increment {f : ℝ → ℂ} {u : ℝ → ℝ} {a b : ℝ} - (hab : a≤b) (hf : ContinuousOn f (Icc a b)) - (hd : ∀ c∈Ico a b, HasDerivWithinAt f (1/(f c-(u c:ℂ))) (Ici c) c) : - (f a).im^2≤(f b).im^2+2*(b-a) := by - let g (c : ℝ) := -(f c).im^2-2*c - let v (c : ℝ) := 2*(f c).im^2/Complex.normSq (f c-(u c:ℂ))-2 - have hg : ContinuousOn g (Icc a b) := - ((Complex.continuous_im.comp_continuousOn hf).pow 2).neg.sub - (continuousOn_const.mul continuousOn_id) - have hgd (c : ℝ) (hc : c∈Ico a b) : HasDerivWithinAt g (v c) (Ici c) c := by - have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt c (hd c hc) - change HasDerivWithinAt (fun x => (f x).im) (1/(f c-(u c:ℂ))).im (Ici c) c at hi - have hh := (hi.pow 2).neg.sub ((hasDerivWithinAt_id c (Ici c)).const_mul 2) - convert! hh using 1 - dsimp only [v] - simp only [Complex.div_im,one_im,one_re,zero_mul,one_mul,sub_im,ofReal_im,sub_zero] - ring - have hb := image_le_of_deriv_right_le_deriv_boundary hg hgd - (B := fun _ => g a) (B' := fun _ => 0) le_rfl continuousOn_const - (fun c _ => hasDerivWithinAt_const c (Ici c) (g a)) - (fun c _ => height_kernel_square_bound (f c) (u c)) (show b∈Icc a b from ⟨hab,le_rfl⟩) - dsimp only [g] at hb - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.CapacityPointwiseLoewner +import OAI.Probability.SLE.Geometry.EndpointMonotonicity + +namespace OAI + +/-! The squared height can decrease by at most twice elapsed capacity. +The proof uses pointwise right derivatives and includes both endpoints. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma height_kernel_square_bound (z : ℂ) (u : ℝ) : + 2*z.im^2/Complex.normSq (z-(u:ℂ))-2≤0 := by + have hsq : z.im^2≤Complex.normSq (z-(u:ℂ)) := by + simp only [Complex.normSq_apply,sub_im,ofReal_im,sub_zero] + nlinarith [sq_nonneg (z-(u:ℂ)).re] + by_cases hn : Complex.normSq (z-(u:ℂ))=0 + · simp only [hn,div_zero,zero_sub] + norm_num + · have hpos := lt_of_le_of_ne (Complex.normSq_nonneg (z-(u:ℂ))) (Ne.symm hn) + have hb : 2*z.im^2/Complex.normSq (z-(u:ℂ))≤2 := + (div_le_iff₀ hpos).mpr (by nlinarith) + linarith + +theorem height_square_increment {f : ℝ → ℂ} {u : ℝ → ℝ} {a b : ℝ} + (hab : a≤b) (hf : ContinuousOn f (Icc a b)) + (hd : ∀ c∈Ico a b, HasDerivWithinAt f (1/(f c-(u c:ℂ))) (Ici c) c) : + (f a).im^2≤(f b).im^2+2*(b-a) := by + let g (c : ℝ) := -(f c).im^2-2*c + let v (c : ℝ) := 2*(f c).im^2/Complex.normSq (f c-(u c:ℂ))-2 + have hg : ContinuousOn g (Icc a b) := + ((Complex.continuous_im.comp_continuousOn hf).pow 2).neg.sub + (continuousOn_const.mul continuousOn_id) + have hgd (c : ℝ) (hc : c∈Ico a b) : HasDerivWithinAt g (v c) (Ici c) c := by + have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt c (hd c hc) + change HasDerivWithinAt (fun x => (f x).im) (1/(f c-(u c:ℂ))).im (Ici c) c at hi + have hh := (hi.pow 2).neg.sub ((hasDerivWithinAt_id c (Ici c)).const_mul 2) + convert! hh using 1 + dsimp only [v] + simp only [Complex.div_im,one_im,one_re,zero_mul,one_mul,sub_im,ofReal_im,sub_zero] + ring + have hb := image_le_of_deriv_right_le_deriv_boundary hg hgd + (B := fun _ => g a) (B' := fun _ => 0) le_rfl continuousOn_const + (fun c _ => hasDerivWithinAt_const c (Ici c) (g a)) + (fun c _ => height_kernel_square_bound (f c) (u c)) (show b∈Icc a b from ⟨hab,le_rfl⟩) + dsimp only [g] at hb + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ForwardInfinityBounds.lean b/lean/OAI/Probability/SLE/Loewner/ForwardInfinityBounds.lean index 2014c9273d276b61936fe5e79bfe1f0eebbc410e..8908a50e898b0fa72ae0f51da7543d1acde1a8ed 100644 --- a/lean/OAI/Probability/SLE/Loewner/ForwardInfinityBounds.lean +++ b/lean/OAI/Probability/SLE/Loewner/ForwardInfinityBounds.lean @@ -1,79 +1,79 @@ -import Mathlib - -namespace OAI - -/-! Uniform bounds for forward Loewner trajectories far from the origin. -Only the bounded driver on the fixed time interval is used. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -lemma forward_far_center_lower {q z : ℂ} {u M A r : ℝ} - (hr : r=‖z‖) (hu : |u| ≤ M) (hq : ‖q-z‖ ≤ A) - (hsize : 2*(M+A) ≤ r) : r/2 ≤ ‖q-(u : ℂ)‖ := by - have h1 := norm_le_norm_sub_add z q - have h2 := norm_le_norm_sub_add q (u : ℂ) - rw [norm_sub_rev z q] at h1 - rw [Complex.norm_real,Real.norm_eq_abs] at h2 - linarith - -theorem forward_far_displacement {T M : ℝ} (hT : 0 ≤ T) (hM : 0 ≤ M) - (q : ℝ → ℂ) (u : ℝ → ℝ) (z : ℂ) - (hc : ContinuousOn q (Icc 0 T)) (hq0 : q 0=z) - (hd : ∀ t ∈ Ico 0 T, HasDerivWithinAt q - (2/(q t-(u t : ℂ))) (Ici t) t) - (hu : ∀ t ∈ Icc 0 T, |u t| ≤ M) - (hlarge : 8*(M+T+1) ≤ ‖z‖) : - (∀ t ∈ Icc 0 T, ‖q t-z‖ ≤ (4/‖z‖)*t) ∧ - ∀ t ∈ Icc 0 T, ‖z‖/2 ≤ ‖q t-(u t : ℂ)‖ := by - let r : ℝ := ‖z‖ - have hr : 0 < r := by dsimp [r]; nlinarith - have hr8 : 8 ≤ r := by dsimp [r]; nlinarith - have hfac : 8/r ≤ 1 := (div_le_one hr).mpr hr8 - have hbnd (t : ℝ) (ht : t ∈ Icc 0 T) : 1+(8/r)*t ≤ 1+T := by - have hh := mul_le_mul_of_nonneg_right hfac ht.1 - nlinarith [ht.2] - have hsize : 2*(M+(1+T)) ≤ r := by dsimp [r]; nlinarith - have hrough : ∀ t ∈ Icc 0 T, ‖q t-z‖ ≤ 1+(8/r)*t := by - apply image_norm_le_of_norm_deriv_right_lt_deriv_boundary - (hc.sub continuousOn_const) (fun t ht => (hd t ht).sub_const z) - (B := fun t => 1+(8/r)*t) (B' := fun _ => 8/r) - · simp [hq0] - · intro t - convert! ((hasDerivAt_id t).const_mul (8/r)).const_add 1 using 1 - simp - · intro t ht he - have hcenter := forward_far_center_lower (q := q t) (z := z) - (u := u t) (M := M) (A := 1+T) rfl (hu t (Ico_subset_Icc_self ht)) - (he.le.trans (hbnd t (Ico_subset_Icc_self ht))) hsize - have hnorm : ‖2/(q t-(u t : ℂ))‖ ≤ 4/r := by - rw [norm_div] - norm_num only [norm_ofNat] - have hh := div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 2) - (half_pos hr) hcenter - convert! hh using 1 - field_simp - ring - exact hnorm.trans_lt (div_lt_div_of_pos_right (by norm_num) hr) - have hcenter (t : ℝ) (ht : t ∈ Icc 0 T) : r/2 ≤ ‖q t-(u t : ℂ)‖ := - forward_far_center_lower rfl (hu t ht) ((hrough t ht).trans (hbnd t ht)) hsize - refine ⟨?_,hcenter⟩ - have hderiv (t : ℝ) (ht : t ∈ Ico 0 T) : ‖2/(q t-(u t : ℂ))‖ ≤ 4/r := by - rw [norm_div] - norm_num only [norm_ofNat] - have hh := div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 2) - (half_pos hr) (hcenter t (Ico_subset_Icc_self ht)) - convert! hh using 1 - field_simp - ring - intro t ht - simpa only [hq0,sub_zero] using - norm_image_sub_le_of_norm_deriv_right_le_segment hc hd hderiv t ht - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! Uniform bounds for forward Loewner trajectories far from the origin. +Only the bounded driver on the fixed time interval is used. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +lemma forward_far_center_lower {q z : ℂ} {u M A r : ℝ} + (hr : r=‖z‖) (hu : |u| ≤ M) (hq : ‖q-z‖ ≤ A) + (hsize : 2*(M+A) ≤ r) : r/2 ≤ ‖q-(u : ℂ)‖ := by + have h1 := norm_le_norm_sub_add z q + have h2 := norm_le_norm_sub_add q (u : ℂ) + rw [norm_sub_rev z q] at h1 + rw [Complex.norm_real,Real.norm_eq_abs] at h2 + linarith + +theorem forward_far_displacement {T M : ℝ} (hT : 0 ≤ T) (hM : 0 ≤ M) + (q : ℝ → ℂ) (u : ℝ → ℝ) (z : ℂ) + (hc : ContinuousOn q (Icc 0 T)) (hq0 : q 0=z) + (hd : ∀ t ∈ Ico 0 T, HasDerivWithinAt q + (2/(q t-(u t : ℂ))) (Ici t) t) + (hu : ∀ t ∈ Icc 0 T, |u t| ≤ M) + (hlarge : 8*(M+T+1) ≤ ‖z‖) : + (∀ t ∈ Icc 0 T, ‖q t-z‖ ≤ (4/‖z‖)*t) ∧ + ∀ t ∈ Icc 0 T, ‖z‖/2 ≤ ‖q t-(u t : ℂ)‖ := by + let r : ℝ := ‖z‖ + have hr : 0 < r := by dsimp [r]; nlinarith + have hr8 : 8 ≤ r := by dsimp [r]; nlinarith + have hfac : 8/r ≤ 1 := (div_le_one hr).mpr hr8 + have hbnd (t : ℝ) (ht : t ∈ Icc 0 T) : 1+(8/r)*t ≤ 1+T := by + have hh := mul_le_mul_of_nonneg_right hfac ht.1 + nlinarith [ht.2] + have hsize : 2*(M+(1+T)) ≤ r := by dsimp [r]; nlinarith + have hrough : ∀ t ∈ Icc 0 T, ‖q t-z‖ ≤ 1+(8/r)*t := by + apply image_norm_le_of_norm_deriv_right_lt_deriv_boundary + (hc.sub continuousOn_const) (fun t ht => (hd t ht).sub_const z) + (B := fun t => 1+(8/r)*t) (B' := fun _ => 8/r) + · simp [hq0] + · intro t + convert! ((hasDerivAt_id t).const_mul (8/r)).const_add 1 using 1 + simp + · intro t ht he + have hcenter := forward_far_center_lower (q := q t) (z := z) + (u := u t) (M := M) (A := 1+T) rfl (hu t (Ico_subset_Icc_self ht)) + (he.le.trans (hbnd t (Ico_subset_Icc_self ht))) hsize + have hnorm : ‖2/(q t-(u t : ℂ))‖ ≤ 4/r := by + rw [norm_div] + norm_num only [norm_ofNat] + have hh := div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 2) + (half_pos hr) hcenter + convert! hh using 1 + field_simp + ring + exact hnorm.trans_lt (div_lt_div_of_pos_right (by norm_num) hr) + have hcenter (t : ℝ) (ht : t ∈ Icc 0 T) : r/2 ≤ ‖q t-(u t : ℂ)‖ := + forward_far_center_lower rfl (hu t ht) ((hrough t ht).trans (hbnd t ht)) hsize + refine ⟨?_,hcenter⟩ + have hderiv (t : ℝ) (ht : t ∈ Ico 0 T) : ‖2/(q t-(u t : ℂ))‖ ≤ 4/r := by + rw [norm_div] + norm_num only [norm_ofNat] + have hh := div_le_div_of_nonneg_left (by norm_num : (0:ℝ) ≤ 2) + (half_pos hr) (hcenter t (Ico_subset_Icc_self ht)) + convert! hh using 1 + field_simp + ring + intro t ht + simpa only [hq0,sub_zero] using + norm_image_sub_le_of_norm_deriv_right_le_segment hc hd hderiv t ht + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/FreshCapacityWitness.lean b/lean/OAI/Probability/SLE/Loewner/FreshCapacityWitness.lean index 83eb5c5c05b63e0f47e3c3716789282901e424c2..dba37f90575934e086595670e77ec9a926069fa6 100644 --- a/lean/OAI/Probability/SLE/Loewner/FreshCapacityWitness.lean +++ b/lean/OAI/Probability/SLE/Loewner/FreshCapacityWitness.lean @@ -1,48 +1,48 @@ -import OAI.Probability.SLE.Geometry.CanonicalRestartWitness - -namespace OAI - -/-! The published trace-generation input is used only on the single -continuous Brownian path law. Strong Markov supplies the fresh witness. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge - -local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem fresh_capacity_witness_exists - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) : - ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), - IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) - β G' F' := by - obtain ⟨Φ,Gc,Fc,_,hW⟩ := exists_measurable_canonical_witness _ κ hpublished - have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _ - (fun t => (measurable_pi_apply t).comp - (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) - have hLaw : P.map (sourceRestartDriverPath B hc τ)=P.map (sourceDriverPath B hc) := by - simpa only [Measure.restrict_univ,measure_univ,one_smul] using - source_restart_continuous_driver_law hB hm hc τ hτ (A := univ) MeasurableSet.univ - filter_upwards [canonical_witness_pullback hW _ hY hLaw] with ω hω - refine ⟨Φ (sourceRestartDriverPath B hc τ ω),Gc (sourceRestartDriverPath B hc τ ω), - Fc (sourceRestartDriverPath B hc τ ω),?_⟩ - convert hω using 1 - funext t - change Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω = - Real.sqrt κ*(B (τ ω+t) ω-B (τ ω) ω) - ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalRestartWitness + +namespace OAI + +/-! The published trace-generation input is used only on the single +continuous Brownian path law. Strong Markov supplies the fresh witness. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge + +local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem fresh_capacity_witness_exists + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) : + ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), + IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) + β G' F' := by + obtain ⟨Φ,Gc,Fc,_,hW⟩ := exists_measurable_canonical_witness _ κ hpublished + have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _ + (fun t => (measurable_pi_apply t).comp + (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) + have hLaw : P.map (sourceRestartDriverPath B hc τ)=P.map (sourceDriverPath B hc) := by + simpa only [Measure.restrict_univ,measure_univ,one_smul] using + source_restart_continuous_driver_law hB hm hc τ hτ (A := univ) MeasurableSet.univ + filter_upwards [canonical_witness_pullback hW _ hY hLaw] with ω hω + refine ⟨Φ (sourceRestartDriverPath B hc τ ω),Gc (sourceRestartDriverPath B hc τ ω), + Fc (sourceRestartDriverPath B hc τ ω),?_⟩ + convert hω using 1 + funext t + change Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω = + Real.sqrt κ*(B (τ ω+t) ω-B (τ ω) ω) + ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/FreshTipMapOut.lean b/lean/OAI/Probability/SLE/Loewner/FreshTipMapOut.lean index 79eba7b6696c82d19f6d2f4a167c264f5cf78724..083e59e40cb9b4130c0120cba4cf1f5f837e4ecf 100644 --- a/lean/OAI/Probability/SLE/Loewner/FreshTipMapOut.lean +++ b/lean/OAI/Probability/SLE/Loewner/FreshTipMapOut.lean @@ -1,139 +1,139 @@ -import OAI.Probability.SLE.PhysicalTransport -import OAI.Probability.SLE.Gauge.RibbonSurvival - -namespace OAI - -/-! Mapping the actual geometric continuation into the restarted half-plane. -At a first height crossing the tip is fresh. The fresh-tip chart limit then maps a simple access path continuously to a simple path -starting at the restarted driving point. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal unitInterval - -namespace SLEExactGauge - -theorem tip_survives_prior_of_strict_height - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (hheight : 0 < (γ T).im) - (hfirst : ∀ u : ℝ≥0, u < T → (γ u).im < (γ T).im) : - ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s := by - intro s hs - have hfree : range (upwardRay (γ T)) ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by - rintro z ⟨u, rfl⟩ - have him : (upwardRay (γ T) u).im = (γ T).im + (u : ℝ) := by simp [upwardRay] - refine ⟨by change 0 < (upwardRay (γ T) u).im; rw [him]; positivity, ?_⟩ - rintro ⟨v, hv, he⟩ - have hh := hfirst v (hv.2.trans_lt hs) - have he' := congrArg Complex.im he - rw [him] at he' - linarith [u.coe_nonneg] - have hh := survives_of_unbounded_free_curve γ s (upwardRay (γ T)) - (upwardRay_continuous _) hfree (upwardRay_unbounded _) (mem_range_self 0) - simpa only [upwardRay, NNReal.coe_zero, Complex.ofReal_zero, zero_mul, add_zero] using hh - -/-- Extend the map only at the accessible tip. This is sufficient to map -the entire simple continuation Path, and does not require a boundary -extension of the forward conformal map at other hull points. -/ -theorem map_out_simple_access_path - (D : Set ℂ) (_hD : IsOpen D) (g : ℂ → ℂ) - (hg : ContinuousOn g D) (hgi : InjOn g D) (hgH : MapsTo g D upperHalfPlane) - (ζ w : ℂ) (u : ℝ) (hζ : ζ ∉ D) - (hlim : Tendsto g (𝓝[D] ζ) (𝓝 (u : ℂ))) - (p : Path ζ w) (hp : Function.Injective p) - (hpD : ∀ t, t ≠ 0 → p t ∈ D) : - ∃ q : Path (0 : ℂ) (g w - (u : ℂ)), Function.Injective q ∧ - (∀ t, t ≠ 0 → q t ∈ upperHalfPlane) ∧ - (∀ t, t ≠ 0 → q t = g (p t) - (u : ℂ)) := by - classical - let gb := Function.update g ζ (u : ℂ) - have hs : insert ζ D \ {ζ} = D := by - ext z - simp only [Set.mem_sdiff, mem_insert_iff, mem_singleton_iff] - constructor - · rintro ⟨rfl | hz, hn⟩ - · exact False.elim (hn rfl) - · exact hz - · intro hz - exact ⟨Or.inr hz, fun he => hζ (he ▸ hz)⟩ - have hgb : ContinuousOn gb (insert ζ D) := by - rw [continuousOn_update_iff, hs] - exact ⟨hg, fun _ => hlim⟩ - have hpS : range p ⊆ insert ζ D := by - rintro z ⟨t, rfl⟩ - by_cases ht : t = 0 - · simpa only [ht, p.source] using mem_insert ζ D - · exact mem_insert_of_mem ζ (hpD t ht) - have hne (t : unitInterval) (ht : t ≠ 0) : p t ≠ ζ := - fun he => hζ (he ▸ hpD t ht) - have heval (t : unitInterval) (ht : t ≠ 0) : gb (p t) = g (p t) := - Function.update_of_ne (hne t ht) _ _ - have hw : gb w = g w := by - simpa only [p.target] using heval 1 one_ne_zero - let q : Path (0 : ℂ) (g w - (u : ℂ)) := { - toFun := fun t => gb (p t) - (u : ℂ) - continuous_toFun := ((hgb.mono hpS).comp_continuous p.continuous - (fun t => mem_range_self t)).sub continuous_const - source' := by simp [gb] - target' := by simp only [p.target, hw] } - have hqH (t : unitInterval) (ht : t ≠ 0) : q t ∈ upperHalfPlane := by - change 0 < (gb (p t) - (u : ℂ)).im - rw [heval t ht] - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, - sub_zero] using hgH (hpD t ht) - refine ⟨q, ?_, hqH, fun t ht => by change gb (p t) - _ = _; rw [heval t ht]⟩ - intro s t he - by_cases hs0 : s = 0 - · subst s - by_contra ht0 - have hh := hqH t (Ne.symm ht0) - rw [← he, q.source] at hh - simp only [upperHalfPlane, mem_ofPred_eq, Complex.zero_im, lt_self_iff_false] at hh - by_cases ht0 : t = 0 - · subst t - have hh := hqH s hs0 - rw [he, q.source] at hh - simp only [upperHalfPlane, mem_ofPred_eq, Complex.zero_im, lt_self_iff_false] at hh - apply hp - apply hgi (hpD s hs0) (hpD t ht0) - have he' : gb (p s) - (u : ℂ) = gb (p t) - (u : ℂ) := he - rw [heval s hs0, heval t ht0] at he' - exact sub_left_injective he' - -/-- Actual capacity charts at a first height crossing map the physical -simple continuation to a supported simple path in the restarted half-plane, -with the conformal map-back identity at every positive path time. -/ -theorem IsCapacityTwoWitness.map_out_access_at_first_height - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) - (T : ℝ≥0) (hT : 0 < T) (hheight : 0 < (γ T).im) - (hfirst : ∀ u : ℝ≥0, u < T → (γ u).im < (γ T).im) - (w : ℂ) (p : Path (γ T) w) (hp : Function.Injective p) - (hpD : ∀ t, t ≠ 0 → p t ∈ survivingDomain γ T) : - ∃ q : Path (0 : ℂ) (G T w - (U T : ℂ)), Function.Injective q ∧ - (∀ t, t ≠ 0 → q t ∈ upperHalfPlane) ∧ - (∀ t, t ≠ 0 → F T (q t + (U T : ℂ)) = p t) := by - have hgi : InjOn (G T) (survivingDomain γ T) := by - intro z hz v hv he - calc - z = F T (G T z) := (hw.2.2.2.2.2.2.2.1 T z hz).symm - _ = F T (G T v) := congrArg (F T) he - _ = v := hw.2.2.2.2.2.2.2.1 T v hv - have htip : γ T ∉ survivingDomain γ T := by - intro hh - exact hh.1.2 (mem_image_of_mem γ ⟨zero_le, le_rfl⟩) - obtain ⟨q, hq, hqH, hqeq⟩ := map_out_simple_access_path - (survivingDomain γ T) (isOpen_survivingDomain hw.1 T) (G T) - (hw.2.2.2.1 T).continuousOn hgi (hw.2.2.2.2.2.1 T) - (γ T) w (U T) htip - (hw.fresh_tip_chart_tendsto hU hT - (tip_survives_prior_of_strict_height γ T hheight hfirst)) p hp hpD - refine ⟨q, hq, hqH, ?_⟩ - intro t ht - rw [hqeq t ht, sub_add_cancel] - exact hw.2.2.2.2.2.2.2.1 T (p t) (hpD t ht) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.PhysicalTransport +import OAI.Probability.SLE.Gauge.RibbonSurvival + +namespace OAI + +/-! Mapping the actual geometric continuation into the restarted half-plane. +At a first height crossing the tip is fresh. The fresh-tip chart limit then maps a simple access path continuously to a simple path +starting at the restarted driving point. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal unitInterval + +namespace SLEExactGauge + +theorem tip_survives_prior_of_strict_height + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (hheight : 0 < (γ T).im) + (hfirst : ∀ u : ℝ≥0, u < T → (γ u).im < (γ T).im) : + ∀ s : ℝ≥0, s < T → γ T ∈ survivingDomain γ s := by + intro s hs + have hfree : range (upwardRay (γ T)) ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by + rintro z ⟨u, rfl⟩ + have him : (upwardRay (γ T) u).im = (γ T).im + (u : ℝ) := by simp [upwardRay] + refine ⟨by change 0 < (upwardRay (γ T) u).im; rw [him]; positivity, ?_⟩ + rintro ⟨v, hv, he⟩ + have hh := hfirst v (hv.2.trans_lt hs) + have he' := congrArg Complex.im he + rw [him] at he' + linarith [u.coe_nonneg] + have hh := survives_of_unbounded_free_curve γ s (upwardRay (γ T)) + (upwardRay_continuous _) hfree (upwardRay_unbounded _) (mem_range_self 0) + simpa only [upwardRay, NNReal.coe_zero, Complex.ofReal_zero, zero_mul, add_zero] using hh + +/-- Extend the map only at the accessible tip. This is sufficient to map +the entire simple continuation Path, and does not require a boundary +extension of the forward conformal map at other hull points. -/ +theorem map_out_simple_access_path + (D : Set ℂ) (_hD : IsOpen D) (g : ℂ → ℂ) + (hg : ContinuousOn g D) (hgi : InjOn g D) (hgH : MapsTo g D upperHalfPlane) + (ζ w : ℂ) (u : ℝ) (hζ : ζ ∉ D) + (hlim : Tendsto g (𝓝[D] ζ) (𝓝 (u : ℂ))) + (p : Path ζ w) (hp : Function.Injective p) + (hpD : ∀ t, t ≠ 0 → p t ∈ D) : + ∃ q : Path (0 : ℂ) (g w - (u : ℂ)), Function.Injective q ∧ + (∀ t, t ≠ 0 → q t ∈ upperHalfPlane) ∧ + (∀ t, t ≠ 0 → q t = g (p t) - (u : ℂ)) := by + classical + let gb := Function.update g ζ (u : ℂ) + have hs : insert ζ D \ {ζ} = D := by + ext z + simp only [Set.mem_sdiff, mem_insert_iff, mem_singleton_iff] + constructor + · rintro ⟨rfl | hz, hn⟩ + · exact False.elim (hn rfl) + · exact hz + · intro hz + exact ⟨Or.inr hz, fun he => hζ (he ▸ hz)⟩ + have hgb : ContinuousOn gb (insert ζ D) := by + rw [continuousOn_update_iff, hs] + exact ⟨hg, fun _ => hlim⟩ + have hpS : range p ⊆ insert ζ D := by + rintro z ⟨t, rfl⟩ + by_cases ht : t = 0 + · simpa only [ht, p.source] using mem_insert ζ D + · exact mem_insert_of_mem ζ (hpD t ht) + have hne (t : unitInterval) (ht : t ≠ 0) : p t ≠ ζ := + fun he => hζ (he ▸ hpD t ht) + have heval (t : unitInterval) (ht : t ≠ 0) : gb (p t) = g (p t) := + Function.update_of_ne (hne t ht) _ _ + have hw : gb w = g w := by + simpa only [p.target] using heval 1 one_ne_zero + let q : Path (0 : ℂ) (g w - (u : ℂ)) := { + toFun := fun t => gb (p t) - (u : ℂ) + continuous_toFun := ((hgb.mono hpS).comp_continuous p.continuous + (fun t => mem_range_self t)).sub continuous_const + source' := by simp [gb] + target' := by simp only [p.target, hw] } + have hqH (t : unitInterval) (ht : t ≠ 0) : q t ∈ upperHalfPlane := by + change 0 < (gb (p t) - (u : ℂ)).im + rw [heval t ht] + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, + sub_zero] using hgH (hpD t ht) + refine ⟨q, ?_, hqH, fun t ht => by change gb (p t) - _ = _; rw [heval t ht]⟩ + intro s t he + by_cases hs0 : s = 0 + · subst s + by_contra ht0 + have hh := hqH t (Ne.symm ht0) + rw [← he, q.source] at hh + simp only [upperHalfPlane, mem_ofPred_eq, Complex.zero_im, lt_self_iff_false] at hh + by_cases ht0 : t = 0 + · subst t + have hh := hqH s hs0 + rw [he, q.source] at hh + simp only [upperHalfPlane, mem_ofPred_eq, Complex.zero_im, lt_self_iff_false] at hh + apply hp + apply hgi (hpD s hs0) (hpD t ht0) + have he' : gb (p s) - (u : ℂ) = gb (p t) - (u : ℂ) := he + rw [heval s hs0, heval t ht0] at he' + exact sub_left_injective he' + +/-- Actual capacity charts at a first height crossing map the physical +simple continuation to a supported simple path in the restarted half-plane, +with the conformal map-back identity at every positive path time. -/ +theorem IsCapacityTwoWitness.map_out_access_at_first_height + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) + (T : ℝ≥0) (hT : 0 < T) (hheight : 0 < (γ T).im) + (hfirst : ∀ u : ℝ≥0, u < T → (γ u).im < (γ T).im) + (w : ℂ) (p : Path (γ T) w) (hp : Function.Injective p) + (hpD : ∀ t, t ≠ 0 → p t ∈ survivingDomain γ T) : + ∃ q : Path (0 : ℂ) (G T w - (U T : ℂ)), Function.Injective q ∧ + (∀ t, t ≠ 0 → q t ∈ upperHalfPlane) ∧ + (∀ t, t ≠ 0 → F T (q t + (U T : ℂ)) = p t) := by + have hgi : InjOn (G T) (survivingDomain γ T) := by + intro z hz v hv he + calc + z = F T (G T z) := (hw.2.2.2.2.2.2.2.1 T z hz).symm + _ = F T (G T v) := congrArg (F T) he + _ = v := hw.2.2.2.2.2.2.2.1 T v hv + have htip : γ T ∉ survivingDomain γ T := by + intro hh + exact hh.1.2 (mem_image_of_mem γ ⟨zero_le, le_rfl⟩) + obtain ⟨q, hq, hqH, hqeq⟩ := map_out_simple_access_path + (survivingDomain γ T) (isOpen_survivingDomain hw.1 T) (G T) + (hw.2.2.2.1 T).continuousOn hgi (hw.2.2.2.2.2.1 T) + (γ T) w (U T) htip + (hw.fresh_tip_chart_tendsto hU hT + (tip_survives_prior_of_strict_height γ T hheight hfirst)) p hp hpD + refine ⟨q, hq, hqH, ?_⟩ + intro t ht + rw [hqeq t ht, sub_add_cancel] + exact hw.2.2.2.2.2.2.2.1 T (p t) (hpD t ht) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/FutureArcThinCrossing.lean b/lean/OAI/Probability/SLE/Loewner/FutureArcThinCrossing.lean index 6086d886a39d2244e9e88945417e8250c6984646..e23abb399c21dc6dde33a7724e4db8445c5ff452 100644 --- a/lean/OAI/Probability/SLE/Loewner/FutureArcThinCrossing.lean +++ b/lean/OAI/Probability/SLE/Loewner/FutureArcThinCrossing.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.Loewner.InverseThinCollapse -import OAI.Probability.SLE.Loewner.FutureSlitHeight -import OAI.Probability.SLE.Loewner.MappedFutureRange - -namespace OAI - -/-! A real crossing of the actual mapped future supplies the small -inverse values required by the Dirichlet-energy criterion. -/ -noncomputable section -open Set Filter Function Complex Metric MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -theorem mappedFutureArc_thin_crossing (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s (q u).re) (Icc (0:unitInterval) 1) := - (Complex.continuous_re.comp q.continuous).continuousOn - have hex : ∃ u∈Icc (0:unitInterval) 1, (q u).re=x := by - rcases hx with hx | hx - · apply intermediate_value_Icc (show (0:unitInterval)≤1 from zero_le_one) hcont - simpa only [hq0,hq1] using ⟨hx.1.le,hx.2.le⟩ - · apply intermediate_value_Icc' (show (0:unitInterval)≤1 from zero_le_one) hcont - simpa only [hq0,hq1] using ⟨hx.1.le,hx.2.le⟩ - obtain ⟨u,_,hux⟩ := hex - have hu : u≠0 := by - intro he - subst u - rw [hq0] at hux - rcases hx with hx | hx <;> linarith [hx.1,hx.2] - have hpos : 0<(q u).im := mappedFutureArc_upper p hi h0 hH hst ht hu - have him : (centeredFutureArc p hi h0 hH hst ht u).im=(q u).im := by - change (q u-(_ : ℂ)).im=(q u).im - simp only [sub_im,ofReal_im,sub_zero] - have hsq := centeredFutureArc_height_sq_le p hi h0 hH hst ht u - rw [him] at hsq - have hcap : 0≤2*(initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s) := - mul_nonneg (by norm_num) (sub_nonneg.mpr (initialCapacity_monotone p hi h0 hH hst.le)) - have hheight : (q u).im≤Real.sqrt (2*(initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s)) := - (Real.le_sqrt hpos.le hcap).mpr hsq - have hcoord : (x:ℂ)+((q u).im:ℂ)*I=q u := by - apply Complex.ext <;> simp [hux] - refine ⟨(q u).im,⟨hpos,hheight⟩,?_⟩ - rw [hcoord,mappedFutureArc_segment p hi h0 hH hst ht hu, - initialInverse_forward_at p hi h0 hH s - (point_after_prefix_mem_domain p hi hH (slitSegmentParameter_gt hst hu))] - exact hp _ ⟨slitSegmentParameter_gt hst hu,slitSegmentParameter_le s t hst.le u⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseThinCollapse +import OAI.Probability.SLE.Loewner.FutureSlitHeight +import OAI.Probability.SLE.Loewner.MappedFutureRange + +namespace OAI + +/-! A real crossing of the actual mapped future supplies the small +inverse values required by the Dirichlet-energy criterion. -/ +noncomputable section +open Set Filter Function Complex Metric MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +theorem mappedFutureArc_thin_crossing (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s (q u).re) (Icc (0:unitInterval) 1) := + (Complex.continuous_re.comp q.continuous).continuousOn + have hex : ∃ u∈Icc (0:unitInterval) 1, (q u).re=x := by + rcases hx with hx | hx + · apply intermediate_value_Icc (show (0:unitInterval)≤1 from zero_le_one) hcont + simpa only [hq0,hq1] using ⟨hx.1.le,hx.2.le⟩ + · apply intermediate_value_Icc' (show (0:unitInterval)≤1 from zero_le_one) hcont + simpa only [hq0,hq1] using ⟨hx.1.le,hx.2.le⟩ + obtain ⟨u,_,hux⟩ := hex + have hu : u≠0 := by + intro he + subst u + rw [hq0] at hux + rcases hx with hx | hx <;> linarith [hx.1,hx.2] + have hpos : 0<(q u).im := mappedFutureArc_upper p hi h0 hH hst ht hu + have him : (centeredFutureArc p hi h0 hH hst ht u).im=(q u).im := by + change (q u-(_ : ℂ)).im=(q u).im + simp only [sub_im,ofReal_im,sub_zero] + have hsq := centeredFutureArc_height_sq_le p hi h0 hH hst ht u + rw [him] at hsq + have hcap : 0≤2*(initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s) := + mul_nonneg (by norm_num) (sub_nonneg.mpr (initialCapacity_monotone p hi h0 hH hst.le)) + have hheight : (q u).im≤Real.sqrt (2*(initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s)) := + (Real.le_sqrt hpos.le hcap).mpr hsq + have hcoord : (x:ℂ)+((q u).im:ℂ)*I=q u := by + apply Complex.ext <;> simp [hux] + refine ⟨(q u).im,⟨hpos,hheight⟩,?_⟩ + rw [hcoord,mappedFutureArc_segment p hi h0 hH hst ht hu, + initialInverse_forward_at p hi h0 hH s + (point_after_prefix_mem_domain p hi hH (slitSegmentParameter_gt hst hu))] + exact hp _ ⟨slitSegmentParameter_gt hst hu,slitSegmentParameter_le s t hst.le u⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/FutureCapacityAdditive.lean b/lean/OAI/Probability/SLE/Loewner/FutureCapacityAdditive.lean index ae5dcdef0232bbdc5747d1fe77cd00959b058a00..8e108fa3b5ae515c460c6be47233849207500fdc 100644 --- a/lean/OAI/Probability/SLE/Loewner/FutureCapacityAdditive.lean +++ b/lean/OAI/Probability/SLE/Loewner/FutureCapacityAdditive.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Loewner.FutureComposedCharts -import OAI.Probability.SLE.Loewner.ShiftedHydrodynamicComposition -import OAI.Probability.SLE.Loewner.SlitChartUniqueness - -namespace OAI - -/-! Removing an initial slit and its exact mapped future adds the two -hydrodynamic capacities. The composed chart is the later normalized -chart, including when the initial time is zero. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -variable (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s (Q.inverse_mapsTo hz).1) - · intro z hz - have hsub : z∈upperHalfPlane \ range (initialSubarc p s) := - ⟨hz.1,fun he => hz.2 (initialSubarc_range_mono p hst.le he)⟩ - have hd := (initialForward_centered_bijOn p hi h0 hH hst ht).mapsTo hz - have hh := Q.height_le _ hd - change (Q.forward (initialForwardChart p hi h0 hH s z-(U:ℂ))).im≤ - (initialForwardChart p hi h0 hH s z-(U:ℂ)).im at hh - simp only [sub_im,ofReal_im,sub_zero] at hh - change ((U:ℂ)+Q.forward (initialForwardChart p hi h0 hH s z-(U:ℂ))).im≤z.im - simpa only [add_im,ofReal_im,zero_add] using - hh.trans (initialForwardChart_height_le p hi h0 hH s hsub) - -theorem futureSlitChart_capacity_eq : - (futureSlitChart p hi h0 hH hst ht).capacity= - initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s := by - have ht0 : t≠0 := ne_of_gt (lt_of_le_of_lt bot_le hst) - have hh := ((composedFutureSlitChart p hi h0 hH hst ht).unique_on_domain - (initialSlitChart p hi h0 hH t ht0)).2.2 - change initialCapacity p hi h0 hH s+(futureSlitChart p hi h0 hH hst ht).capacity= - (initialSlitChart p hi h0 hH t ht0).capacity at hh - rw [initialCapacity,dite_eq_right ht0] - linarith - -theorem futureComposedForward_eq {z : ℂ} - (hz : z∈upperHalfPlane \ range (initialSubarc p t)) : - futureComposedForward p hi h0 hH hst ht z=initialForwardChart p hi h0 hH t z := by - have ht0 : t≠0 := ne_of_gt (lt_of_le_of_lt bot_le hst) - have hh := ((composedFutureSlitChart p hi h0 hH hst ht).unique_on_domain - (initialSlitChart p hi h0 hH t ht0)).1 hz - simpa only [composedFutureSlitChart,initialForwardChart,dite_eq_right ht0] using hh - -theorem futureComposedInverse_eq {z : ℂ} (hz : z∈upperHalfPlane) : - futureComposedInverse p hi h0 hH hst ht z=initialInverseChart p hi h0 hH t z := by - have ht0 : t≠0 := ne_of_gt (lt_of_le_of_lt bot_le hst) - have hh := ((composedFutureSlitChart p hi h0 hH hst ht).unique_on_domain - (initialSlitChart p hi h0 hH t ht0)).2.1 hz - simpa only [composedFutureSlitChart,initialInverseChart,dite_eq_right ht0] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.FutureComposedCharts +import OAI.Probability.SLE.Loewner.ShiftedHydrodynamicComposition +import OAI.Probability.SLE.Loewner.SlitChartUniqueness + +namespace OAI + +/-! Removing an initial slit and its exact mapped future adds the two +hydrodynamic capacities. The composed chart is the later normalized +chart, including when the initial time is zero. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +variable (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s (Q.inverse_mapsTo hz).1) + · intro z hz + have hsub : z∈upperHalfPlane \ range (initialSubarc p s) := + ⟨hz.1,fun he => hz.2 (initialSubarc_range_mono p hst.le he)⟩ + have hd := (initialForward_centered_bijOn p hi h0 hH hst ht).mapsTo hz + have hh := Q.height_le _ hd + change (Q.forward (initialForwardChart p hi h0 hH s z-(U:ℂ))).im≤ + (initialForwardChart p hi h0 hH s z-(U:ℂ)).im at hh + simp only [sub_im,ofReal_im,sub_zero] at hh + change ((U:ℂ)+Q.forward (initialForwardChart p hi h0 hH s z-(U:ℂ))).im≤z.im + simpa only [add_im,ofReal_im,zero_add] using + hh.trans (initialForwardChart_height_le p hi h0 hH s hsub) + +theorem futureSlitChart_capacity_eq : + (futureSlitChart p hi h0 hH hst ht).capacity= + initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s := by + have ht0 : t≠0 := ne_of_gt (lt_of_le_of_lt bot_le hst) + have hh := ((composedFutureSlitChart p hi h0 hH hst ht).unique_on_domain + (initialSlitChart p hi h0 hH t ht0)).2.2 + change initialCapacity p hi h0 hH s+(futureSlitChart p hi h0 hH hst ht).capacity= + (initialSlitChart p hi h0 hH t ht0).capacity at hh + rw [initialCapacity,dite_eq_right ht0] + linarith + +theorem futureComposedForward_eq {z : ℂ} + (hz : z∈upperHalfPlane \ range (initialSubarc p t)) : + futureComposedForward p hi h0 hH hst ht z=initialForwardChart p hi h0 hH t z := by + have ht0 : t≠0 := ne_of_gt (lt_of_le_of_lt bot_le hst) + have hh := ((composedFutureSlitChart p hi h0 hH hst ht).unique_on_domain + (initialSlitChart p hi h0 hH t ht0)).1 hz + simpa only [composedFutureSlitChart,initialForwardChart,dite_eq_right ht0] using hh + +theorem futureComposedInverse_eq {z : ℂ} (hz : z∈upperHalfPlane) : + futureComposedInverse p hi h0 hH hst ht z=initialInverseChart p hi h0 hH t z := by + have ht0 : t≠0 := ne_of_gt (lt_of_le_of_lt bot_le hst) + have hh := ((composedFutureSlitChart p hi h0 hH hst ht).unique_on_domain + (initialSlitChart p hi h0 hH t ht0)).2.1 hz + simpa only [composedFutureSlitChart,initialInverseChart,dite_eq_right ht0] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/FutureComposedCharts.lean b/lean/OAI/Probability/SLE/Loewner/FutureComposedCharts.lean index 49d33647bf7a8b12f074568c9d9d935144ec3b40..a0d0db4b2d6d980cfda477c0fe48e96e68a4f61b 100644 --- a/lean/OAI/Probability/SLE/Loewner/FutureComposedCharts.lean +++ b/lean/OAI/Probability/SLE/Loewner/FutureComposedCharts.lean @@ -1,94 +1,94 @@ -import OAI.Probability.SLE.Loewner.FutureSlitDomain -import OAI.Probability.SLE.Loewner.InitialChartNormalization - -namespace OAI - -/-! Forward and inverse charts obtained by removing a future segment -after an initial slit. The real translation keeps their normalization. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -variable (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s z-(U:ℂ)) upperHalfPlane upperHalfPlane := by - intro z hz - change 0<(z-(U:ℂ)).im - simpa only [sub_im,ofReal_im,sub_zero] using (show 0 (U:ℂ)+z) upperHalfPlane upperHalfPlane := by - intro z hz - change 0<((U:ℂ)+z).im - simpa only [add_im,ofReal_im,zero_add] using (show 0 hz.2 (initialSubarc_range_mono p hst.le he)⟩ - have hA := initialForward_centered_bijOn p hi h0 hH hst ht - have hJH : MapsTo Q.inverse upperHalfPlane upperHalfPlane := - fun _ hz => (Q.inverse_mapsTo hz).1 - have hinner : MapsTo (fun z : ℂ => (U:ℂ)+Q.inverse (z-(U:ℂ))) - upperHalfPlane upperHalfPlane := hadd.comp (hJH.comp hshift) - have hFin : MapsTo (futureComposedInverse p hi h0 hH hst ht) - upperHalfPlane (upperHalfPlane \ range (initialSubarc p t)) := by - intro z hz - have hb := initialInverseChart_mem_domain p hi h0 hH s (hinner hz) - apply (initialForward_centered_mem_domain_iff p hi h0 hH hst ht hb).mp - change initialForwardChart p hi h0 hH s - (initialInverseChart p hi h0 hH s ((U:ℂ)+Q.inverse (z-(U:ℂ))))-(U:ℂ)∈ - upperHalfPlane \ range (centeredFutureArc p hi h0 hH hst ht) - rw [initialForward_inverse p hi h0 hH s (hinner hz),add_sub_cancel_left] - exact Q.inverse_mapsTo (hshift hz) - refine ⟨?_,?_,?_,hFin,?_,?_⟩ - · apply DifferentiableOn.const_add - exact Q.forward_differentiable.comp - (((initialForwardChart_differentiableOn_current p hi h0 hH s).mono hsub).sub_const _) hA.mapsTo - · exact (initialInverseChart_differentiableOn p hi h0 hH s).comp - ((Q.inverse_differentiable.comp (differentiableOn_id.sub_const _) hshift).const_add _) hinner - · intro z hz - exact hadd (Q.bijOn.mapsTo (hA.mapsTo hz)) - · intro z hz - change initialInverseChart p hi h0 hH s ((U:ℂ)+Q.inverse - (((U:ℂ)+Q.forward (initialForwardChart p hi h0 hH s z-(U:ℂ)))-(U:ℂ)))=z - rw [add_sub_cancel_left,Q.left_inverse _ (hA.mapsTo hz),add_sub_cancel, - initialInverse_forward_at p hi h0 hH s (hsub hz)] - · intro z hz - change (U:ℂ)+Q.forward (initialForwardChart p hi h0 hH s - (initialInverseChart p hi h0 hH s ((U:ℂ)+Q.inverse (z-(U:ℂ))))-(U:ℂ))=z - rw [initialForward_inverse p hi h0 hH s (hinner hz),add_sub_cancel_left, - Q.right_inverse _ (hshift hz),add_sub_cancel] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.FutureSlitDomain +import OAI.Probability.SLE.Loewner.InitialChartNormalization + +namespace OAI + +/-! Forward and inverse charts obtained by removing a future segment +after an initial slit. The real translation keeps their normalization. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +variable (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s z-(U:ℂ)) upperHalfPlane upperHalfPlane := by + intro z hz + change 0<(z-(U:ℂ)).im + simpa only [sub_im,ofReal_im,sub_zero] using (show 0 (U:ℂ)+z) upperHalfPlane upperHalfPlane := by + intro z hz + change 0<((U:ℂ)+z).im + simpa only [add_im,ofReal_im,zero_add] using (show 0 hz.2 (initialSubarc_range_mono p hst.le he)⟩ + have hA := initialForward_centered_bijOn p hi h0 hH hst ht + have hJH : MapsTo Q.inverse upperHalfPlane upperHalfPlane := + fun _ hz => (Q.inverse_mapsTo hz).1 + have hinner : MapsTo (fun z : ℂ => (U:ℂ)+Q.inverse (z-(U:ℂ))) + upperHalfPlane upperHalfPlane := hadd.comp (hJH.comp hshift) + have hFin : MapsTo (futureComposedInverse p hi h0 hH hst ht) + upperHalfPlane (upperHalfPlane \ range (initialSubarc p t)) := by + intro z hz + have hb := initialInverseChart_mem_domain p hi h0 hH s (hinner hz) + apply (initialForward_centered_mem_domain_iff p hi h0 hH hst ht hb).mp + change initialForwardChart p hi h0 hH s + (initialInverseChart p hi h0 hH s ((U:ℂ)+Q.inverse (z-(U:ℂ))))-(U:ℂ)∈ + upperHalfPlane \ range (centeredFutureArc p hi h0 hH hst ht) + rw [initialForward_inverse p hi h0 hH s (hinner hz),add_sub_cancel_left] + exact Q.inverse_mapsTo (hshift hz) + refine ⟨?_,?_,?_,hFin,?_,?_⟩ + · apply DifferentiableOn.const_add + exact Q.forward_differentiable.comp + (((initialForwardChart_differentiableOn_current p hi h0 hH s).mono hsub).sub_const _) hA.mapsTo + · exact (initialInverseChart_differentiableOn p hi h0 hH s).comp + ((Q.inverse_differentiable.comp (differentiableOn_id.sub_const _) hshift).const_add _) hinner + · intro z hz + exact hadd (Q.bijOn.mapsTo (hA.mapsTo hz)) + · intro z hz + change initialInverseChart p hi h0 hH s ((U:ℂ)+Q.inverse + (((U:ℂ)+Q.forward (initialForwardChart p hi h0 hH s z-(U:ℂ)))-(U:ℂ)))=z + rw [add_sub_cancel_left,Q.left_inverse _ (hA.mapsTo hz),add_sub_cancel, + initialInverse_forward_at p hi h0 hH s (hsub hz)] + · intro z hz + change (U:ℂ)+Q.forward (initialForwardChart p hi h0 hH s + (initialInverseChart p hi h0 hH s ((U:ℂ)+Q.inverse (z-(U:ℂ))))-(U:ℂ))=z + rw [initialForward_inverse p hi h0 hH s (hinner hz),add_sub_cancel_left, + Q.right_inverse _ (hshift hz),add_sub_cancel] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/FutureSlitDomain.lean b/lean/OAI/Probability/SLE/Loewner/FutureSlitDomain.lean index b7795112cbd15b0779b6821f34dfaf8844a8f917..0ebf0e5e4184f100d29c6cb0ec61502f96d96f49 100644 --- a/lean/OAI/Probability/SLE/Loewner/FutureSlitDomain.lean +++ b/lean/OAI/Probability/SLE/Loewner/FutureSlitDomain.lean @@ -1,94 +1,94 @@ -import OAI.Probability.SLE.Loewner.MappedFutureRange - -namespace OAI - -/-! Exact transport of the later slit complement to the centered future -slit complement. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialInverseChart_mem_domain (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) {w : ℂ} (hw : w∈upperHalfPlane) : - initialInverseChart p hi h0 hH s w∈upperHalfPlane \ range (initialSubarc p s) := by - by_cases hs : s=0 - · subst s - simp only [initialInverseChart_zero,id_eq] - refine ⟨hw,?_⟩ - rintro ⟨u,hu⟩ - have hw0 : w=0 := by simpa only [initialSubarc_apply,zero_mul,h0] using hu.symm - rw [hw0] at hw - exact (lt_irrefl (0:ℝ)) hw - · rw [initialInverseChart,dite_eq_right hs] - exact (initialSlitChart p hi h0 hH s hs).inverse_mapsTo hw - -lemma centeredFutureArc_range_iff (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s z-_) hu⟩ - -lemma initialForward_centered_mem_domain_iff (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s initialForwardChart p hi h0 hH s z- - (initialDrivingValue p hi h0 hH s (hst.trans ht):ℂ)) - (upperHalfPlane \ range (initialSubarc p t)) - (upperHalfPlane \ range (centeredFutureArc p hi h0 hH hst ht)) := by - have hsub : upperHalfPlane \ range (initialSubarc p t) ⊆ - upperHalfPlane \ range (initialSubarc p s) := by - intro z hz - exact ⟨hz.1,fun h => hz.2 (initialSubarc_range_mono p hst.le h)⟩ - refine ⟨fun z hz => (initialForward_centered_mem_domain_iff p hi h0 hH hst ht - (hsub hz)).mpr hz,?_,?_⟩ - · intro z hz w hw he - have hEq := congrArg (initialInverseChart p hi h0 hH s) (sub_left_inj.mp he) - simpa only [initialInverse_forward_at p hi h0 hH s (hsub hz), - initialInverse_forward_at p hi h0 hH s (hsub hw)] using hEq - · intro w hw - let U : ℝ := initialDrivingValue p hi h0 hH s (hst.trans ht) - have hwH : w+(U:ℂ)∈upperHalfPlane := by - change 0<(w+(U:ℂ)).im - simpa only [add_im,ofReal_im,add_zero] using (show 0 z-_) hu⟩ + +lemma initialForward_centered_mem_domain_iff (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s initialForwardChart p hi h0 hH s z- + (initialDrivingValue p hi h0 hH s (hst.trans ht):ℂ)) + (upperHalfPlane \ range (initialSubarc p t)) + (upperHalfPlane \ range (centeredFutureArc p hi h0 hH hst ht)) := by + have hsub : upperHalfPlane \ range (initialSubarc p t) ⊆ + upperHalfPlane \ range (initialSubarc p s) := by + intro z hz + exact ⟨hz.1,fun h => hz.2 (initialSubarc_range_mono p hst.le h)⟩ + refine ⟨fun z hz => (initialForward_centered_mem_domain_iff p hi h0 hH hst ht + (hsub hz)).mpr hz,?_,?_⟩ + · intro z hz w hw he + have hEq := congrArg (initialInverseChart p hi h0 hH s) (sub_left_inj.mp he) + simpa only [initialInverse_forward_at p hi h0 hH s (hsub hz), + initialInverse_forward_at p hi h0 hH s (hsub hw)] using hEq + · intro w hw + let U : ℝ := initialDrivingValue p hi h0 hH s (hst.trans ht) + have hwH : w+(U:ℂ)∈upperHalfPlane := by + change 0<(w+(U:ℂ)).im + simpa only [add_im,ofReal_im,add_zero] using (show 0 ?_⟩ - by_cases hu : u=0 - · subst u - simpa only [centeredFutureArc_zero,norm_zero] using hε - have hp : 0<(u:ℝ) := unitInterval.pos_iff_ne_zero.mpr hu - have hd : 0<(t:ℝ)-(s:ℝ) := sub_pos.mpr hst - have hrpos : 0<((t:ℝ)-(s:ℝ))*(u:ℝ) := mul_pos hd hp - have hrlt : ((t:ℝ)-(s:ℝ))*(u:ℝ)<δ := by - have hm := mul_le_mul_of_nonneg_left u.property.2 hd.le - linarith - have he := hnear (show ((t:ℝ)-(s:ℝ))*(u:ℝ)∈ball (0:ℝ) δ ∩ Ioi 0 from - ⟨by simpa only [mem_ball,Real.dist_eq,sub_zero,abs_of_pos hrpos] using hrlt,hrpos⟩) - change ‖mappedFutureArc p hi h0 hH hst ht u-(_ : ℂ)‖<ε - rw [mappedFutureArc_of_ne_zero p hi h0 hH hst ht hu] - change dist (initialForwardChart p hi h0 hH s - (p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*(u:ℝ))))) - (initialDrivingValue p hi h0 hH s hs:ℂ)<ε at he - simpa only [mappedFutureRaw,dist_eq_norm] using he - -theorem futureSlitChart_capacity_small (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) (hs : s<1) {ε : ℝ} (hε : 0<ε) : - ∃ δ : ℝ, 0<δ ∧ ∀ (t : unitInterval) (hst : s - (futureSlitChart p hi h0 hH hst ht).capacity_le_radius_sq hε - (fun u => (hm t hst ht htd u).le)⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CenteredFutureSlit +import OAI.Probability.SLE.Loewner.CapacityRadiusBound + +namespace OAI + +/-! The complete mapped future slit shrinks to the driving value as its +terminal time decreases to the current time. -/ +noncomputable section +open Set Filter Function Metric Complex +open scoped Topology + +namespace SLEExactGauge + +theorem centeredFutureArc_small (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) (hs : s<1) {ε : ℝ} (hε : 0<ε) : + ∃ δ : ℝ, 0<δ ∧ ∀ (t : unitInterval) (hst : s ?_⟩ + by_cases hu : u=0 + · subst u + simpa only [centeredFutureArc_zero,norm_zero] using hε + have hp : 0<(u:ℝ) := unitInterval.pos_iff_ne_zero.mpr hu + have hd : 0<(t:ℝ)-(s:ℝ) := sub_pos.mpr hst + have hrpos : 0<((t:ℝ)-(s:ℝ))*(u:ℝ) := mul_pos hd hp + have hrlt : ((t:ℝ)-(s:ℝ))*(u:ℝ)<δ := by + have hm := mul_le_mul_of_nonneg_left u.property.2 hd.le + linarith + have he := hnear (show ((t:ℝ)-(s:ℝ))*(u:ℝ)∈ball (0:ℝ) δ ∩ Ioi 0 from + ⟨by simpa only [mem_ball,Real.dist_eq,sub_zero,abs_of_pos hrpos] using hrlt,hrpos⟩) + change ‖mappedFutureArc p hi h0 hH hst ht u-(_ : ℂ)‖<ε + rw [mappedFutureArc_of_ne_zero p hi h0 hH hst ht hu] + change dist (initialForwardChart p hi h0 hH s + (p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*(u:ℝ))))) + (initialDrivingValue p hi h0 hH s hs:ℂ)<ε at he + simpa only [mappedFutureRaw,dist_eq_norm] using he + +theorem futureSlitChart_capacity_small (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) (hs : s<1) {ε : ℝ} (hε : 0<ε) : + ∃ δ : ℝ, 0<δ ∧ ∀ (t : unitInterval) (hst : s + (futureSlitChart p hi h0 hH hst ht).capacity_le_radius_sq hε + (fun u => (hm t hst ht htd u).le)⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/HalfPlaneDisplacementBound.lean b/lean/OAI/Probability/SLE/Loewner/HalfPlaneDisplacementBound.lean index 3d8a5c933c6dc1680b2f1a6a2e7af1051992ca72..6475139c4c3f872f71c2c1773f0989b552a6bbe0 100644 --- a/lean/OAI/Probability/SLE/Loewner/HalfPlaneDisplacementBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/HalfPlaneDisplacementBound.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Mass.HalfPlaneZeroMass - -namespace OAI - -/-! The total hydrodynamic mass bounds a nonnegative-imaginary displacement. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma hydrodynamic_vertical_limits {f : ℂ → ℂ} {c : ℝ} - (hcoef : Tendsto (fun z : ℂ => z*f z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-c:ℂ))) : - Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0) ∧ - Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 c) := by - have hp := hcoef.comp verticalInfinity_tendsto - have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by - simpa only [Metric.cobounded_eq_cocompact] using - (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) - have hh := hp.mul (hi.comp (verticalInfinity_tendsto.mono_right inf_le_left)) - have hzero : Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0) := by - simp only [mul_zero] at hh - apply hh.congr - intro n - have hn : verticalInfinity n ≠ 0 := by - intro he - have hv := verticalInfinity_mem n - simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hv - simp only [Function.comp_apply] - field_simp [hn] - refine ⟨hzero,?_⟩ - have ht := (Complex.continuous_re.tendsto (-c:ℂ) |>.comp hp).neg - simpa only [Function.comp_apply,verticalInfinity,mul_re,mul_im,ofReal_re,ofReal_im, - I_re,I_im,mul_zero,zero_mul,add_zero,mul_one,neg_re,neg_neg,neg_zero,zero_sub] using ht - -lemma nonnegative_im_hydrodynamic_bound {f : ℂ → ℂ} {c : ℝ} - (hf : DifferentiableOn ℂ f upperHalfPlane) - (hm : ∀ z∈upperHalfPlane, 0≤(f z).im) (hc : 0≤c) - (hcoef : Tendsto (fun z : ℂ => z*f z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-c:ℂ))) - {z : ℂ} (hz : z∈upperHalfPlane) : ‖f z‖≤c/z.im := by - obtain ⟨hzero,hmass⟩ := hydrodynamic_vertical_limits hcoef - rcases holomorphic_nonnegative_im_dichotomy hf hm with ⟨a,ha⟩ | hpos - · have ht : Tendsto (fun _ : ℕ => a) atTop (𝓝 0) := - hzero.congr (fun n => ha _ (verticalInfinity_mem n)) - have ha0 : a=0 := tendsto_nhds_unique tendsto_const_nhds ht - rw [ha z hz,ha0,norm_zero] - exact div_nonneg hc (show 0 z*f z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-c:ℂ))) : + Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0) ∧ + Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 c) := by + have hp := hcoef.comp verticalInfinity_tendsto + have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by + simpa only [Metric.cobounded_eq_cocompact] using + (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) + have hh := hp.mul (hi.comp (verticalInfinity_tendsto.mono_right inf_le_left)) + have hzero : Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0) := by + simp only [mul_zero] at hh + apply hh.congr + intro n + have hn : verticalInfinity n ≠ 0 := by + intro he + have hv := verticalInfinity_mem n + simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hv + simp only [Function.comp_apply] + field_simp [hn] + refine ⟨hzero,?_⟩ + have ht := (Complex.continuous_re.tendsto (-c:ℂ) |>.comp hp).neg + simpa only [Function.comp_apply,verticalInfinity,mul_re,mul_im,ofReal_re,ofReal_im, + I_re,I_im,mul_zero,zero_mul,add_zero,mul_one,neg_re,neg_neg,neg_zero,zero_sub] using ht + +lemma nonnegative_im_hydrodynamic_bound {f : ℂ → ℂ} {c : ℝ} + (hf : DifferentiableOn ℂ f upperHalfPlane) + (hm : ∀ z∈upperHalfPlane, 0≤(f z).im) (hc : 0≤c) + (hcoef : Tendsto (fun z : ℂ => z*f z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-c:ℂ))) + {z : ℂ} (hz : z∈upperHalfPlane) : ‖f z‖≤c/z.im := by + obtain ⟨hzero,hmass⟩ := hydrodynamic_vertical_limits hcoef + rcases holomorphic_nonnegative_im_dichotomy hf hm with ⟨a,ha⟩ | hpos + · have ht : Tendsto (fun _ : ℕ => a) atTop (𝓝 0) := + hzero.congr (fun n => ha _ (verticalInfinity_mem n)) + have ha0 : a=0 := tendsto_nhds_unique tendsto_const_nhds ht + rw [ha z hz,ha0,norm_zero] + exact div_nonneg hc (show 0 g (w+Complex.equivRealProdCLM.symm (planarWiener t b))) (𝓝[<] τ) (𝓝 (x:ℂ))} - -theorem planarHalfPlaneExitArc_bound - (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D) - (w : ℂ) (hw : w∈D) {g : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) - (hgH : MapsTo g D upperHalfPlane) (T : ℝ≥0) (positive : Bool) : - (W.prod W) (planarHalfPlaneExitArc D g w T positive) ≤ - ENNReal.ofReal (halfPlaneExitWeight positive (g w)) := by - have hU := planarTranslatedDomain_open hD w - have h0 : (0:ℝ×ℝ)∈planarTranslatedDomain D w := by simpa [planarTranslatedDomain] using hw - let e := Classical.choice (exists_smoothDomainExhaustion _ hU 0 h0) - let u (z : ℝ×ℝ) := halfPlaneExitWeight positive (g (w+Complex.equivRealProdCLM.symm z)) - obtain ⟨hu,hhar⟩ := halfPlaneExitWeight_planar positive hD hg hgH w - have hub (z) (hz : z∈planarTranslatedDomain D w) : |u z|≤(1:ℝ≥0) := by - have hh := halfPlaneExitWeight_bounds positive (hgH hz) - rw [abs_of_nonneg hh.1] - exact hh.2 - have hup (z) (hz : z∈planarTranslatedDomain D w) : 0 ≤ u z := - (halfPlaneExitWeight_bounds positive (hgH hz)).1 - have hh := e.nonnegative_harmonic_exit_bound W hW T hU u hu hhar 1 hub hup - (planarHalfPlaneExitArc D g w T positive) 1 (by norm_num) (by - rintro b ⟨τ,hτ,hτT,hp,hx,x,hs,hlim⟩ - refine ⟨τ,hτ,hτT,hp,hx,?_⟩ - cases positive - · have him : ∀ᶠ t in 𝓝[<] τ, 0 ≤ (g (w+Complex.equivRealProdCLM.symm (planarWiener t b))).im := - (show ∀ᶠ t in 𝓝[<] τ, t<τ from self_mem_nhdsWithin).mono - (fun t ht => (hgH (hp t ht)).le) - exact halfPlaneExitWeight_negative_limit hs him hlim - · exact halfPlaneExitWeight_positive_limit hs hlim) - simpa [u] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.HalfPlaneExitWeights +import OAI.Probability.SLE.Geometry.HarmonicExitProbability + +namespace OAI + +/-! The two half-plane angular upper bounds for finite physical exits. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Target + +def planarHalfPlaneExitArc (D : Set ℂ) (g : ℂ → ℂ) (w : ℂ) (T : ℝ≥0) (positive : Bool) : + Set (Driver×Driver) := + {b | ∃ τ : ℝ≥0, 0 < τ ∧ τ ≤ T ∧ + (∀ t<τ, w+Complex.equivRealProdCLM.symm (planarWiener t b)∈D) ∧ + w+Complex.equivRealProdCLM.symm (planarWiener τ b)∉D ∧ + ∃ x : ℝ, (if positive then 0 g (w+Complex.equivRealProdCLM.symm (planarWiener t b))) (𝓝[<] τ) (𝓝 (x:ℂ))} + +theorem planarHalfPlaneExitArc_bound + (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D) + (w : ℂ) (hw : w∈D) {g : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) + (hgH : MapsTo g D upperHalfPlane) (T : ℝ≥0) (positive : Bool) : + (W.prod W) (planarHalfPlaneExitArc D g w T positive) ≤ + ENNReal.ofReal (halfPlaneExitWeight positive (g w)) := by + have hU := planarTranslatedDomain_open hD w + have h0 : (0:ℝ×ℝ)∈planarTranslatedDomain D w := by simpa [planarTranslatedDomain] using hw + let e := Classical.choice (exists_smoothDomainExhaustion _ hU 0 h0) + let u (z : ℝ×ℝ) := halfPlaneExitWeight positive (g (w+Complex.equivRealProdCLM.symm z)) + obtain ⟨hu,hhar⟩ := halfPlaneExitWeight_planar positive hD hg hgH w + have hub (z) (hz : z∈planarTranslatedDomain D w) : |u z|≤(1:ℝ≥0) := by + have hh := halfPlaneExitWeight_bounds positive (hgH hz) + rw [abs_of_nonneg hh.1] + exact hh.2 + have hup (z) (hz : z∈planarTranslatedDomain D w) : 0 ≤ u z := + (halfPlaneExitWeight_bounds positive (hgH hz)).1 + have hh := e.nonnegative_harmonic_exit_bound W hW T hU u hu hhar 1 hub hup + (planarHalfPlaneExitArc D g w T positive) 1 (by norm_num) (by + rintro b ⟨τ,hτ,hτT,hp,hx,x,hs,hlim⟩ + refine ⟨τ,hτ,hτT,hp,hx,?_⟩ + cases positive + · have him : ∀ᶠ t in 𝓝[<] τ, 0 ≤ (g (w+Complex.equivRealProdCLM.symm (planarWiener t b))).im := + (show ∀ᶠ t in 𝓝[<] τ, t<τ from self_mem_nhdsWithin).mono + (fun t ht => (hgH (hp t ht)).le) + exact halfPlaneExitWeight_negative_limit hs him hlim + · exact halfPlaneExitWeight_positive_limit hs hlim) + simpa [u] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/HalfPlaneExitWeights.lean b/lean/OAI/Probability/SLE/Loewner/HalfPlaneExitWeights.lean index e5c58b140d2624dc8b99f8d4271a7e6d7a77191d..4af40588324873d8f74e739cac2695f8a27aef89 100644 --- a/lean/OAI/Probability/SLE/Loewner/HalfPlaneExitWeights.lean +++ b/lean/OAI/Probability/SLE/Loewner/HalfPlaneExitWeights.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.Stochastic.PlanarHolomorphicTests -import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv - -namespace OAI - -/-! The bounded angular harmonic functions for the two sides of the -real boundary, together with their one-sided boundary limits. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -def halfPlaneExitWeight (positive : Bool) (z : ℂ) : ℝ := - if positive then 1-z.arg/Real.pi else z.arg/Real.pi - -lemma halfPlaneExitWeight_bounds (s : Bool) {z : ℂ} (hz : 0 < z.im) : - halfPlaneExitWeight s z∈Icc (0:ℝ) 1 := by - have h0 : 0 ≤ z.arg/Real.pi := div_nonneg (Complex.arg_nonneg_iff.mpr hz.le) Real.pi_pos.le - have h1 : z.arg/Real.pi ≤ 1 := (div_le_one Real.pi_pos).mpr (Complex.arg_le_pi z) - cases s <;> simp only [halfPlaneExitWeight,Bool.false_eq_true,↓reduceIte,mem_Icc] <;> - constructor <;> linarith - -lemma halfPlaneExitWeight_planar (s : Bool) {D : Set ℂ} (hD : IsOpen D) - {g : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) (hgH : MapsTo g D upperHalfPlane) (w : ℂ) : - ContDiffOn ℝ (⊤ : ℕ∞) - (fun z => halfPlaneExitWeight s (g (w+Complex.equivRealProdCLM.symm z))) (planarTranslatedDomain D w) ∧ - ∀ z∈planarTranslatedDomain D w, - planarGenerator (fun z => halfPlaneExitWeight s (g (w+Complex.equivRealProdCLM.symm z))) z=0 := by - have hlog : DifferentiableOn ℂ Complex.log upperHalfPlane := by - intro z hz - exact (Complex.differentiableAt_log (Complex.mem_slitPlane_iff.mpr (Or.inr (ne_of_gt hz)))).differentiableWithinAt - let Q (z : ℂ) := Complex.log (g z)/(Real.pi:ℂ) - have hQ : DifferentiableOn ℂ Q D := (hlog.comp hg hgH).div_const _ - cases s - · have hh := (holomorphic_planar_components hD hQ w).2 - simpa only [halfPlaneExitWeight,Bool.false_eq_true,↓reduceIte,Q,Complex.div_ofReal_im,Complex.log_im] using hh - · have hQc : DifferentiableOn ℂ (fun z => Complex.I-Q z) D := (differentiableOn_const Complex.I).sub hQ - have hh := (holomorphic_planar_components hD hQc w).2 - simpa only [halfPlaneExitWeight,↓reduceIte,Q,Complex.sub_im,Complex.I_im, - Complex.div_ofReal_im,Complex.log_im] using hh - -lemma halfPlaneExitWeight_negative_limit {A : Type*} {l : Filter A} - {ζ : A → ℂ} {x : ℝ} (hx : x<0) (hζ : ∀ᶠ t in l, 0 ≤ (ζ t).im) - (hlim : Tendsto ζ l (𝓝 (x:ℂ))) : - Tendsto (fun t => halfPlaneExitWeight false (ζ t)) l (𝓝 1) := by - have hh := (Complex.tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero - (z := (x:ℂ)) (by simpa using hx) (by simp)).comp - (tendsto_nhdsWithin_iff.mpr ⟨hlim,hζ⟩) - simpa only [halfPlaneExitWeight,Bool.false_eq_true,↓reduceIte,Function.comp_apply, - div_self Real.pi_ne_zero] using hh.div_const Real.pi - -lemma halfPlaneExitWeight_positive_limit {A : Type*} {l : Filter A} - {ζ : A → ℂ} {x : ℝ} (hx : 0 halfPlaneExitWeight true (ζ t)) l (𝓝 1) := by - have hh := (Complex.continuousAt_arg (Complex.mem_slitPlane_iff.mpr - (Or.inl (by simpa using hx)))).tendsto.comp hlim - have hq := hh.div_const Real.pi - simpa only [halfPlaneExitWeight,↓reduceIte,Function.comp_apply, - Complex.arg_ofReal_of_nonneg hx.le,zero_div,sub_zero] using tendsto_const_nhds.sub hq - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.PlanarHolomorphicTests +import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv + +namespace OAI + +/-! The bounded angular harmonic functions for the two sides of the +real boundary, together with their one-sided boundary limits. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +def halfPlaneExitWeight (positive : Bool) (z : ℂ) : ℝ := + if positive then 1-z.arg/Real.pi else z.arg/Real.pi + +lemma halfPlaneExitWeight_bounds (s : Bool) {z : ℂ} (hz : 0 < z.im) : + halfPlaneExitWeight s z∈Icc (0:ℝ) 1 := by + have h0 : 0 ≤ z.arg/Real.pi := div_nonneg (Complex.arg_nonneg_iff.mpr hz.le) Real.pi_pos.le + have h1 : z.arg/Real.pi ≤ 1 := (div_le_one Real.pi_pos).mpr (Complex.arg_le_pi z) + cases s <;> simp only [halfPlaneExitWeight,Bool.false_eq_true,↓reduceIte,mem_Icc] <;> + constructor <;> linarith + +lemma halfPlaneExitWeight_planar (s : Bool) {D : Set ℂ} (hD : IsOpen D) + {g : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) (hgH : MapsTo g D upperHalfPlane) (w : ℂ) : + ContDiffOn ℝ (⊤ : ℕ∞) + (fun z => halfPlaneExitWeight s (g (w+Complex.equivRealProdCLM.symm z))) (planarTranslatedDomain D w) ∧ + ∀ z∈planarTranslatedDomain D w, + planarGenerator (fun z => halfPlaneExitWeight s (g (w+Complex.equivRealProdCLM.symm z))) z=0 := by + have hlog : DifferentiableOn ℂ Complex.log upperHalfPlane := by + intro z hz + exact (Complex.differentiableAt_log (Complex.mem_slitPlane_iff.mpr (Or.inr (ne_of_gt hz)))).differentiableWithinAt + let Q (z : ℂ) := Complex.log (g z)/(Real.pi:ℂ) + have hQ : DifferentiableOn ℂ Q D := (hlog.comp hg hgH).div_const _ + cases s + · have hh := (holomorphic_planar_components hD hQ w).2 + simpa only [halfPlaneExitWeight,Bool.false_eq_true,↓reduceIte,Q,Complex.div_ofReal_im,Complex.log_im] using hh + · have hQc : DifferentiableOn ℂ (fun z => Complex.I-Q z) D := (differentiableOn_const Complex.I).sub hQ + have hh := (holomorphic_planar_components hD hQc w).2 + simpa only [halfPlaneExitWeight,↓reduceIte,Q,Complex.sub_im,Complex.I_im, + Complex.div_ofReal_im,Complex.log_im] using hh + +lemma halfPlaneExitWeight_negative_limit {A : Type*} {l : Filter A} + {ζ : A → ℂ} {x : ℝ} (hx : x<0) (hζ : ∀ᶠ t in l, 0 ≤ (ζ t).im) + (hlim : Tendsto ζ l (𝓝 (x:ℂ))) : + Tendsto (fun t => halfPlaneExitWeight false (ζ t)) l (𝓝 1) := by + have hh := (Complex.tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero + (z := (x:ℂ)) (by simpa using hx) (by simp)).comp + (tendsto_nhdsWithin_iff.mpr ⟨hlim,hζ⟩) + simpa only [halfPlaneExitWeight,Bool.false_eq_true,↓reduceIte,Function.comp_apply, + div_self Real.pi_ne_zero] using hh.div_const Real.pi + +lemma halfPlaneExitWeight_positive_limit {A : Type*} {l : Filter A} + {ζ : A → ℂ} {x : ℝ} (hx : 0 halfPlaneExitWeight true (ζ t)) l (𝓝 1) := by + have hh := (Complex.continuousAt_arg (Complex.mem_slitPlane_iff.mpr + (Or.inl (by simpa using hx)))).tendsto.comp hlim + have hq := hh.div_const Real.pi + simpa only [halfPlaneExitWeight,↓reduceIte,Function.comp_apply, + Complex.arg_ofReal_of_nonneg hx.le,zero_div,sub_zero] using tendsto_const_nhds.sub hq + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/HalfPlaneFiniteExit.lean b/lean/OAI/Probability/SLE/Loewner/HalfPlaneFiniteExit.lean index ba90fba180481f93ca4b11939f3079b443063d98..08861cd92459e318f1e8b2f8fc89042eb1e9d138 100644 --- a/lean/OAI/Probability/SLE/Loewner/HalfPlaneFiniteExit.lean +++ b/lean/OAI/Probability/SLE/Loewner/HalfPlaneFiniteExit.lean @@ -1,79 +1,79 @@ -import OAI.Probability.SLE.Geometry.NoBoundaryAtoms -import OAI.Probability.SLE.Geometry.HalfPlaneChartBoundary - -namespace OAI - -/-! Finite nonzero real limits of conformal half-plane maps at actual -finite physical Brownian exits. -/ -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Target - -theorem halfPlane_finite_exit_limit - (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D) - {g f : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) - (hf : DifferentiableOn ℂ f upperHalfPlane) - (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) - (hinv : ∀ z∈D, f (g z)=z) (w : ℂ) (hw : w∈D) (T : ℝ≥0) : - ∀ᵐ b ∂W.prod W, ∀ τ : ℝ≥0, 0 < τ → τ ≤ T → - (∀ t<τ, w+Complex.equivRealProdCLM.symm (planarWiener t b)∈D) → - w+Complex.equivRealProdCLM.symm (planarWiener τ b)∉D → - ∃ x : ℝ, x≠0 ∧ - Tendsto (fun t => g (w+Complex.equivRealProdCLM.symm (planarWiener t b))) - (𝓝[<] τ) (𝓝 (x:ℂ)) := by - let v := g w - have hv : 0 < v.im := hgH hw - let F := halfPlaneToDisk v ∘ g - let H := f ∘ diskToHalfPlane v - have hF : DifferentiableOn ℂ F D := (halfPlaneToDisk_differentiable hv).comp hg hgH - have hFD : MapsTo F D (ball 0 1) := (halfPlaneToDisk_maps hv).comp hgH - have hH : ContinuousOn H (ball 0 1) := hf.continuousOn.comp - (diskToHalfPlane_differentiable v).continuousOn (diskToHalfPlane_maps hv) - have hHD : MapsTo H (ball 0 1) D := hfD.comp (diskToHalfPlane_maps hv) - have hFG (z) (hz : z∈D) : diskToHalfPlane v (F z)=g z := - diskToHalfPlane_halfPlaneToDisk hv (hgH hz) - have hHF (z) (hz : z∈D) : H (F z)=z := by - change f (diskToHalfPlane v (F z))=z - rw [hFG z hz,hinv z hz] - have hF0 : F w=0 := by simp [F,v,halfPlaneToDisk] - have hlim := conformal_disk_finite_exit_limit W hW hD w hw hF hFD hH hHD hHF T - have hnull1 := planarDiskExitAtom_null W hW hD w hw hF hFD hF0 1 (by simp) T - have hnull0 := planarDiskExitAtom_null W hW hD w hw hF hFD hF0 - (halfPlaneToDisk v 0) (halfPlaneToDisk_zero_norm hv) T - have hno1 : ∀ᵐ b ∂W.prod W, b∉planarDiskExitAtom D F w 1 T := by - simpa only [ae_iff,not_not,ofPred_mem_eq] using hnull1 - have hno0 : ∀ᵐ b ∂W.prod W, b∉planarDiskExitAtom D F w (halfPlaneToDisk v 0) T := by - simpa only [ae_iff,not_not,ofPred_mem_eq] using hnull0 - filter_upwards [hlim,hno1,hno0] with b hb hb1 hb0 - intro τ hτ hτT hp hx - obtain ⟨c,hcn,hc⟩ := hb τ hτ hτT hp hx - have hc1 : c≠1 := by - intro he - exact hb1 ⟨τ,hτ,hτT,hp,hx,he ▸ hc⟩ - have hc0 : c≠halfPlaneToDisk v 0 := by - intro he - exact hb0 ⟨τ,hτ,hτT,hp,hx,he ▸ hc⟩ - have hgconv := (diskToHalfPlane_continuousAt (v := v) hc1).tendsto.comp hc - have heq : (fun t => diskToHalfPlane v (F (w+Complex.equivRealProdCLM.symm (planarWiener t b)))) - =ᶠ[𝓝[<] τ] (fun t => g (w+Complex.equivRealProdCLM.symm (planarWiener t b))) := by - filter_upwards [self_mem_nhdsWithin] with t ht - exact hFG _ (hp t ht) - have hgconv' := hgconv.congr' heq - let z := diskToHalfPlane v c - have hzI : z.im=0 := diskToHalfPlane_boundary_im hcn - have hz0 : z≠0 := diskToHalfPlane_boundary_ne_zero hv hc1 hc0 - have hzreal : z=(z.re:ℂ) := by apply Complex.ext <;> simp [hzI] - refine ⟨z.re,?_,?_⟩ - · intro he - exact hz0 (by rw [hzreal,he,Complex.ofReal_zero]) - · change Tendsto _ (𝓝[<] τ) (𝓝 z) at hgconv' - rwa [hzreal] at hgconv' - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.NoBoundaryAtoms +import OAI.Probability.SLE.Geometry.HalfPlaneChartBoundary + +namespace OAI + +/-! Finite nonzero real limits of conformal half-plane maps at actual +finite physical Brownian exits. -/ +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Target + +theorem halfPlane_finite_exit_limit + (W : WienerLaw) (hW : IsWiener W) {D : Set ℂ} (hD : IsOpen D) + {g f : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) + (hf : DifferentiableOn ℂ f upperHalfPlane) + (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) + (hinv : ∀ z∈D, f (g z)=z) (w : ℂ) (hw : w∈D) (T : ℝ≥0) : + ∀ᵐ b ∂W.prod W, ∀ τ : ℝ≥0, 0 < τ → τ ≤ T → + (∀ t<τ, w+Complex.equivRealProdCLM.symm (planarWiener t b)∈D) → + w+Complex.equivRealProdCLM.symm (planarWiener τ b)∉D → + ∃ x : ℝ, x≠0 ∧ + Tendsto (fun t => g (w+Complex.equivRealProdCLM.symm (planarWiener t b))) + (𝓝[<] τ) (𝓝 (x:ℂ)) := by + let v := g w + have hv : 0 < v.im := hgH hw + let F := halfPlaneToDisk v ∘ g + let H := f ∘ diskToHalfPlane v + have hF : DifferentiableOn ℂ F D := (halfPlaneToDisk_differentiable hv).comp hg hgH + have hFD : MapsTo F D (ball 0 1) := (halfPlaneToDisk_maps hv).comp hgH + have hH : ContinuousOn H (ball 0 1) := hf.continuousOn.comp + (diskToHalfPlane_differentiable v).continuousOn (diskToHalfPlane_maps hv) + have hHD : MapsTo H (ball 0 1) D := hfD.comp (diskToHalfPlane_maps hv) + have hFG (z) (hz : z∈D) : diskToHalfPlane v (F z)=g z := + diskToHalfPlane_halfPlaneToDisk hv (hgH hz) + have hHF (z) (hz : z∈D) : H (F z)=z := by + change f (diskToHalfPlane v (F z))=z + rw [hFG z hz,hinv z hz] + have hF0 : F w=0 := by simp [F,v,halfPlaneToDisk] + have hlim := conformal_disk_finite_exit_limit W hW hD w hw hF hFD hH hHD hHF T + have hnull1 := planarDiskExitAtom_null W hW hD w hw hF hFD hF0 1 (by simp) T + have hnull0 := planarDiskExitAtom_null W hW hD w hw hF hFD hF0 + (halfPlaneToDisk v 0) (halfPlaneToDisk_zero_norm hv) T + have hno1 : ∀ᵐ b ∂W.prod W, b∉planarDiskExitAtom D F w 1 T := by + simpa only [ae_iff,not_not,ofPred_mem_eq] using hnull1 + have hno0 : ∀ᵐ b ∂W.prod W, b∉planarDiskExitAtom D F w (halfPlaneToDisk v 0) T := by + simpa only [ae_iff,not_not,ofPred_mem_eq] using hnull0 + filter_upwards [hlim,hno1,hno0] with b hb hb1 hb0 + intro τ hτ hτT hp hx + obtain ⟨c,hcn,hc⟩ := hb τ hτ hτT hp hx + have hc1 : c≠1 := by + intro he + exact hb1 ⟨τ,hτ,hτT,hp,hx,he ▸ hc⟩ + have hc0 : c≠halfPlaneToDisk v 0 := by + intro he + exact hb0 ⟨τ,hτ,hτT,hp,hx,he ▸ hc⟩ + have hgconv := (diskToHalfPlane_continuousAt (v := v) hc1).tendsto.comp hc + have heq : (fun t => diskToHalfPlane v (F (w+Complex.equivRealProdCLM.symm (planarWiener t b)))) + =ᶠ[𝓝[<] τ] (fun t => g (w+Complex.equivRealProdCLM.symm (planarWiener t b))) := by + filter_upwards [self_mem_nhdsWithin] with t ht + exact hFG _ (hp t ht) + have hgconv' := hgconv.congr' heq + let z := diskToHalfPlane v c + have hzI : z.im=0 := diskToHalfPlane_boundary_im hcn + have hz0 : z≠0 := diskToHalfPlane_boundary_ne_zero hv hc1 hc0 + have hzreal : z=(z.re:ℂ) := by apply Complex.ext <;> simp [hzI] + refine ⟨z.re,?_,?_⟩ + · intro he + exact hz0 (by rw [hzreal,he,Complex.ofReal_zero]) + · change Tendsto _ (𝓝[<] τ) (𝓝 z) at hgconv' + rwa [hzreal] at hgconv' + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/HydrodynamicAsymptotics.lean b/lean/OAI/Probability/SLE/Loewner/HydrodynamicAsymptotics.lean index 2d24aa42a247f8373e39f117acb8a5d3e2d4be55..978a089edcded65a6e619cb3fb47479cdfd1abac 100644 --- a/lean/OAI/Probability/SLE/Loewner/HydrodynamicAsymptotics.lean +++ b/lean/OAI/Probability/SLE/Loewner/HydrodynamicAsymptotics.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Loewner.HydrodynamicCoefficient - -namespace OAI - -/-! Elementary consequences of hydrodynamic normalization, including -the inverse coefficient and escape of the normalized inverse to infinity. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma normalized_ratio_tendsto_cocompact {F : ℂ → ℂ} - (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : - Tendsto F (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (cocompact ℂ) := by - have hr : Tendsto (fun z : ℂ => ‖F z/z‖) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - simpa only [norm_one] using hF.norm - have hn := Filter.Tendsto.pos_mul_atTop (by norm_num : (0:ℝ)<1) hr - (tendsto_norm_cocompact_atTop.mono_left (show cocompact ℂ ⊓ 𝓟 upperHalfPlane ≤ cocompact ℂ from inf_le_left)) - have hnorm : Tendsto (fun z : ℂ => ‖F z‖) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) atTop := by - apply hn.congr' - filter_upwards [(show ∀ᶠ z : ℂ in cocompact ℂ, z ≠ 0 from - (isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact).filter_mono - (show cocompact ℂ ⊓ 𝓟 upperHalfPlane ≤ cocompact ℂ from inf_le_left)] with z hz - rw [norm_div,div_mul_cancel₀ _ (norm_ne_zero_iff.mpr hz)] - have hb := tendsto_norm_atTop_iff_cobounded.mp hnorm - simpa only [Metric.cobounded_eq_cocompact] using hb - -lemma inverse_hydrodynamic_coefficient {G F : ℂ → ℂ} {c : ℂ} - (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) - (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) - (hGF : ∀ z ∈ upperHalfPlane, G (F z)=z) : - Tendsto (fun z : ℂ => z*(F z-z)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-c)) := by - have he := normalized_ratio_tendsto_cocompact hF - have hq : Tendsto (fun z : ℂ => z/F z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - simpa only [inv_div,inv_one] using hF.inv₀ one_ne_zero - have ht := (hq.mul (hG.comp he)).neg - simp only [one_mul] at ht - apply ht.congr' - filter_upwards [he.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact), - (show ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, z∈upperHalfPlane from - eventually_inf_principal.mpr (Eventually.of_forall (fun _ h => h)))] with z hz hzH - simp only [Function.comp_apply,hGF z hzH] - have hz0 : F z ≠ 0 := hz - field_simp [hz0] - ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.HydrodynamicCoefficient + +namespace OAI + +/-! Elementary consequences of hydrodynamic normalization, including +the inverse coefficient and escape of the normalized inverse to infinity. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +lemma normalized_ratio_tendsto_cocompact {F : ℂ → ℂ} + (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : + Tendsto F (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (cocompact ℂ) := by + have hr : Tendsto (fun z : ℂ => ‖F z/z‖) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + simpa only [norm_one] using hF.norm + have hn := Filter.Tendsto.pos_mul_atTop (by norm_num : (0:ℝ)<1) hr + (tendsto_norm_cocompact_atTop.mono_left (show cocompact ℂ ⊓ 𝓟 upperHalfPlane ≤ cocompact ℂ from inf_le_left)) + have hnorm : Tendsto (fun z : ℂ => ‖F z‖) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) atTop := by + apply hn.congr' + filter_upwards [(show ∀ᶠ z : ℂ in cocompact ℂ, z ≠ 0 from + (isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact).filter_mono + (show cocompact ℂ ⊓ 𝓟 upperHalfPlane ≤ cocompact ℂ from inf_le_left)] with z hz + rw [norm_div,div_mul_cancel₀ _ (norm_ne_zero_iff.mpr hz)] + have hb := tendsto_norm_atTop_iff_cobounded.mp hnorm + simpa only [Metric.cobounded_eq_cocompact] using hb + +lemma inverse_hydrodynamic_coefficient {G F : ℂ → ℂ} {c : ℂ} + (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) + (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) + (hGF : ∀ z ∈ upperHalfPlane, G (F z)=z) : + Tendsto (fun z : ℂ => z*(F z-z)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-c)) := by + have he := normalized_ratio_tendsto_cocompact hF + have hq : Tendsto (fun z : ℂ => z/F z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + simpa only [inv_div,inv_one] using hF.inv₀ one_ne_zero + have ht := (hq.mul (hG.comp he)).neg + simp only [one_mul] at ht + apply ht.congr' + filter_upwards [he.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact), + (show ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, z∈upperHalfPlane from + eventually_inf_principal.mpr (Eventually.of_forall (fun _ h => h)))] with z hz hzH + simp only [Function.comp_apply,hGF z hzH] + have hz0 : F z ≠ 0 := hz + field_simp [hz0] + ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/HydrodynamicCoefficient.lean b/lean/OAI/Probability/SLE/Loewner/HydrodynamicCoefficient.lean index 85b9bf0cc3ba014d893aec99a738d569775cb28f..ddb633854eb61e33f957dc8ae489168fa1284476 100644 --- a/lean/OAI/Probability/SLE/Loewner/HydrodynamicCoefficient.lean +++ b/lean/OAI/Probability/SLE/Loewner/HydrodynamicCoefficient.lean @@ -1,43 +1,43 @@ -import OAI.Probability.SLE.Loewner.HydrodynamicHeight - -namespace OAI - -/-! A height-decreasing half-plane chart has nonnegative hydrodynamic -coefficient. The argument uses the actual Laurent limit on the imaginary axis. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma hydrodynamic_coefficient_nonneg {G : ℂ → ℂ} {c : ℝ} - (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) - (hheight : ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, (G z).im ≤ z.im) : 0≤c := by - have hv := verticalInfinity_tendsto - have hr := Complex.continuous_re.tendsto (c:ℂ) |>.comp (hc.comp (hv.mono_right inf_le_left)) - have hn : ∀ᶠ n in atTop, 0 ≤ (verticalInfinity n*(G (verticalInfinity n)-verticalInfinity n)).re := by - filter_upwards [hv.eventually hheight] with n hn - have hy : 0 < (n:ℝ)+1 := by positivity - simp only [verticalInfinity,mul_im,mul_re,ofReal_re,ofReal_im,I_re,I_im,mul_zero, - add_zero,mul_one,sub_im,sub_re,zero_sub] at hn ⊢ - nlinarith - have hh : 0 ≤ (c:ℂ).re := ge_of_tendsto hr (by - simpa only [Function.comp_apply] using hn) - simpa only [ofReal_re] using hh - -lemma normalized_inverse_height - {D : Set ℂ} {G F : ℂ → ℂ} - (hG : MapsTo G D upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) (hmap : MapsTo F upperHalfPlane upperHalfPlane) - (hleft : ∀ z ∈ D, F (G z)=z) - (hn : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : - ∀ z ∈ D, (G z).im ≤ z.im := by - intro z hz - have hh := normalized_halfPlane_height hF hmap hn (hG hz) - rwa [hleft z hz] at hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.HydrodynamicHeight + +namespace OAI + +/-! A height-decreasing half-plane chart has nonnegative hydrodynamic +coefficient. The argument uses the actual Laurent limit on the imaginary axis. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +lemma hydrodynamic_coefficient_nonneg {G : ℂ → ℂ} {c : ℝ} + (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) + (hheight : ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, (G z).im ≤ z.im) : 0≤c := by + have hv := verticalInfinity_tendsto + have hr := Complex.continuous_re.tendsto (c:ℂ) |>.comp (hc.comp (hv.mono_right inf_le_left)) + have hn : ∀ᶠ n in atTop, 0 ≤ (verticalInfinity n*(G (verticalInfinity n)-verticalInfinity n)).re := by + filter_upwards [hv.eventually hheight] with n hn + have hy : 0 < (n:ℝ)+1 := by positivity + simp only [verticalInfinity,mul_im,mul_re,ofReal_re,ofReal_im,I_re,I_im,mul_zero, + add_zero,mul_one,sub_im,sub_re,zero_sub] at hn ⊢ + nlinarith + have hh : 0 ≤ (c:ℂ).re := ge_of_tendsto hr (by + simpa only [Function.comp_apply] using hn) + simpa only [ofReal_re] using hh + +lemma normalized_inverse_height + {D : Set ℂ} {G F : ℂ → ℂ} + (hG : MapsTo G D upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) (hmap : MapsTo F upperHalfPlane upperHalfPlane) + (hleft : ∀ z ∈ D, F (G z)=z) + (hn : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : + ∀ z ∈ D, (G z).im ≤ z.im := by + intro z hz + have hh := normalized_halfPlane_height hF hmap hn (hG hz) + rwa [hleft z hz] at hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/HydrodynamicComposition.lean b/lean/OAI/Probability/SLE/Loewner/HydrodynamicComposition.lean index 802b5c6bbe9800c9479a9625a25da8a5ec7c7374..f7927d0cabdfe5994706f8e067f347198646dbcc 100644 --- a/lean/OAI/Probability/SLE/Loewner/HydrodynamicComposition.lean +++ b/lean/OAI/Probability/SLE/Loewner/HydrodynamicComposition.lean @@ -1,97 +1,97 @@ -import OAI.Probability.SLE.Loewner.HydrodynamicAsymptotics - -namespace OAI - -/-! The coefficient of a normalized composition subtracts capacities. -Schwarz–Pick then makes capacity monotone under inclusion of slit domains. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma hydrodynamic_ratio_limit {G : ℂ → ℂ} {c : ℂ} - (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) : - Tendsto (fun z : ℂ => G z/z) (cocompact ℂ) (𝓝 1) := by - have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by - simpa only [Metric.cobounded_eq_cocompact] using - (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) - have hh := (hG.mul (hi.pow 2)).const_add 1 - simp only [zero_pow two_ne_zero,mul_zero,add_zero] at hh - apply hh.congr' - filter_upwards [(isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact] with z hz - have hz0 : z ≠ 0 := hz - field_simp [hz0] - ring - -lemma hydrodynamic_composition_coefficient {G H F : ℂ → ℂ} {c d : ℂ} - (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) - (hH : Tendsto (fun z : ℂ => z*(H z-z)) (cocompact ℂ) (𝓝 d)) - (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) - (hHF : ∀ z ∈ upperHalfPlane, H (F z)=z) : - Tendsto (fun z : ℂ => z*(G (F z)-z)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (c-d)) := by - have hescape := normalized_ratio_tendsto_cocompact hF - have hquot : Tendsto (fun z : ℂ => z/F z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - simpa only [inv_div,inv_one] using hF.inv₀ one_ne_zero - have hh := (hquot.mul (hG.comp hescape)).add (inverse_hydrodynamic_coefficient hH hF hHF) - simp only [one_mul,sub_eq_add_neg] at hh ⊢ - apply hh.congr' - filter_upwards [hescape.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] with z hz - simp only [Function.comp_apply] - have hz0 : F z ≠ 0 := hz - field_simp [hz0] - ring - -lemma hydrodynamic_composition_ratio {G F : ℂ → ℂ} {c : ℂ} - (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) - (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : - Tendsto (fun z : ℂ => G (F z)/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - have he := normalized_ratio_tendsto_cocompact hF - have hh := ((hydrodynamic_ratio_limit hG).comp he).mul hF - simp only [mul_one] at hh - apply hh.congr' - filter_upwards [he.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] with z hz - simp only [Function.comp_apply] - have hz0 : F z ≠ 0 := hz - field_simp [hz0] - - -lemma normalized_selfMap_coefficient_nonpos {G : ℂ → ℂ} {c : ℝ} - (hd : DifferentiableOn ℂ G upperHalfPlane) (hm : MapsTo G upperHalfPlane upperHalfPlane) - (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (c:ℂ))) - (hn : Tendsto (fun z : ℂ => G z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : c≤0 := by - have hr := Complex.continuous_re.tendsto (c:ℂ) |>.comp (hc.comp verticalInfinity_tendsto) - have hh : ∀ᶠ n in atTop, (verticalInfinity n*(G (verticalInfinity n)-verticalInfinity n)).re ≤ 0 := by - apply Eventually.of_forall - intro n - have hheight := normalized_halfPlane_height hd hm hn (verticalInfinity_mem n) - have hy : 0 < (n:ℝ)+1 := by positivity - simp only [verticalInfinity,mul_im,mul_re,ofReal_re,ofReal_im,I_re,I_im,mul_zero, - add_zero,mul_one,sub_im,sub_re,zero_sub] at hheight ⊢ - nlinarith - have hle : (c:ℂ).re ≤ 0 := le_of_tendsto hr (by - simpa only [Function.comp_apply] using hh) - simpa only [ofReal_re] using hle - -theorem nested_hydrodynamic_coefficient_le - {D : Set ℂ} {G H F : ℂ → ℂ} {c d : ℝ} - (hG : DifferentiableOn ℂ G D) (hGm : MapsTo G D upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) (hFm : MapsTo F upperHalfPlane D) - (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) - (hd : Tendsto (fun z : ℂ => z*(H z-z)) (cocompact ℂ) (𝓝 (d:ℂ))) - (hn : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) - (hHF : ∀ z ∈ upperHalfPlane, H (F z)=z) : c≤d := by - have hcomp := hydrodynamic_composition_coefficient hc hd hn hHF - have hnorm := hydrodynamic_composition_ratio hc hn - have hle := normalized_selfMap_coefficient_nonpos (hG.comp hF hFm) - (hGm.comp hFm) (c := c-d) (by - convert! hcomp using 1 - simp only [ofReal_sub]) - (by convert! hnorm using 1) - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.HydrodynamicAsymptotics + +namespace OAI + +/-! The coefficient of a normalized composition subtracts capacities. +Schwarz–Pick then makes capacity monotone under inclusion of slit domains. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +lemma hydrodynamic_ratio_limit {G : ℂ → ℂ} {c : ℂ} + (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) : + Tendsto (fun z : ℂ => G z/z) (cocompact ℂ) (𝓝 1) := by + have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by + simpa only [Metric.cobounded_eq_cocompact] using + (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) + have hh := (hG.mul (hi.pow 2)).const_add 1 + simp only [zero_pow two_ne_zero,mul_zero,add_zero] at hh + apply hh.congr' + filter_upwards [(isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact] with z hz + have hz0 : z ≠ 0 := hz + field_simp [hz0] + ring + +lemma hydrodynamic_composition_coefficient {G H F : ℂ → ℂ} {c d : ℂ} + (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) + (hH : Tendsto (fun z : ℂ => z*(H z-z)) (cocompact ℂ) (𝓝 d)) + (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) + (hHF : ∀ z ∈ upperHalfPlane, H (F z)=z) : + Tendsto (fun z : ℂ => z*(G (F z)-z)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (c-d)) := by + have hescape := normalized_ratio_tendsto_cocompact hF + have hquot : Tendsto (fun z : ℂ => z/F z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + simpa only [inv_div,inv_one] using hF.inv₀ one_ne_zero + have hh := (hquot.mul (hG.comp hescape)).add (inverse_hydrodynamic_coefficient hH hF hHF) + simp only [one_mul,sub_eq_add_neg] at hh ⊢ + apply hh.congr' + filter_upwards [hescape.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] with z hz + simp only [Function.comp_apply] + have hz0 : F z ≠ 0 := hz + field_simp [hz0] + ring + +lemma hydrodynamic_composition_ratio {G F : ℂ → ℂ} {c : ℂ} + (hG : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 c)) + (hF : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : + Tendsto (fun z : ℂ => G (F z)/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + have he := normalized_ratio_tendsto_cocompact hF + have hh := ((hydrodynamic_ratio_limit hG).comp he).mul hF + simp only [mul_one] at hh + apply hh.congr' + filter_upwards [he.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] with z hz + simp only [Function.comp_apply] + have hz0 : F z ≠ 0 := hz + field_simp [hz0] + + +lemma normalized_selfMap_coefficient_nonpos {G : ℂ → ℂ} {c : ℝ} + (hd : DifferentiableOn ℂ G upperHalfPlane) (hm : MapsTo G upperHalfPlane upperHalfPlane) + (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (c:ℂ))) + (hn : Tendsto (fun z : ℂ => G z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) : c≤0 := by + have hr := Complex.continuous_re.tendsto (c:ℂ) |>.comp (hc.comp verticalInfinity_tendsto) + have hh : ∀ᶠ n in atTop, (verticalInfinity n*(G (verticalInfinity n)-verticalInfinity n)).re ≤ 0 := by + apply Eventually.of_forall + intro n + have hheight := normalized_halfPlane_height hd hm hn (verticalInfinity_mem n) + have hy : 0 < (n:ℝ)+1 := by positivity + simp only [verticalInfinity,mul_im,mul_re,ofReal_re,ofReal_im,I_re,I_im,mul_zero, + add_zero,mul_one,sub_im,sub_re,zero_sub] at hheight ⊢ + nlinarith + have hle : (c:ℂ).re ≤ 0 := le_of_tendsto hr (by + simpa only [Function.comp_apply] using hh) + simpa only [ofReal_re] using hle + +theorem nested_hydrodynamic_coefficient_le + {D : Set ℂ} {G H F : ℂ → ℂ} {c d : ℝ} + (hG : DifferentiableOn ℂ G D) (hGm : MapsTo G D upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) (hFm : MapsTo F upperHalfPlane D) + (hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ))) + (hd : Tendsto (fun z : ℂ => z*(H z-z)) (cocompact ℂ) (𝓝 (d:ℂ))) + (hn : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) + (hHF : ∀ z ∈ upperHalfPlane, H (F z)=z) : c≤d := by + have hcomp := hydrodynamic_composition_coefficient hc hd hn hHF + have hnorm := hydrodynamic_composition_ratio hc hn + have hle := normalized_selfMap_coefficient_nonpos (hG.comp hF hFm) + (hGm.comp hFm) (c := c-d) (by + convert! hcomp using 1 + simp only [ofReal_sub]) + (by convert! hnorm using 1) + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/HydrodynamicHeight.lean b/lean/OAI/Probability/SLE/Loewner/HydrodynamicHeight.lean index 18a1910c359275c9dd7773cf172406cbea864337..b5b3018990b6aaa20252047e3d40ea4670780ef0 100644 --- a/lean/OAI/Probability/SLE/Loewner/HydrodynamicHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/HydrodynamicHeight.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Geometry.HalfPlaneJulia -import OAI.Probability.SLE.Loewner.SlitInverseAtInfinity - -namespace OAI - -/-! Normalization at infinity forces a half-plane self-map to increase -height. This compares the inverse maps of nested slit domains. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -def verticalInfinity (n : ℕ) : ℂ := (((n:ℝ)+1:ℝ):ℂ)*I - -lemma verticalHeights_tendsto : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop := by - convert! (tendsto_natCast_atTop_atTop (R := ℝ)).comp (tendsto_add_atTop_nat 1) using 1 - ext n - simp only [Function.comp_apply,Nat.cast_add,Nat.cast_one] - -lemma verticalInfinity_mem (n : ℕ) : verticalInfinity n ∈ upperHalfPlane := by - simp only [verticalInfinity,upperHalfPlane,mem_ofPred_eq,mul_im,ofReal_re,I_im,mul_one, - ofReal_im,I_re,mul_zero,add_zero] - positivity - -lemma verticalInfinity_tendsto : - Tendsto verticalInfinity atTop (cocompact ℂ ⊓ 𝓟 upperHalfPlane) := by - apply tendsto_inf.mpr - constructor - · rw [← Metric.cobounded_eq_cocompact,← tendsto_norm_atTop_iff_cobounded] - have hh : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop := - verticalHeights_tendsto - have hnorm : (fun n : ℕ => ‖verticalInfinity n‖)=(fun n : ℕ => (n:ℝ)+1) := by - funext n - simp only [verticalInfinity,norm_mul,Complex.norm_real,Real.norm_eq_abs,norm_I, - mul_one,abs_of_pos (by positivity : 0<(n:ℝ)+1)] - rw [hnorm] - exact hh - · exact tendsto_principal.mpr (Eventually.of_forall verticalInfinity_mem) - -theorem normalized_halfPlane_height {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (hmap : MapsTo F upperHalfPlane upperHalfPlane) - (hn : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) - {z : ℂ} (hz : z ∈ upperHalfPlane) : z.im ≤ (F z).im := by - have hr := hn.comp verticalInfinity_tendsto - have he (n : ℕ) : F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ) = - (F (verticalInfinity n)/verticalInfinity n)*I := by - have hne : (((n:ℝ)+1:ℝ):ℂ) ≠ 0 := ofReal_ne_zero.mpr (by positivity) - simp only [verticalInfinity] - field_simp - have hlim : Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I) := by - simpa only [Function.comp_apply,he,one_mul] using hr.mul_const I - exact halfPlane_julia_at_infinity hF hmap (fun n => (n:ℝ)+1) - (fun n => by positivity) - (tendsto_inv_atTop_zero.comp (verticalHeights_tendsto)) hlim hz - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HalfPlaneJulia +import OAI.Probability.SLE.Loewner.SlitInverseAtInfinity + +namespace OAI + +/-! Normalization at infinity forces a half-plane self-map to increase +height. This compares the inverse maps of nested slit domains. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +def verticalInfinity (n : ℕ) : ℂ := (((n:ℝ)+1:ℝ):ℂ)*I + +lemma verticalHeights_tendsto : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop := by + convert! (tendsto_natCast_atTop_atTop (R := ℝ)).comp (tendsto_add_atTop_nat 1) using 1 + ext n + simp only [Function.comp_apply,Nat.cast_add,Nat.cast_one] + +lemma verticalInfinity_mem (n : ℕ) : verticalInfinity n ∈ upperHalfPlane := by + simp only [verticalInfinity,upperHalfPlane,mem_ofPred_eq,mul_im,ofReal_re,I_im,mul_one, + ofReal_im,I_re,mul_zero,add_zero] + positivity + +lemma verticalInfinity_tendsto : + Tendsto verticalInfinity atTop (cocompact ℂ ⊓ 𝓟 upperHalfPlane) := by + apply tendsto_inf.mpr + constructor + · rw [← Metric.cobounded_eq_cocompact,← tendsto_norm_atTop_iff_cobounded] + have hh : Tendsto (fun n : ℕ => (n:ℝ)+1) atTop atTop := + verticalHeights_tendsto + have hnorm : (fun n : ℕ => ‖verticalInfinity n‖)=(fun n : ℕ => (n:ℝ)+1) := by + funext n + simp only [verticalInfinity,norm_mul,Complex.norm_real,Real.norm_eq_abs,norm_I, + mul_one,abs_of_pos (by positivity : 0<(n:ℝ)+1)] + rw [hnorm] + exact hh + · exact tendsto_principal.mpr (Eventually.of_forall verticalInfinity_mem) + +theorem normalized_halfPlane_height {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (hmap : MapsTo F upperHalfPlane upperHalfPlane) + (hn : Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) + {z : ℂ} (hz : z ∈ upperHalfPlane) : z.im ≤ (F z).im := by + have hr := hn.comp verticalInfinity_tendsto + have he (n : ℕ) : F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ) = + (F (verticalInfinity n)/verticalInfinity n)*I := by + have hne : (((n:ℝ)+1:ℝ):ℂ) ≠ 0 := ofReal_ne_zero.mpr (by positivity) + simp only [verticalInfinity] + field_simp + have hlim : Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I) := by + simpa only [Function.comp_apply,he,one_mul] using hr.mul_const I + exact halfPlane_julia_at_infinity hF hmap (fun n => (n:ℝ)+1) + (fun n => by positivity) + (tendsto_inv_atTop_zero.comp (verticalHeights_tendsto)) hlim hz + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialCapacity.lean b/lean/OAI/Probability/SLE/Loewner/InitialCapacity.lean index 153c0014675745f4a0585a36517c92454af7f0ad..874e85e60f7bc4982081559d9b4e7092c7795e19 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialCapacity.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialCapacity.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Loewner.InitialSubarc - -namespace OAI - -/-! A monotone positive capacity parameter on the exact initial subarcs. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -structure SimpleSlitChart (p : C(unitInterval,ℂ)) where - forward : ℂ → ℂ - inverse : ℂ → ℂ - capacity : ℝ - capacity_pos : 0 z*(forward z-z)) (cocompact ℂ) (𝓝 (capacity:ℂ)) - inverse_normalized : Tendsto (fun z : ℂ => inverse z/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) - height_le : ∀ z ∈ upperHalfPlane \ range p, (forward z).im≤z.im - -lemma exists_simpleSlitChart (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : Nonempty (SimpleSlitChart p) := by - obtain ⟨G,F,c,hc,hb,hdG,hdF,hm,hl,hr,hlim,hn,hh⟩ := - simple_slit_exists_positive_capacity_chart p hi h0 hH - exact ⟨⟨G,F,c,hc,hb,hdG,hdF,hm,hl,hr,hlim,hn,hh⟩⟩ - -noncomputable def initialSlitChart (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) (ht : t ≠ 0) : SimpleSlitChart (initialSubarc p t) := - Classical.choice (exists_simpleSlitChart _ (initialSubarc_injective hi ht) - (by simpa only [initialSubarc_zero] using h0) (initialSubarc_upper hH ht)) - -lemma SimpleSlitChart.capacity_le_of_range_subset - {p q : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (Q : SimpleSlitChart q) - (hpq : range p ⊆ range q) : P.capacity≤Q.capacity := by - have hdom : upperHalfPlane \ range q ⊆ upperHalfPlane \ range p := by - intro z hz - exact ⟨hz.1,fun hp => hz.2 (hpq hp)⟩ - exact nested_hydrodynamic_coefficient_le P.forward_differentiable P.bijOn.mapsTo - Q.inverse_differentiable (Q.inverse_mapsTo.mono_right hdom) - P.coefficient Q.coefficient Q.inverse_normalized Q.right_inverse - -noncomputable def initialCapacity (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) (t : unitInterval) : ℝ := - if ht : t=0 then 0 else (initialSlitChart p hi h0 hH t ht).capacity - -@[simp] lemma initialCapacity_zero (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : initialCapacity p hi h0 hH 0=0 := by - simp [initialCapacity] - -lemma initialCapacity_pos (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : unitInterval} (ht : t ≠ 0) : 0 < initialCapacity p hi h0 hH t := by - rw [initialCapacity,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).capacity_pos - -lemma initialCapacity_nonneg (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) (t : unitInterval) : - 0 ≤ initialCapacity p hi h0 hH t := by - by_cases ht : t=0 - · simp only [ht,initialCapacity_zero,le_refl] - · exact (initialCapacity_pos p hi h0 hH ht).le - -theorem initialCapacity_monotone (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - Monotone (initialCapacity p hi h0 hH) := by - intro s t hst - by_cases hs : s=0 - · rw [hs,initialCapacity_zero] - exact initialCapacity_nonneg p hi h0 hH t - have ht : t≠0 := by - intro ht - exact hs (le_antisymm (ht ▸ hst) bot_le) - rw [initialCapacity,dite_eq_right hs,initialCapacity,dite_eq_right ht] - exact SimpleSlitChart.capacity_le_of_range_subset _ _ (initialSubarc_range_mono p hst) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialSubarc + +namespace OAI + +/-! A monotone positive capacity parameter on the exact initial subarcs. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +structure SimpleSlitChart (p : C(unitInterval,ℂ)) where + forward : ℂ → ℂ + inverse : ℂ → ℂ + capacity : ℝ + capacity_pos : 0 z*(forward z-z)) (cocompact ℂ) (𝓝 (capacity:ℂ)) + inverse_normalized : Tendsto (fun z : ℂ => inverse z/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) + height_le : ∀ z ∈ upperHalfPlane \ range p, (forward z).im≤z.im + +lemma exists_simpleSlitChart (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : Nonempty (SimpleSlitChart p) := by + obtain ⟨G,F,c,hc,hb,hdG,hdF,hm,hl,hr,hlim,hn,hh⟩ := + simple_slit_exists_positive_capacity_chart p hi h0 hH + exact ⟨⟨G,F,c,hc,hb,hdG,hdF,hm,hl,hr,hlim,hn,hh⟩⟩ + +noncomputable def initialSlitChart (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) (ht : t ≠ 0) : SimpleSlitChart (initialSubarc p t) := + Classical.choice (exists_simpleSlitChart _ (initialSubarc_injective hi ht) + (by simpa only [initialSubarc_zero] using h0) (initialSubarc_upper hH ht)) + +lemma SimpleSlitChart.capacity_le_of_range_subset + {p q : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (Q : SimpleSlitChart q) + (hpq : range p ⊆ range q) : P.capacity≤Q.capacity := by + have hdom : upperHalfPlane \ range q ⊆ upperHalfPlane \ range p := by + intro z hz + exact ⟨hz.1,fun hp => hz.2 (hpq hp)⟩ + exact nested_hydrodynamic_coefficient_le P.forward_differentiable P.bijOn.mapsTo + Q.inverse_differentiable (Q.inverse_mapsTo.mono_right hdom) + P.coefficient Q.coefficient Q.inverse_normalized Q.right_inverse + +noncomputable def initialCapacity (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) (t : unitInterval) : ℝ := + if ht : t=0 then 0 else (initialSlitChart p hi h0 hH t ht).capacity + +@[simp] lemma initialCapacity_zero (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : initialCapacity p hi h0 hH 0=0 := by + simp [initialCapacity] + +lemma initialCapacity_pos (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : unitInterval} (ht : t ≠ 0) : 0 < initialCapacity p hi h0 hH t := by + rw [initialCapacity,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).capacity_pos + +lemma initialCapacity_nonneg (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) (t : unitInterval) : + 0 ≤ initialCapacity p hi h0 hH t := by + by_cases ht : t=0 + · simp only [ht,initialCapacity_zero,le_refl] + · exact (initialCapacity_pos p hi h0 hH ht).le + +theorem initialCapacity_monotone (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + Monotone (initialCapacity p hi h0 hH) := by + intro s t hst + by_cases hs : s=0 + · rw [hs,initialCapacity_zero] + exact initialCapacity_nonneg p hi h0 hH t + have ht : t≠0 := by + intro ht + exact hs (le_antisymm (ht ▸ hst) bot_le) + rw [initialCapacity,dite_eq_right hs,initialCapacity,dite_eq_right ht] + exact SimpleSlitChart.capacity_le_of_range_subset _ _ (initialSubarc_range_mono p hst) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialCapacityAtZero.lean b/lean/OAI/Probability/SLE/Loewner/InitialCapacityAtZero.lean index 1fb1bba591d88bfd6423abccd4d0f3ef93bb9d4d..fc6cf8ac5eb5673bad28420879a4434bcd80a850 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialCapacityAtZero.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialCapacityAtZero.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Loewner.CapacityRadiusBound - -namespace OAI - -/-! The capacity of initial subarcs tends to zero at the starting point, -by the explicit half-disk capacity bound. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -theorem initialCapacity_continuousAt_zero (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ContinuousAt (initialCapacity p hi h0 hH) 0 := by - rw [Metric.continuousAt_iff] - intro ε hε - let R : ℝ := Real.sqrt ε/2 - have hR : 0 (x:ℝ)) - ((initialCapacityHomeomorph p hi h0 hH).apply_symm_apply c) - exact hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialCapacityContinuity +import OAI.Probability.SLE.Loewner.InitialCapacityStrict +import Mathlib.Topology.Order.IntermediateValue +import Mathlib.Order.Hom.Set + +namespace OAI + +/-! The capacity of a simple attached slit is a genuine continuous clock, +with an exact continuous inverse on its full range. -/ +noncomputable section +open Set Function +open scoped Topology + +namespace SLEExactGauge + +lemma initialCapacity_range (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + range (initialCapacity p hi h0 hH)=Icc 0 (initialCapacity p hi h0 hH 1) := by + have hh := (initialCapacity_continuous p hi h0 hH).image_Icc_of_strictMono + (initialCapacity_strictMono p hi h0 hH) (a := 0) (b := 1) + have hIcc : Icc (0:unitInterval) 1=univ := by + ext x + simp only [mem_Icc,mem_univ,iff_true] + exact ⟨bot_le,le_top⟩ + simpa only [hIcc,image_univ,initialCapacity_zero] using hh + +noncomputable def initialCapacityHomeomorph (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + unitInterval ≃ₜ Icc 0 (initialCapacity p hi h0 hH 1) := + (((initialCapacity_strictMono p hi h0 hH).orderIso (initialCapacity p hi h0 hH)).trans + (Set.orderIsoOfEq _ _ (initialCapacity_range p hi h0 hH))).toHomeomorph + +@[simp] lemma initialCapacityHomeomorph_apply (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) : + ((initialCapacityHomeomorph p hi h0 hH t : Icc 0 (initialCapacity p hi h0 hH 1)):ℝ)= + initialCapacity p hi h0 hH t := rfl + +@[simp] lemma initialCapacityHomeomorph_symm_capacity (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (c : Icc 0 (initialCapacity p hi h0 hH 1)) : + initialCapacity p hi h0 hH ((initialCapacityHomeomorph p hi h0 hH).symm c)=(c:ℝ) := by + have hh := congrArg (fun x : Icc 0 (initialCapacity p hi h0 hH 1) => (x:ℝ)) + ((initialCapacityHomeomorph p hi h0 hH).apply_symm_apply c) + exact hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialCapacityContinuity.lean b/lean/OAI/Probability/SLE/Loewner/InitialCapacityContinuity.lean index 1a0ca8c8ae8b92cfb07bbf2275bbbaa78d893672..991142859949d7938b386be58b9b98a9165acf2b 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialCapacityContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialCapacityContinuity.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Loewner.InitialCapacityTerminal -import OAI.Probability.SLE.Loewner.InitialForwardComparison -import OAI.Probability.SLE.Mass.VerticalDisplacementMass - -namespace OAI - -/-! Continuity of the full capacity parameter. The capacity of the initial -slit and its complementary capacity relative to the final slit give the two -inequalities for every subsequential limit. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialCapacity_limit_value_at (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} {c : ℝ} - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (hs : s≠0) (ht : Tendsto t atTop (𝓝 s)) - (hcap : Tendsto (fun n => initialCapacity p hi h0 hH (t n)) atTop (𝓝 c)) : - c=initialCapacity p hi h0 hH s := by - let P := initialSlitChart p hi h0 hH s hs - have heF : EqOn F P.inverse upperHalfPlane := by - simpa only [initialInverseChart,dite_eq_right hs] using - initialInverse_limit_eq_at p hi h0 hH hF hlim hs ht - have hlow : P.capacity≤c := by - apply hydrodynamic_mass_le_of_displacement - (inverse_hydrodynamic_coefficient P.coefficient P.inverse_normalized P.right_inverse) - intro z hz - rw [← heF hz] - exact initialInverse_limit_displacement p hi h0 hH hlim hcap hz - obtain ⟨Q⟩ := exists_simpleSlitChart p hi h0 hH - obtain ⟨φ,hφ,hforward⟩ := initialInverse_limit_forward_on_final p hi h0 hH hF hlim hs ht - have hbound (z : ℂ) (hz : z∈upperHalfPlane) : - ‖P.forward (Q.inverse z)-z‖≤(Q.capacity-c)/z.im := by - have hleft := ((hforward (Q.inverse z) (Q.inverse_mapsTo hz)).sub_const z).norm - have hright := ((tendsto_const_nhds (x := Q.capacity)).sub - (hcap.comp hφ.tendsto_atTop)).div_const z.im - exact le_of_tendsto_of_tendsto hleft hright - (Eventually.of_forall fun n => initialForward_comparison_to_final p hi h0 hH Q _ hz) - have hcomp : Tendsto (fun z : ℂ => z*(P.forward (Q.inverse z)-z)) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-((Q.capacity-P.capacity:ℝ):ℂ))) := by - have he : -((Q.capacity-P.capacity:ℝ):ℂ)=(P.capacity:ℂ)-Q.capacity := by - push_cast - ring - rw [he] - exact hydrodynamic_composition_coefficient P.coefficient Q.coefficient - Q.inverse_normalized Q.right_inverse - have hupper := hydrodynamic_mass_le_of_displacement hcomp hbound - rw [initialCapacity,dite_eq_right hs] - change c=P.capacity - exact le_antisymm (by linarith) hlow - -theorem initialCapacity_continuousAt (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (s : unitInterval) : ContinuousAt (initialCapacity p hi h0 hH) s := by - by_cases hs : s=0 - · subst s - exact initialCapacity_continuousAt_zero p hi h0 hH - apply tendsto_of_subseq_tendsto - intro t ht - have hb (n : ℕ) : initialCapacity p hi h0 hH (t n)∈ - Icc 0 (initialCapacity p hi h0 hH 1) := - ⟨initialCapacity_nonneg p hi h0 hH _,initialCapacity_monotone p hi h0 hH le_top⟩ - obtain ⟨c,_,φ,hφ,hcap⟩ := isCompact_Icc.tendsto_subseq hb - obtain ⟨ψ,F,hψ,hF,hFlim⟩ := initialInverseChart_subsequence p hi h0 hH (t ∘ φ) - have ht' : Tendsto (fun n => t (φ (ψ n))) atTop (𝓝 s) := - (ht.comp hφ.tendsto_atTop).comp hψ.tendsto_atTop - have hc' := hcap.comp hψ.tendsto_atTop - have he := initialCapacity_limit_value_at p hi h0 hH hF hFlim hs ht' hc' - refine ⟨φ ∘ ψ,?_⟩ - change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) - atTop (𝓝 (initialCapacity p hi h0 hH s)) - change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) atTop (𝓝 c) at hc' - rwa [he] at hc' - -theorem initialCapacity_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - Continuous (initialCapacity p hi h0 hH) := - continuous_iff_continuousAt.mpr (initialCapacity_continuousAt p hi h0 hH) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialCapacityTerminal +import OAI.Probability.SLE.Loewner.InitialForwardComparison +import OAI.Probability.SLE.Mass.VerticalDisplacementMass + +namespace OAI + +/-! Continuity of the full capacity parameter. The capacity of the initial +slit and its complementary capacity relative to the final slit give the two +inequalities for every subsequential limit. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma initialCapacity_limit_value_at (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} {c : ℝ} + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (hs : s≠0) (ht : Tendsto t atTop (𝓝 s)) + (hcap : Tendsto (fun n => initialCapacity p hi h0 hH (t n)) atTop (𝓝 c)) : + c=initialCapacity p hi h0 hH s := by + let P := initialSlitChart p hi h0 hH s hs + have heF : EqOn F P.inverse upperHalfPlane := by + simpa only [initialInverseChart,dite_eq_right hs] using + initialInverse_limit_eq_at p hi h0 hH hF hlim hs ht + have hlow : P.capacity≤c := by + apply hydrodynamic_mass_le_of_displacement + (inverse_hydrodynamic_coefficient P.coefficient P.inverse_normalized P.right_inverse) + intro z hz + rw [← heF hz] + exact initialInverse_limit_displacement p hi h0 hH hlim hcap hz + obtain ⟨Q⟩ := exists_simpleSlitChart p hi h0 hH + obtain ⟨φ,hφ,hforward⟩ := initialInverse_limit_forward_on_final p hi h0 hH hF hlim hs ht + have hbound (z : ℂ) (hz : z∈upperHalfPlane) : + ‖P.forward (Q.inverse z)-z‖≤(Q.capacity-c)/z.im := by + have hleft := ((hforward (Q.inverse z) (Q.inverse_mapsTo hz)).sub_const z).norm + have hright := ((tendsto_const_nhds (x := Q.capacity)).sub + (hcap.comp hφ.tendsto_atTop)).div_const z.im + exact le_of_tendsto_of_tendsto hleft hright + (Eventually.of_forall fun n => initialForward_comparison_to_final p hi h0 hH Q _ hz) + have hcomp : Tendsto (fun z : ℂ => z*(P.forward (Q.inverse z)-z)) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-((Q.capacity-P.capacity:ℝ):ℂ))) := by + have he : -((Q.capacity-P.capacity:ℝ):ℂ)=(P.capacity:ℂ)-Q.capacity := by + push_cast + ring + rw [he] + exact hydrodynamic_composition_coefficient P.coefficient Q.coefficient + Q.inverse_normalized Q.right_inverse + have hupper := hydrodynamic_mass_le_of_displacement hcomp hbound + rw [initialCapacity,dite_eq_right hs] + change c=P.capacity + exact le_antisymm (by linarith) hlow + +theorem initialCapacity_continuousAt (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (s : unitInterval) : ContinuousAt (initialCapacity p hi h0 hH) s := by + by_cases hs : s=0 + · subst s + exact initialCapacity_continuousAt_zero p hi h0 hH + apply tendsto_of_subseq_tendsto + intro t ht + have hb (n : ℕ) : initialCapacity p hi h0 hH (t n)∈ + Icc 0 (initialCapacity p hi h0 hH 1) := + ⟨initialCapacity_nonneg p hi h0 hH _,initialCapacity_monotone p hi h0 hH le_top⟩ + obtain ⟨c,_,φ,hφ,hcap⟩ := isCompact_Icc.tendsto_subseq hb + obtain ⟨ψ,F,hψ,hF,hFlim⟩ := initialInverseChart_subsequence p hi h0 hH (t ∘ φ) + have ht' : Tendsto (fun n => t (φ (ψ n))) atTop (𝓝 s) := + (ht.comp hφ.tendsto_atTop).comp hψ.tendsto_atTop + have hc' := hcap.comp hψ.tendsto_atTop + have he := initialCapacity_limit_value_at p hi h0 hH hF hFlim hs ht' hc' + refine ⟨φ ∘ ψ,?_⟩ + change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) + atTop (𝓝 (initialCapacity p hi h0 hH s)) + change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) atTop (𝓝 c) at hc' + rwa [he] at hc' + +theorem initialCapacity_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + Continuous (initialCapacity p hi h0 hH) := + continuous_iff_continuousAt.mpr (initialCapacity_continuousAt p hi h0 hH) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialCapacityStrict.lean b/lean/OAI/Probability/SLE/Loewner/InitialCapacityStrict.lean index ae0bd992fd3d0ed00859eacc712afcfc80f67a47..6f75257c99e97b8df6182dee49bc0d3edb0a24c0 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialCapacityStrict.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialCapacityStrict.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Loewner.InitialCapacityAtZero - -namespace OAI - -/-! Capacity strictly increases when a simple attached slit is extended. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.capacity_lt_of_domain_difference - {p q : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (Q : SimpleSlitChart q) - (hpq : range p ⊆ range q) {z : ℂ} - (hz : z∈upperHalfPlane \ range p) (hzq : z∈range q) : - P.capacity ⟨hw.1,fun hp => hw.2 (hpq hp)⟩ - have hm := Q.inverse_mapsTo.mono_right hdom - have hd := P.forward_differentiable.comp Q.inverse_differentiable hm - have hh := hydrodynamic_composition_coefficient P.coefficient Q.coefficient - Q.inverse_normalized Q.right_inverse - have hcoef : Tendsto (fun w : ℂ => w*(P.forward (Q.inverse w)-w)) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by - simpa only [he,sub_self] using hh - have hn := hydrodynamic_composition_ratio P.coefficient Q.inverse_normalized - have heq := holomorphic_zero_hydrodynamic_mass (hd.sub differentiableOn_id) - (fun w hw => by - have hb := normalized_halfPlane_height hd (P.bijOn.mapsTo.comp hm) hn hw - change 0 ≤ (P.forward (Q.inverse w)-w).im - simpa only [Function.comp_apply,Complex.sub_im] using sub_nonneg.mpr hb) hcoef - have hwH := P.bijOn.mapsTo hz - have hsame : Q.inverse (P.forward z)=z := by - apply P.bijOn.injOn (hm hwH) hz - exact sub_eq_zero.mp (heq (P.forward z) hwH) - exact (Q.inverse_mapsTo hwH).2 (hsame.symm ▸ hzq) - -theorem initialCapacity_strictMono (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - StrictMono (initialCapacity p hi h0 hH) := by - intro s t hst - have ht : t≠0 := ne_of_gt (bot_le.trans_lt hst) - by_cases hs : s=0 - · rw [hs,initialCapacity_zero] - exact initialCapacity_pos p hi h0 hH ht - rw [initialCapacity,dite_eq_right hs,initialCapacity,dite_eq_right ht] - apply SimpleSlitChart.capacity_lt_of_domain_difference _ _ - (initialSubarc_range_mono p hst.le) (z := p t) - · refine ⟨hH t ht,?_⟩ - rw [initialSubarc_range] - rintro ⟨u,hu,he⟩ - have hut := hi he - exact (not_le_of_gt hst) (hut ▸ hu.2) - · rw [initialSubarc_range] - exact mem_image_of_mem p ⟨bot_le,le_rfl⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialCapacityAtZero + +namespace OAI + +/-! Capacity strictly increases when a simple attached slit is extended. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.capacity_lt_of_domain_difference + {p q : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (Q : SimpleSlitChart q) + (hpq : range p ⊆ range q) {z : ℂ} + (hz : z∈upperHalfPlane \ range p) (hzq : z∈range q) : + P.capacity ⟨hw.1,fun hp => hw.2 (hpq hp)⟩ + have hm := Q.inverse_mapsTo.mono_right hdom + have hd := P.forward_differentiable.comp Q.inverse_differentiable hm + have hh := hydrodynamic_composition_coefficient P.coefficient Q.coefficient + Q.inverse_normalized Q.right_inverse + have hcoef : Tendsto (fun w : ℂ => w*(P.forward (Q.inverse w)-w)) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by + simpa only [he,sub_self] using hh + have hn := hydrodynamic_composition_ratio P.coefficient Q.inverse_normalized + have heq := holomorphic_zero_hydrodynamic_mass (hd.sub differentiableOn_id) + (fun w hw => by + have hb := normalized_halfPlane_height hd (P.bijOn.mapsTo.comp hm) hn hw + change 0 ≤ (P.forward (Q.inverse w)-w).im + simpa only [Function.comp_apply,Complex.sub_im] using sub_nonneg.mpr hb) hcoef + have hwH := P.bijOn.mapsTo hz + have hsame : Q.inverse (P.forward z)=z := by + apply P.bijOn.injOn (hm hwH) hz + exact sub_eq_zero.mp (heq (P.forward z) hwH) + exact (Q.inverse_mapsTo hwH).2 (hsame.symm ▸ hzq) + +theorem initialCapacity_strictMono (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + StrictMono (initialCapacity p hi h0 hH) := by + intro s t hst + have ht : t≠0 := ne_of_gt (bot_le.trans_lt hst) + by_cases hs : s=0 + · rw [hs,initialCapacity_zero] + exact initialCapacity_pos p hi h0 hH ht + rw [initialCapacity,dite_eq_right hs,initialCapacity,dite_eq_right ht] + apply SimpleSlitChart.capacity_lt_of_domain_difference _ _ + (initialSubarc_range_mono p hst.le) (z := p t) + · refine ⟨hH t ht,?_⟩ + rw [initialSubarc_range] + rintro ⟨u,hu,he⟩ + have hut := hi he + exact (not_le_of_gt hst) (hut ▸ hu.2) + · rw [initialSubarc_range] + exact mem_image_of_mem p ⟨bot_le,le_rfl⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialCapacityTerminal.lean b/lean/OAI/Probability/SLE/Loewner/InitialCapacityTerminal.lean index 94299ed337f67336c0ec58a201f3ddec656f6265..a09229b2e1aad9cb510e320a0adff1c8d6560cb6 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialCapacityTerminal.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialCapacityTerminal.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Loewner.SlitChartLimitUniqueness - -namespace OAI - -/-! Continuity of initial-subarc capacity at its terminal time. Compactness -of the actual inverse charts identifies every scalar subsequential limit. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.capacity_eq_initial_final {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - P.capacity = initialCapacity p hi h0 hH 1 := by - have h1 : (1:unitInterval)≠0 := by norm_num - rw [initialCapacity,dite_eq_right h1] - have hr : range (initialSubarc p 1)=range p := by - ext z - simp only [mem_range,initialSubarc_apply,one_mul] - exact le_antisymm - (P.capacity_le_of_range_subset (initialSlitChart p hi h0 hH 1 h1) hr.ge) - ((initialSlitChart p hi h0 hH 1 h1).capacity_le_of_range_subset P hr.le) - -lemma initialCapacity_terminal_limit_value (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} {c : ℝ} - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) - (hcap : Tendsto (fun n => initialCapacity p hi h0 hH (t n)) atTop (𝓝 c)) : - c = initialCapacity p hi h0 hH 1 := by - obtain ⟨P⟩ := exists_simpleSlitChart p hi h0 hH - have heq := initialInverse_limit_eq_final p hi h0 hH P hF hlim ht - have hcoef := inverse_hydrodynamic_coefficient P.coefficient P.inverse_normalized P.right_inverse - have hmass := (hydrodynamic_vertical_limits (c := P.capacity) hcoef).2 - have hle : P.capacity≤c := by - apply le_of_tendsto hmass - apply Eventually.of_forall - intro n - have hb := initialInverse_limit_displacement p hi h0 hH hlim hcap (verticalInfinity_mem n) - rw [heq (verticalInfinity_mem n)] at hb - have him := (Complex.im_le_norm (P.inverse (verticalInfinity n)-verticalInfinity n)).trans hb - have hy : 0<(n:ℝ)+1 := by positivity - have hbound := (le_div_iff₀ hy).mp (show - (P.inverse (verticalInfinity n)-verticalInfinity n).im≤c/((n:ℝ)+1) from by - simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero] - using him) - nlinarith - have hge : c ≤ initialCapacity p hi h0 hH 1 := by - apply le_of_tendsto hcap - exact Eventually.of_forall fun n => initialCapacity_monotone p hi h0 hH - (show t n≤1 from le_top) - rw [P.capacity_eq_initial_final hi h0 hH] at hle - exact le_antisymm hge hle - -theorem initialCapacity_continuousAt_one (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ContinuousAt (initialCapacity p hi h0 hH) 1 := by - apply tendsto_of_subseq_tendsto - intro t ht - have hb (n : ℕ) : initialCapacity p hi h0 hH (t n)∈ - Icc 0 (initialCapacity p hi h0 hH 1) := - ⟨initialCapacity_nonneg p hi h0 hH _,initialCapacity_monotone p hi h0 hH le_top⟩ - obtain ⟨c,_,φ,hφ,hcap⟩ := isCompact_Icc.tendsto_subseq hb - obtain ⟨ψ,F,hψ,hF,hFlim⟩ := initialInverseChart_subsequence p hi h0 hH (t ∘ φ) - have ht' : Tendsto (fun n => t (φ (ψ n))) atTop (𝓝 1) := - (ht.comp hφ.tendsto_atTop).comp hψ.tendsto_atTop - have hc' := hcap.comp hψ.tendsto_atTop - have he := initialCapacity_terminal_limit_value p hi h0 hH hF hFlim ht' hc' - refine ⟨φ ∘ ψ,?_⟩ - change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) - atTop (𝓝 (initialCapacity p hi h0 hH 1)) - change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) atTop (𝓝 c) at hc' - rwa [he] at hc' - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartLimitUniqueness + +namespace OAI + +/-! Continuity of initial-subarc capacity at its terminal time. Compactness +of the actual inverse charts identifies every scalar subsequential limit. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.capacity_eq_initial_final {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + P.capacity = initialCapacity p hi h0 hH 1 := by + have h1 : (1:unitInterval)≠0 := by norm_num + rw [initialCapacity,dite_eq_right h1] + have hr : range (initialSubarc p 1)=range p := by + ext z + simp only [mem_range,initialSubarc_apply,one_mul] + exact le_antisymm + (P.capacity_le_of_range_subset (initialSlitChart p hi h0 hH 1 h1) hr.ge) + ((initialSlitChart p hi h0 hH 1 h1).capacity_le_of_range_subset P hr.le) + +lemma initialCapacity_terminal_limit_value (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} {c : ℝ} + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) + (hcap : Tendsto (fun n => initialCapacity p hi h0 hH (t n)) atTop (𝓝 c)) : + c = initialCapacity p hi h0 hH 1 := by + obtain ⟨P⟩ := exists_simpleSlitChart p hi h0 hH + have heq := initialInverse_limit_eq_final p hi h0 hH P hF hlim ht + have hcoef := inverse_hydrodynamic_coefficient P.coefficient P.inverse_normalized P.right_inverse + have hmass := (hydrodynamic_vertical_limits (c := P.capacity) hcoef).2 + have hle : P.capacity≤c := by + apply le_of_tendsto hmass + apply Eventually.of_forall + intro n + have hb := initialInverse_limit_displacement p hi h0 hH hlim hcap (verticalInfinity_mem n) + rw [heq (verticalInfinity_mem n)] at hb + have him := (Complex.im_le_norm (P.inverse (verticalInfinity n)-verticalInfinity n)).trans hb + have hy : 0<(n:ℝ)+1 := by positivity + have hbound := (le_div_iff₀ hy).mp (show + (P.inverse (verticalInfinity n)-verticalInfinity n).im≤c/((n:ℝ)+1) from by + simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero] + using him) + nlinarith + have hge : c ≤ initialCapacity p hi h0 hH 1 := by + apply le_of_tendsto hcap + exact Eventually.of_forall fun n => initialCapacity_monotone p hi h0 hH + (show t n≤1 from le_top) + rw [P.capacity_eq_initial_final hi h0 hH] at hle + exact le_antisymm hge hle + +theorem initialCapacity_continuousAt_one (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + ContinuousAt (initialCapacity p hi h0 hH) 1 := by + apply tendsto_of_subseq_tendsto + intro t ht + have hb (n : ℕ) : initialCapacity p hi h0 hH (t n)∈ + Icc 0 (initialCapacity p hi h0 hH 1) := + ⟨initialCapacity_nonneg p hi h0 hH _,initialCapacity_monotone p hi h0 hH le_top⟩ + obtain ⟨c,_,φ,hφ,hcap⟩ := isCompact_Icc.tendsto_subseq hb + obtain ⟨ψ,F,hψ,hF,hFlim⟩ := initialInverseChart_subsequence p hi h0 hH (t ∘ φ) + have ht' : Tendsto (fun n => t (φ (ψ n))) atTop (𝓝 1) := + (ht.comp hφ.tendsto_atTop).comp hψ.tendsto_atTop + have hc' := hcap.comp hψ.tendsto_atTop + have he := initialCapacity_terminal_limit_value p hi h0 hH hF hFlim ht' hc' + refine ⟨φ ∘ ψ,?_⟩ + change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) + atTop (𝓝 (initialCapacity p hi h0 hH 1)) + change Tendsto (fun n => initialCapacity p hi h0 hH (t (φ (ψ n)))) atTop (𝓝 c) at hc' + rwa [he] at hc' + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialChartIdentities.lean b/lean/OAI/Probability/SLE/Loewner/InitialChartIdentities.lean index efa3616950bc8d5c6298ac1470d09fcf5ab00b00..5e2ec695c040edd3015a0c6c22ce8eb317d51f21 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialChartIdentities.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialChartIdentities.lean @@ -1,34 +1,34 @@ -import OAI.Probability.SLE.Loewner.InitialForwardChart - -namespace OAI - -/-! Forward identities on the domain at the current time, rather than -only on the common final domain. -/ -noncomputable section -open Set Function - -namespace SLEExactGauge - -lemma initialForwardChart_mem_upper (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : - initialForwardChart p hi h0 hH s z∈upperHalfPlane := by - by_cases hs : s=0 - · simpa only [hs,initialForwardChart_zero,id_eq] using hz.1 - · rw [initialForwardChart,dite_eq_right hs] - exact (initialSlitChart p hi h0 hH s hs).bijOn.mapsTo hz - -lemma initialInverse_forward_at (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : - initialInverseChart p hi h0 hH s (initialForwardChart p hi h0 hH s z)=z := by - by_cases hs : s=0 - · simp only [hs,initialForwardChart_zero,initialInverseChart_zero,id_eq] - · rw [initialForwardChart,dite_eq_right hs,initialInverseChart,dite_eq_right hs] - exact (initialSlitChart p hi h0 hH s hs).left_inverse z hz - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialForwardChart + +namespace OAI + +/-! Forward identities on the domain at the current time, rather than +only on the common final domain. -/ +noncomputable section +open Set Function + +namespace SLEExactGauge + +lemma initialForwardChart_mem_upper (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : + initialForwardChart p hi h0 hH s z∈upperHalfPlane := by + by_cases hs : s=0 + · simpa only [hs,initialForwardChart_zero,id_eq] using hz.1 + · rw [initialForwardChart,dite_eq_right hs] + exact (initialSlitChart p hi h0 hH s hs).bijOn.mapsTo hz + +lemma initialInverse_forward_at (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : + initialInverseChart p hi h0 hH s (initialForwardChart p hi h0 hH s z)=z := by + by_cases hs : s=0 + · simp only [hs,initialForwardChart_zero,initialInverseChart_zero,id_eq] + · rw [initialForwardChart,dite_eq_right hs,initialInverseChart,dite_eq_right hs] + exact (initialSlitChart p hi h0 hH s hs).left_inverse z hz + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialChartNormalization.lean b/lean/OAI/Probability/SLE/Loewner/InitialChartNormalization.lean index 9a3785aa4f35115e7a0c81d3d8e1282d600e122c..5373b7dfba6a36ce9ff6e816e58443c66aa430c4 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialChartNormalization.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialChartNormalization.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Loewner.InitialChartIdentities -import OAI.Probability.SLE.Loewner.RadiusHullInverse - -namespace OAI - -/-! Normalization and height of the initial chart family, including its -identity chart at time zero. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialForwardChart_differentiableOn_current (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) : - DifferentiableOn ℂ (initialForwardChart p hi h0 hH s) - (upperHalfPlane \ range (initialSubarc p s)) := by - by_cases hs : s=0 - · simp only [hs,initialForwardChart_zero] - exact differentiableOn_id - · simpa only [initialForwardChart,dite_eq_right hs] using - (initialSlitChart p hi h0 hH s hs).forward_differentiable - -lemma initialForwardChart_coefficient (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) : - Tendsto (fun z : ℂ => z*(initialForwardChart p hi h0 hH s z-z)) (cocompact ℂ) - (𝓝 (initialCapacity p hi h0 hH s:ℂ)) := by - by_cases hs : s=0 - · simp only [hs,initialForwardChart_zero,id_eq,sub_self,mul_zero,initialCapacity_zero,ofReal_zero] - exact tendsto_const_nhds - · simpa only [initialForwardChart,initialCapacity,dite_eq_right hs] using - (initialSlitChart p hi h0 hH s hs).coefficient - -lemma initialInverseChart_normalized (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) : - Tendsto (fun z : ℂ => initialInverseChart p hi h0 hH s z/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - by_cases hs : s=0 - · apply ratio_one_of_bounded_displacement (R := 0) - intro z _ - simp only [hs,initialInverseChart_zero,id_eq,sub_self,norm_zero,le_refl] - · simpa only [initialInverseChart,dite_eq_right hs] using - (initialSlitChart p hi h0 hH s hs).inverse_normalized - -lemma initialForwardChart_height_le (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : - (initialForwardChart p hi h0 hH s z).im≤z.im := by - by_cases hs : s=0 - · simp only [hs,initialForwardChart_zero,id_eq,le_refl] - · simpa only [initialForwardChart,dite_eq_right hs] using - (initialSlitChart p hi h0 hH s hs).height_le z hz - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialChartIdentities +import OAI.Probability.SLE.Loewner.RadiusHullInverse + +namespace OAI + +/-! Normalization and height of the initial chart family, including its +identity chart at time zero. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma initialForwardChart_differentiableOn_current (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) : + DifferentiableOn ℂ (initialForwardChart p hi h0 hH s) + (upperHalfPlane \ range (initialSubarc p s)) := by + by_cases hs : s=0 + · simp only [hs,initialForwardChart_zero] + exact differentiableOn_id + · simpa only [initialForwardChart,dite_eq_right hs] using + (initialSlitChart p hi h0 hH s hs).forward_differentiable + +lemma initialForwardChart_coefficient (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) : + Tendsto (fun z : ℂ => z*(initialForwardChart p hi h0 hH s z-z)) (cocompact ℂ) + (𝓝 (initialCapacity p hi h0 hH s:ℂ)) := by + by_cases hs : s=0 + · simp only [hs,initialForwardChart_zero,id_eq,sub_self,mul_zero,initialCapacity_zero,ofReal_zero] + exact tendsto_const_nhds + · simpa only [initialForwardChart,initialCapacity,dite_eq_right hs] using + (initialSlitChart p hi h0 hH s hs).coefficient + +lemma initialInverseChart_normalized (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) : + Tendsto (fun z : ℂ => initialInverseChart p hi h0 hH s z/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + by_cases hs : s=0 + · apply ratio_one_of_bounded_displacement (R := 0) + intro z _ + simp only [hs,initialInverseChart_zero,id_eq,sub_self,norm_zero,le_refl] + · simpa only [initialInverseChart,dite_eq_right hs] using + (initialSlitChart p hi h0 hH s hs).inverse_normalized + +lemma initialForwardChart_height_le (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : + (initialForwardChart p hi h0 hH s z).im≤z.im := by + by_cases hs : s=0 + · simp only [hs,initialForwardChart_zero,id_eq,le_refl] + · simpa only [initialForwardChart,dite_eq_right hs] using + (initialSlitChart p hi h0 hH s hs).height_le z hz + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialConeFromInterval.lean b/lean/OAI/Probability/SLE/Loewner/InitialConeFromInterval.lean index 316e54662fb0948919b68c5aee21a910b5d1efbe..d063e6053be996ee9edff822e568c73ae606b664 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialConeFromInterval.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialConeFromInterval.lean @@ -1,41 +1,41 @@ -import OAI.Probability.SLE.Loewner.LocalReverseCone -import OAI.Probability.SLE.LoewnerCharts - -namespace OAI - -/-! The initial cone only uses the driving derivative inside its interval; -no continuation of that derivative beyond the terminal time is needed. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -lemma exists_initial_reverse_cone_of_interval_derivative {T L : ℝ} - (hT : 0 < T) (hL : 0 ≤ L) (q : ℝ → ℂ) (u u' : ℝ → ℝ) - (hqc : ContinuousOn q (Icc 0 T)) - (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) - (hq0 : q 0=(u 0 : ℂ)) - (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Ici t) t) - (hqp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) - (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) : - ∃ a : ℝ, 0 < a ∧ a ≤ T ∧ - ∀ t ∈ Icc 0 a, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2 := by - have hhalf : T/2 < T := half_lt_self hT - have hsub : Icc 0 (T/2) ⊆ Icc 0 T := Icc_subset_Icc_right hhalf.le - have hpos (t : ℝ) (ht : t ∈ Ioc 0 (T/2)) : t ∈ Ioc 0 T := - ⟨ht.1,ht.2.trans hhalf.le⟩ - obtain ⟨a,ha,haT,hcone⟩ := exists_initial_reverse_cone (half_pos hT) hL q u u' - (hqc.mono hsub) (fun t ht => (hu t (hsub ht)).continuousWithinAt.mono hsub) hq0 - (fun t ht => hq t (hpos t ht)) - (fun t ht => hasDerivWithinAt_Ici_of_Icc ⟨ht.1.le,ht.2.trans_lt hhalf⟩ - (hu t ⟨ht.1.le,ht.2.trans hhalf.le⟩)) - (fun t ht => hqp t (hpos t ht)) (fun t ht => hb t ⟨ht.1.le,ht.2.trans hhalf.le⟩) - exact ⟨a,ha,haT.trans hhalf.le,hcone⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LocalReverseCone +import OAI.Probability.SLE.LoewnerCharts + +namespace OAI + +/-! The initial cone only uses the driving derivative inside its interval; +no continuation of that derivative beyond the terminal time is needed. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +lemma exists_initial_reverse_cone_of_interval_derivative {T L : ℝ} + (hT : 0 < T) (hL : 0 ≤ L) (q : ℝ → ℂ) (u u' : ℝ → ℝ) + (hqc : ContinuousOn q (Icc 0 T)) + (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) + (hq0 : q 0=(u 0 : ℂ)) + (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Ici t) t) + (hqp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) + (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) : + ∃ a : ℝ, 0 < a ∧ a ≤ T ∧ + ∀ t ∈ Icc 0 a, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2 := by + have hhalf : T/2 < T := half_lt_self hT + have hsub : Icc 0 (T/2) ⊆ Icc 0 T := Icc_subset_Icc_right hhalf.le + have hpos (t : ℝ) (ht : t ∈ Ioc 0 (T/2)) : t ∈ Ioc 0 T := + ⟨ht.1,ht.2.trans hhalf.le⟩ + obtain ⟨a,ha,haT,hcone⟩ := exists_initial_reverse_cone (half_pos hT) hL q u u' + (hqc.mono hsub) (fun t ht => (hu t (hsub ht)).continuousWithinAt.mono hsub) hq0 + (fun t ht => hq t (hpos t ht)) + (fun t ht => hasDerivWithinAt_Ici_of_Icc ⟨ht.1.le,ht.2.trans_lt hhalf⟩ + (hu t ⟨ht.1.le,ht.2.trans hhalf.le⟩)) + (fun t ht => hqp t (hpos t ht)) (fun t ht => hb t ⟨ht.1.le,ht.2.trans hhalf.le⟩) + exact ⟨a,ha,haT.trans hhalf.le,hcone⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialDrivingBound.lean b/lean/OAI/Probability/SLE/Loewner/InitialDrivingBound.lean index 61697492486c4aee9ab654f58a29cd445161d0b7..5d2ee805d21a645250a2269f47a90fd3b6075b3a 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialDrivingBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialDrivingBound.lean @@ -1,42 +1,42 @@ -import OAI.Probability.SLE.Loewner.SlitUniformForwardBound -import OAI.Probability.SLE.Loewner.InitialDrivingLimit - -namespace OAI - -/-! Every simple slit in a fixed disk has uniformly bounded driving -coordinates, including when the tip approaches either end. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem initialDrivingValue_bound (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 hp _) hz ((hp _).le.trans (by linarith)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitUniformForwardBound +import OAI.Probability.SLE.Loewner.InitialDrivingLimit + +namespace OAI + +/-! Every simple slit in a fixed disk has uniformly bounded driving +coordinates, including when the tip approaches either end. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +theorem initialDrivingValue_bound (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 hp _) hz ((hp _).le.trans (by linarith)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialDrivingContinuity.lean b/lean/OAI/Probability/SLE/Loewner/InitialDrivingContinuity.lean index 657f12d0560edfd2ddda64f84d43163be696da9e..122b30fd575fb5f601c7db19e4977a12673b7701 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialDrivingContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialDrivingContinuity.lean @@ -1,71 +1,71 @@ -import OAI.Probability.SLE.Stochastic.DrivingLeftCluster - -namespace OAI - -/-! Full continuity of the actual slit driving value before the final -parameter, derived from the inverse energy and boundary noncollapse. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -theorem initialDriving_tendsto_from_left (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 initialDriving p hi h0 hH (t n)) atTop - (𝓝 (initialDriving p hi h0 hH s)) := by - apply tendsto_of_subseq_tendsto - intro ns hns - let M := (512*Real.pi+9)*R - have hbound (n : ℕ) : initialDriving p hi h0 hH (t (ns n))∈Icc (-M) M := by - rw [initialDriving_of_lt p hi h0 hH ((hts (ns n)).trans hs)] - exact abs_le.mp (initialDrivingValue_bound p hi h0 hH hR hp _ _) - obtain ⟨a,_,φ,hφ,hlim⟩ := isCompact_Icc.tendsto_subseq hbound - have he := initialDriving_left_cluster p hi h0 hH hR hp hs0 hs - (fun n => hts (ns (φ n))) ((ht.comp hns).comp hφ.tendsto_atTop) hlim - exact ⟨φ,by simpa only [Function.comp_def,he] using hlim⟩ - -theorem initialDriving_continuousWithinAt_left (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s : unitInterval} (hs : s<1) : - ContinuousWithinAt (initialDriving p hi h0 hH) (Iio s) s := by - by_cases hs0 : s=0 - · subst s - have hem : Iio (0:unitInterval)=∅ := by - ext x - simp only [mem_Iio,mem_empty_iff_false,iff_false] - exact not_lt_of_ge (show (0:unitInterval)≤x from x.property.1) - rw [hem] - exact continuousWithinAt_of_notMem_closure (by simp) - obtain ⟨R0,hRp⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le - let R := max R0 0+1 - have hR : 0 t (n+N)) atTop (𝓝 s) := - (ht.mono_right nhdsWithin_le_nhds).comp (tendsto_add_atTop_nat N) - have hres := initialDriving_tendsto_from_left p hi h0 hH hR - hpR hs0 hs - (fun n => hN (n+N) (by omega)) ht' - exact (tendsto_add_atTop_iff_nat N).mp hres - -theorem initialDriving_continuousAt (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s : unitInterval} (hs : s<1) : - ContinuousAt (initialDriving p hi h0 hH) s := by - exact continuousAt_iff_continuous_left'_right'.mpr - ⟨initialDriving_continuousWithinAt_left p hi h0 hH hs, - initialDriving_continuousWithinAt_right p hi h0 hH hs⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.DrivingLeftCluster + +namespace OAI + +/-! Full continuity of the actual slit driving value before the final +parameter, derived from the inverse energy and boundary noncollapse. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +theorem initialDriving_tendsto_from_left (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 initialDriving p hi h0 hH (t n)) atTop + (𝓝 (initialDriving p hi h0 hH s)) := by + apply tendsto_of_subseq_tendsto + intro ns hns + let M := (512*Real.pi+9)*R + have hbound (n : ℕ) : initialDriving p hi h0 hH (t (ns n))∈Icc (-M) M := by + rw [initialDriving_of_lt p hi h0 hH ((hts (ns n)).trans hs)] + exact abs_le.mp (initialDrivingValue_bound p hi h0 hH hR hp _ _) + obtain ⟨a,_,φ,hφ,hlim⟩ := isCompact_Icc.tendsto_subseq hbound + have he := initialDriving_left_cluster p hi h0 hH hR hp hs0 hs + (fun n => hts (ns (φ n))) ((ht.comp hns).comp hφ.tendsto_atTop) hlim + exact ⟨φ,by simpa only [Function.comp_def,he] using hlim⟩ + +theorem initialDriving_continuousWithinAt_left (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s : unitInterval} (hs : s<1) : + ContinuousWithinAt (initialDriving p hi h0 hH) (Iio s) s := by + by_cases hs0 : s=0 + · subst s + have hem : Iio (0:unitInterval)=∅ := by + ext x + simp only [mem_Iio,mem_empty_iff_false,iff_false] + exact not_lt_of_ge (show (0:unitInterval)≤x from x.property.1) + rw [hem] + exact continuousWithinAt_of_notMem_closure (by simp) + obtain ⟨R0,hRp⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le + let R := max R0 0+1 + have hR : 0 t (n+N)) atTop (𝓝 s) := + (ht.mono_right nhdsWithin_le_nhds).comp (tendsto_add_atTop_nat N) + have hres := initialDriving_tendsto_from_left p hi h0 hH hR + hpR hs0 hs + (fun n => hN (n+N) (by omega)) ht' + exact (tendsto_add_atTop_iff_nat N).mp hres + +theorem initialDriving_continuousAt (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s : unitInterval} (hs : s<1) : + ContinuousAt (initialDriving p hi h0 hH) s := by + exact continuousAt_iff_continuous_left'_right'.mpr + ⟨initialDriving_continuousWithinAt_left p hi h0 hH hs, + initialDriving_continuousWithinAt_right p hi h0 hH hs⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialDrivingIncrement.lean b/lean/OAI/Probability/SLE/Loewner/InitialDrivingIncrement.lean index eee8b8e823b7a6319c093d75b65464e4a3e446a0..c81f89719ad32afe39ef9e1f65a38d939179b8c3 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialDrivingIncrement.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialDrivingIncrement.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.Loewner.InitialDrivingBound -import OAI.Probability.SLE.Loewner.FutureCapacityAdditive - -namespace OAI - -/-! A small mapped future slit forces its two driving values to be -close, without any assumed continuity of the driver. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem initialDriving_increment_bound (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s p (slitFutureParameter t r)) (𝓝 0) (𝓝 (p t)) := by - have hcont : Continuous (fun r : ℝ => p (slitFutureParameter t r)) := - p.continuous.comp (slitFutureParameter_continuous t) - simpa only [slitFutureParameter_zero] using hcont.tendsto 0 - have hwlim : Tendsto w (𝓝 0) (𝓝 (centeredFutureArc p hi h0 hH hst ht 1)) := by - have hh := (hG.tendsto.comp hf).sub_const (U:ℂ) - have he : centeredFutureArc p hi h0 hH hst ht 1= - initialForwardChart p hi h0 hH s (p t)-(U:ℂ) := by - change mappedFutureArc p hi h0 hH hst ht 1-(U:ℂ)=_ - rw [mappedFutureArc_segment p hi h0 hH hst ht one_ne_zero,slitSegmentParameter_one] - rwa [he] - have hnorm : ∀ᶠ r : ℝ in 𝓝[>] 0, ‖w r‖<2*R := by - exact (hwlim.norm.eventually (gt_mem_nhds ((hp 1).trans (by linarith)))).filter_mono inf_le_left - have hl := ((initialDrivingValue_spec p hi h0 hH t ht).sub_const (U:ℂ)).norm - have hl' : Tendsto (fun r : ℝ => ‖initialForwardChart p hi h0 hH t - (p (slitFutureParameter t r))-(U:ℂ)‖) (𝓝[>] 0) - (𝓝 |initialDrivingValue p hi h0 hH t ht-U|) := by - simpa only [←ofReal_sub,Complex.norm_real,Real.norm_eq_abs] using hl - apply le_of_tendsto hl' - have hd : 0<1-(t:ℝ) := sub_pos.mpr ht - filter_upwards [self_mem_nhdsWithin,hnorm, - Filter.Eventually.filter_mono inf_le_left (gt_mem_nhds hd)] with r hr hwr hrd - have hz := slitFuturePoint_mem_domain p hi hH t (show r∈Ioo 0 (1-(t:ℝ)) from ⟨hr,hrd⟩) - have hw := (initialForward_centered_bijOn p hi h0 hH hst ht).mapsTo hz - have hb := Q.forward_uniform_near - (centeredFutureArc_injective p hi h0 hH hst ht) - (centeredFutureArc_zero p hi h0 hH hst ht) - (fun _ hu => centeredFutureArc_upper p hi h0 hH hst ht hu) hR hp hw hwr.le - have he : (U:ℂ)+Q.forward (w r)=initialForwardChart p hi h0 hH t - (p (slitFutureParameter t r)) := futureComposedForward_eq p hi h0 hH hst ht hz - have he' : Q.forward (w r)=initialForwardChart p hi h0 hH t - (p (slitFutureParameter t r))-(U:ℂ) := by linear_combination he - rwa [he'] at hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialDrivingBound +import OAI.Probability.SLE.Loewner.FutureCapacityAdditive + +namespace OAI + +/-! A small mapped future slit forces its two driving values to be +close, without any assumed continuity of the driver. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +theorem initialDriving_increment_bound (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s p (slitFutureParameter t r)) (𝓝 0) (𝓝 (p t)) := by + have hcont : Continuous (fun r : ℝ => p (slitFutureParameter t r)) := + p.continuous.comp (slitFutureParameter_continuous t) + simpa only [slitFutureParameter_zero] using hcont.tendsto 0 + have hwlim : Tendsto w (𝓝 0) (𝓝 (centeredFutureArc p hi h0 hH hst ht 1)) := by + have hh := (hG.tendsto.comp hf).sub_const (U:ℂ) + have he : centeredFutureArc p hi h0 hH hst ht 1= + initialForwardChart p hi h0 hH s (p t)-(U:ℂ) := by + change mappedFutureArc p hi h0 hH hst ht 1-(U:ℂ)=_ + rw [mappedFutureArc_segment p hi h0 hH hst ht one_ne_zero,slitSegmentParameter_one] + rwa [he] + have hnorm : ∀ᶠ r : ℝ in 𝓝[>] 0, ‖w r‖<2*R := by + exact (hwlim.norm.eventually (gt_mem_nhds ((hp 1).trans (by linarith)))).filter_mono inf_le_left + have hl := ((initialDrivingValue_spec p hi h0 hH t ht).sub_const (U:ℂ)).norm + have hl' : Tendsto (fun r : ℝ => ‖initialForwardChart p hi h0 hH t + (p (slitFutureParameter t r))-(U:ℂ)‖) (𝓝[>] 0) + (𝓝 |initialDrivingValue p hi h0 hH t ht-U|) := by + simpa only [←ofReal_sub,Complex.norm_real,Real.norm_eq_abs] using hl + apply le_of_tendsto hl' + have hd : 0<1-(t:ℝ) := sub_pos.mpr ht + filter_upwards [self_mem_nhdsWithin,hnorm, + Filter.Eventually.filter_mono inf_le_left (gt_mem_nhds hd)] with r hr hwr hrd + have hz := slitFuturePoint_mem_domain p hi hH t (show r∈Ioo 0 (1-(t:ℝ)) from ⟨hr,hrd⟩) + have hw := (initialForward_centered_bijOn p hi h0 hH hst ht).mapsTo hz + have hb := Q.forward_uniform_near + (centeredFutureArc_injective p hi h0 hH hst ht) + (centeredFutureArc_zero p hi h0 hH hst ht) + (fun _ hu => centeredFutureArc_upper p hi h0 hH hst ht hu) hR hp hw hwr.le + have he : (U:ℂ)+Q.forward (w r)=initialForwardChart p hi h0 hH t + (p (slitFutureParameter t r)) := futureComposedForward_eq p hi h0 hH hst ht hz + have he' : Q.forward (w r)=initialForwardChart p hi h0 hH t + (p (slitFutureParameter t r))-(U:ℂ) := by linear_combination he + rwa [he'] at hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialDrivingLimit.lean b/lean/OAI/Probability/SLE/Loewner/InitialDrivingLimit.lean index 686a70aa51b163163c4c68d564e631e3a5e85eca..2164056316be2a90fc7f866909fb2a220a98a052 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialDrivingLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialDrivingLimit.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.Loewner.InitialSlitFuture -import OAI.Probability.SLE.Loewner.SlitForwardEndpoint - -namespace OAI - -/-! The mapped future of a simple slit approaches a real boundary point. -This supplies the driving value directly from the prescribed arc. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem exists_initial_driving_limit (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) (hs : s<1) : - ∃ a : ℝ, Tendsto - (fun r : ℝ => initialForwardChart p hi h0 hH s (p (slitFutureParameter s r))) - (𝓝[>] (0:ℝ)) (𝓝 (a:ℂ)) := by - have hD : 0<1-(s:ℝ) := sub_pos.mpr (show (s:ℝ)<1 from hs) - have hC : Continuous (fun r : ℝ => p (slitFutureParameter s r)) := - p.continuous.comp (slitFutureParameter_continuous s) - have hz : Tendsto (fun r : ℝ => p (slitFutureParameter s r)) - (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by - have hc : Tendsto (fun r : ℝ => p (slitFutureParameter s r)) - (𝓝 (0:ℝ)) (𝓝 (p s)) := by - simpa only [slitFutureParameter_zero] using hC.tendsto 0 - exact hc.mono_left nhdsWithin_le_nhds - by_cases hs0 : s=0 - · refine ⟨0,?_⟩ - simpa only [hs0,initialForwardChart_zero,id_eq,h0,ofReal_zero] using hz - · simpa only [initialForwardChart,dite_eq_right hs0] using - (initialSlitChart p hi h0 hH s hs0).exists_forward_endpoint_limit hD hC.continuousOn - hz (fun _ hr => slitFuturePoint_mem_domain p hi hH s hr) (slit_tip_not_mem_domain p s) - -noncomputable def initialDrivingValue (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) (hs : s<1) : ℝ := - Classical.choose (exists_initial_driving_limit p hi h0 hH s hs) - -theorem initialDrivingValue_spec (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) (hs : s<1) : - Tendsto (fun r : ℝ => initialForwardChart p hi h0 hH s (p (slitFutureParameter s r))) - (𝓝[>] (0:ℝ)) (𝓝 ((initialDrivingValue p hi h0 hH s hs):ℂ)) := - Classical.choose_spec (exists_initial_driving_limit p hi h0 hH s hs) - -@[simp] theorem initialDrivingValue_zero (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) : - initialDrivingValue p hi h0 hH 0 zero_lt_one=0 := by - have hz := initialDrivingValue_spec p hi h0 hH 0 zero_lt_one - have hC := (p.continuous.comp (slitFutureParameter_continuous 0)).tendsto 0 - |>.mono_left (show 𝓝[>] (0:ℝ)≤𝓝 (0:ℝ) from inf_le_left) - simp only [initialForwardChart_zero,id_eq] at hz - change Tendsto (fun r => p (slitFutureParameter 0 r)) (𝓝[>] (0:ℝ)) - (𝓝 (p (slitFutureParameter 0 0))) at hC - have he := tendsto_nhds_unique hz hC - simpa only [slitFutureParameter_zero,h0,ofReal_zero,initialForwardChart_zero,id_eq, - ofReal_eq_zero] using he - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialSlitFuture +import OAI.Probability.SLE.Loewner.SlitForwardEndpoint + +namespace OAI + +/-! The mapped future of a simple slit approaches a real boundary point. +This supplies the driving value directly from the prescribed arc. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +theorem exists_initial_driving_limit (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) (hs : s<1) : + ∃ a : ℝ, Tendsto + (fun r : ℝ => initialForwardChart p hi h0 hH s (p (slitFutureParameter s r))) + (𝓝[>] (0:ℝ)) (𝓝 (a:ℂ)) := by + have hD : 0<1-(s:ℝ) := sub_pos.mpr (show (s:ℝ)<1 from hs) + have hC : Continuous (fun r : ℝ => p (slitFutureParameter s r)) := + p.continuous.comp (slitFutureParameter_continuous s) + have hz : Tendsto (fun r : ℝ => p (slitFutureParameter s r)) + (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by + have hc : Tendsto (fun r : ℝ => p (slitFutureParameter s r)) + (𝓝 (0:ℝ)) (𝓝 (p s)) := by + simpa only [slitFutureParameter_zero] using hC.tendsto 0 + exact hc.mono_left nhdsWithin_le_nhds + by_cases hs0 : s=0 + · refine ⟨0,?_⟩ + simpa only [hs0,initialForwardChart_zero,id_eq,h0,ofReal_zero] using hz + · simpa only [initialForwardChart,dite_eq_right hs0] using + (initialSlitChart p hi h0 hH s hs0).exists_forward_endpoint_limit hD hC.continuousOn + hz (fun _ hr => slitFuturePoint_mem_domain p hi hH s hr) (slit_tip_not_mem_domain p s) + +noncomputable def initialDrivingValue (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) (hs : s<1) : ℝ := + Classical.choose (exists_initial_driving_limit p hi h0 hH s hs) + +theorem initialDrivingValue_spec (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) (hs : s<1) : + Tendsto (fun r : ℝ => initialForwardChart p hi h0 hH s (p (slitFutureParameter s r))) + (𝓝[>] (0:ℝ)) (𝓝 ((initialDrivingValue p hi h0 hH s hs):ℂ)) := + Classical.choose_spec (exists_initial_driving_limit p hi h0 hH s hs) + +@[simp] theorem initialDrivingValue_zero (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) : + initialDrivingValue p hi h0 hH 0 zero_lt_one=0 := by + have hz := initialDrivingValue_spec p hi h0 hH 0 zero_lt_one + have hC := (p.continuous.comp (slitFutureParameter_continuous 0)).tendsto 0 + |>.mono_left (show 𝓝[>] (0:ℝ)≤𝓝 (0:ℝ) from inf_le_left) + simp only [initialForwardChart_zero,id_eq] at hz + change Tendsto (fun r => p (slitFutureParameter 0 r)) (𝓝[>] (0:ℝ)) + (𝓝 (p (slitFutureParameter 0 0))) at hC + have he := tendsto_nhds_unique hz hC + simpa only [slitFutureParameter_zero,h0,ofReal_zero,initialForwardChart_zero,id_eq, + ofReal_eq_zero] using he + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialDrivingRightContinuity.lean b/lean/OAI/Probability/SLE/Loewner/InitialDrivingRightContinuity.lean index 9792d60a6ce9b51acb195e9c2cc218f3be02f28a..dbe14abcc54e1e8eeeb94dff5a0662fcea39a14c 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialDrivingRightContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialDrivingRightContinuity.lean @@ -1,55 +1,55 @@ -import OAI.Probability.SLE.Loewner.InitialDrivingIncrement -import OAI.Probability.SLE.Loewner.FutureSlitRadius - -namespace OAI - -/-! Right continuity of the actual driving value follows from shrinking -mapped future slits and the uniform boundary-coordinate estimate. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -noncomputable def initialDriving (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) : ℝ := - if hs : s<1 then initialDrivingValue p hi h0 hH s hs else 0 - -lemma initialDriving_of_lt (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s : unitInterval} (hs : s<1) : - initialDriving p hi h0 hH s=initialDrivingValue p hi h0 hH s hs := dite_eq_left hs - -theorem initialDriving_continuousWithinAt_right (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s : unitInterval} (hs : s<1) : - ContinuousWithinAt (initialDriving p hi h0 hH) (Ioi s) s := by - apply Metric.tendsto_nhds.mpr - intro ε hε - let C : ℝ := 512*Real.pi+9 - have hC : 0 (t:ℝ)-(s:ℝ)) (𝓝[>] s) (𝓝 0) := by - have hc : Continuous (fun t : unitInterval => (t:ℝ)-(s:ℝ)) := - continuous_subtype_val.sub continuous_const - have hh := (hc.tendsto s).mono_left (show 𝓝[>] s≤𝓝 s from inf_le_left) - simpa only [sub_self] using hh - filter_upwards [self_mem_nhdsWithin,htime.eventually (gt_mem_nhds hδ), - Filter.Eventually.filter_mono inf_le_left (gt_mem_nhds hs)] with t hst htd ht - rw [Real.dist_eq,initialDriving_of_lt p hi h0 hH ht,initialDriving_of_lt p hi h0 hH hs] - have hb := initialDriving_increment_bound p hi h0 hH hst ht hR (hsmall t hst ht htd) - apply hb.trans_lt - change C*R<ε - dsimp only [R] - have he : C*(ε/(2*C))=ε/2 := by field_simp - rw [he] - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialDrivingIncrement +import OAI.Probability.SLE.Loewner.FutureSlitRadius + +namespace OAI + +/-! Right continuity of the actual driving value follows from shrinking +mapped future slits and the uniform boundary-coordinate estimate. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +noncomputable def initialDriving (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) : ℝ := + if hs : s<1 then initialDrivingValue p hi h0 hH s hs else 0 + +lemma initialDriving_of_lt (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s : unitInterval} (hs : s<1) : + initialDriving p hi h0 hH s=initialDrivingValue p hi h0 hH s hs := dite_eq_left hs + +theorem initialDriving_continuousWithinAt_right (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s : unitInterval} (hs : s<1) : + ContinuousWithinAt (initialDriving p hi h0 hH) (Ioi s) s := by + apply Metric.tendsto_nhds.mpr + intro ε hε + let C : ℝ := 512*Real.pi+9 + have hC : 0 (t:ℝ)-(s:ℝ)) (𝓝[>] s) (𝓝 0) := by + have hc : Continuous (fun t : unitInterval => (t:ℝ)-(s:ℝ)) := + continuous_subtype_val.sub continuous_const + have hh := (hc.tendsto s).mono_left (show 𝓝[>] s≤𝓝 s from inf_le_left) + simpa only [sub_self] using hh + filter_upwards [self_mem_nhdsWithin,htime.eventually (gt_mem_nhds hδ), + Filter.Eventually.filter_mono inf_le_left (gt_mem_nhds hs)] with t hst htd ht + rw [Real.dist_eq,initialDriving_of_lt p hi h0 hH ht,initialDriving_of_lt p hi h0 hH hs] + have hb := initialDriving_increment_bound p hi h0 hH hst ht hR (hsmall t hst ht htd) + apply hb.trans_lt + change C*R<ε + dsimp only [R] + have he : C*(ε/(2*C))=ε/2 := by field_simp + rw [he] + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialDrivingTip.lean b/lean/OAI/Probability/SLE/Loewner/InitialDrivingTip.lean index 44da184bda8a4be3fa7fcd94a32361015143660e..ce171ddea03a483bf8a29ffabd470c493588b8de 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialDrivingTip.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialDrivingTip.lean @@ -1,75 +1,75 @@ -import OAI.Probability.SLE.Loewner.CapacityLoewnerIntegral -import OAI.Probability.SLE.Loewner.SlitChartBoundaryExtension - -namespace OAI - -/-! The radial inverse limit at the proved driving value is the prescribed -simple-arc tip. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem initialInverse_radial_driver_limit (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s : unitInterval} (hs : s<1) : - Tendsto (fun y : ℝ => initialInverseChart p hi h0 hH s - ((initialDrivingValue p hi h0 hH s hs:ℂ)+(y:ℂ)*I)) - (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by - by_cases hs0 : s=0 - · subst s - simp only [initialInverseChart_zero,initialDrivingValue_zero,ofReal_zero,zero_add,h0,id_eq] - have hh := (Complex.continuous_ofReal.tendsto 0).mul_const I - simpa only [ofReal_zero,zero_mul] using hh.mono_left nhdsWithin_le_nhds - obtain ⟨B,hB,hBF⟩ := (initialSlitChart p hi h0 hH s hs0).exists_boundary_extension - have hBe : ∀ z∈upperHalfPlane, B z=initialInverseChart p hi h0 hH s z := by - intro z hz - simpa only [initialInverseChart,dite_eq_right hs0] using hBF z hz - let u := initialDrivingValue p hi h0 hH s hs - have hG := initialDrivingValue_spec p hi h0 hH s hs - have hGmem : ∀ᶠ r : ℝ in 𝓝[>] (0:ℝ), - initialForwardChart p hi h0 hH s (p (slitFutureParameter s r))∈{z : ℂ | 0≤z.im} := by - filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds - (gt_mem_nhds (sub_pos.mpr (show (s:ℝ)<1 from hs)))] with r hr hrd - exact (show 0<(initialForwardChart p hi h0 hH s - (p (slitFutureParameter s r))).im from initialForwardChart_mem_upper p hi h0 hH s - (slitFuturePoint_mem_domain p hi hH s ⟨hr,hrd⟩)).le - have hBg := (hB (u:ℂ) (by simp)).tendsto.comp - (tendsto_nhdsWithin_iff.mpr ⟨hG,hGmem⟩) - have hphysical : Tendsto (fun r : ℝ => p (slitFutureParameter s r)) - (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by - have hc : Continuous (fun r : ℝ => p (slitFutureParameter s r)) := - p.continuous.comp (slitFutureParameter_continuous s) - simpa only [slitFutureParameter_zero] using - (hc.tendsto 0).mono_left (show 𝓝[>] (0:ℝ)≤𝓝 (0:ℝ) from nhdsWithin_le_nhds) - have heq : B (u:ℂ)=p s := by - apply tendsto_nhds_unique hBg - apply hphysical.congr' - filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds - (gt_mem_nhds (sub_pos.mpr (show (s:ℝ)<1 from hs)))] with r hr hrd - have hd := slitFuturePoint_mem_domain p hi hH s (show r∈Ioo 0 (1-(s:ℝ)) from ⟨hr,hrd⟩) - symm - change B (initialForwardChart p hi h0 hH s (p (slitFutureParameter s r)))= - p (slitFutureParameter s r) - rw [hBe _ (initialForwardChart_mem_upper p hi h0 hH s hd),initialInverse_forward_at p hi h0 hH s hd] - have hzlim : Tendsto (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (u:ℂ)) := by - have hc : Continuous (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) := - continuous_const.add (Complex.continuous_ofReal.mul continuous_const) - simpa only [ofReal_zero,zero_mul,add_zero] using - (hc.tendsto 0).mono_left - (show 𝓝[>] (0:ℝ)≤𝓝 (0:ℝ) from nhdsWithin_le_nhds) - have hzmem : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), (u:ℂ)+(y:ℂ)*I∈{z : ℂ | 0≤z.im} := by - filter_upwards [self_mem_nhdsWithin] with y hy - simpa using hy.le - have hh := (hB (u:ℂ) (by simp)).tendsto.comp (tendsto_nhdsWithin_iff.mpr ⟨hzlim,hzmem⟩) - rw [heq] at hh - apply hh.congr' - filter_upwards [self_mem_nhdsWithin] with y hy - exact hBe _ (by simpa [upperHalfPlane] using hy) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.CapacityLoewnerIntegral +import OAI.Probability.SLE.Loewner.SlitChartBoundaryExtension + +namespace OAI + +/-! The radial inverse limit at the proved driving value is the prescribed +simple-arc tip. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +theorem initialInverse_radial_driver_limit (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s : unitInterval} (hs : s<1) : + Tendsto (fun y : ℝ => initialInverseChart p hi h0 hH s + ((initialDrivingValue p hi h0 hH s hs:ℂ)+(y:ℂ)*I)) + (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by + by_cases hs0 : s=0 + · subst s + simp only [initialInverseChart_zero,initialDrivingValue_zero,ofReal_zero,zero_add,h0,id_eq] + have hh := (Complex.continuous_ofReal.tendsto 0).mul_const I + simpa only [ofReal_zero,zero_mul] using hh.mono_left nhdsWithin_le_nhds + obtain ⟨B,hB,hBF⟩ := (initialSlitChart p hi h0 hH s hs0).exists_boundary_extension + have hBe : ∀ z∈upperHalfPlane, B z=initialInverseChart p hi h0 hH s z := by + intro z hz + simpa only [initialInverseChart,dite_eq_right hs0] using hBF z hz + let u := initialDrivingValue p hi h0 hH s hs + have hG := initialDrivingValue_spec p hi h0 hH s hs + have hGmem : ∀ᶠ r : ℝ in 𝓝[>] (0:ℝ), + initialForwardChart p hi h0 hH s (p (slitFutureParameter s r))∈{z : ℂ | 0≤z.im} := by + filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds + (gt_mem_nhds (sub_pos.mpr (show (s:ℝ)<1 from hs)))] with r hr hrd + exact (show 0<(initialForwardChart p hi h0 hH s + (p (slitFutureParameter s r))).im from initialForwardChart_mem_upper p hi h0 hH s + (slitFuturePoint_mem_domain p hi hH s ⟨hr,hrd⟩)).le + have hBg := (hB (u:ℂ) (by simp)).tendsto.comp + (tendsto_nhdsWithin_iff.mpr ⟨hG,hGmem⟩) + have hphysical : Tendsto (fun r : ℝ => p (slitFutureParameter s r)) + (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by + have hc : Continuous (fun r : ℝ => p (slitFutureParameter s r)) := + p.continuous.comp (slitFutureParameter_continuous s) + simpa only [slitFutureParameter_zero] using + (hc.tendsto 0).mono_left (show 𝓝[>] (0:ℝ)≤𝓝 (0:ℝ) from nhdsWithin_le_nhds) + have heq : B (u:ℂ)=p s := by + apply tendsto_nhds_unique hBg + apply hphysical.congr' + filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds + (gt_mem_nhds (sub_pos.mpr (show (s:ℝ)<1 from hs)))] with r hr hrd + have hd := slitFuturePoint_mem_domain p hi hH s (show r∈Ioo 0 (1-(s:ℝ)) from ⟨hr,hrd⟩) + symm + change B (initialForwardChart p hi h0 hH s (p (slitFutureParameter s r)))= + p (slitFutureParameter s r) + rw [hBe _ (initialForwardChart_mem_upper p hi h0 hH s hd),initialInverse_forward_at p hi h0 hH s hd] + have hzlim : Tendsto (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (u:ℂ)) := by + have hc : Continuous (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) := + continuous_const.add (Complex.continuous_ofReal.mul continuous_const) + simpa only [ofReal_zero,zero_mul,add_zero] using + (hc.tendsto 0).mono_left + (show 𝓝[>] (0:ℝ)≤𝓝 (0:ℝ) from nhdsWithin_le_nhds) + have hzmem : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), (u:ℂ)+(y:ℂ)*I∈{z : ℂ | 0≤z.im} := by + filter_upwards [self_mem_nhdsWithin] with y hy + simpa using hy.le + have hh := (hB (u:ℂ) (by simp)).tendsto.comp (tendsto_nhdsWithin_iff.mpr ⟨hzlim,hzmem⟩) + rw [heq] at hh + apply hh.congr' + filter_upwards [self_mem_nhdsWithin] with y hy + exact hBe _ (by simpa [upperHalfPlane] using hy) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardBaseBounds.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardBaseBounds.lean index c15f3c2ca397d98b41fa88ff8878c1014d3501cc..e0fe6855795ffd32e1a4f39449be42a07204f88d 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardBaseBounds.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardBaseBounds.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Loewner.InitialForwardChart -import OAI.Probability.SLE.Loewner.SlitChartVerticalBounds - -namespace OAI - -/-! Uniform interior control at a single exterior base point for every initial chart. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma initialForwardChart_base_bounds (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {R : ℝ} (hR : 0 linarith - · rw [initialForwardChart,dite_eq_right ht] - apply (initialSlitChart p hi h0 hH t ht).vertical_base_bounds hR - intro u - exact hp (t*u) - -lemma initialForward_base_mem_domain (p : C(unitInterval,ℂ)) {R : ℝ} - (hR : 0 linarith + · rw [initialForwardChart,dite_eq_right ht] + apply (initialSlitChart p hi h0 hH t ht).vertical_base_bounds hR + intro u + exact hp (t*u) + +lemma initialForward_base_mem_domain (p : C(unitInterval,ℂ)) {R : ℝ} + (hR : 0 hz.2 (initialSubarc_range_subset p t h)⟩ - -lemma initialForwardChart_differentiableOn (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) : - DifferentiableOn ℂ (initialForwardChart p hi h0 hH t) (upperHalfPlane \ range p) := by - by_cases ht : t=0 - · simp only [ht,initialForwardChart_zero] - exact differentiableOn_id - · rw [initialForwardChart,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).forward_differentiable.mono - (fun _ hz => ⟨hz.1,fun h => hz.2 (initialSubarc_range_subset p t h)⟩) - -lemma initialInverse_forward_on_final (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : - initialInverseChart p hi h0 hH t (initialForwardChart p hi h0 hH t z)=z := by - by_cases ht : t=0 - · simp [ht] - · rw [initialInverseChart,dite_eq_right ht,initialForwardChart,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).left_inverse _ - ⟨hz.1,fun h => hz.2 (initialSubarc_range_subset p t h)⟩ - -lemma initialForward_inverse (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane) : - initialForwardChart p hi h0 hH t (initialInverseChart p hi h0 hH t z)=z := by - by_cases ht : t=0 - · simp [ht] - · rw [initialForwardChart,dite_eq_right ht,initialInverseChart,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).right_inverse _ hz - -lemma halfPlaneCayley_differentiableOn : DifferentiableOn ℂ halfPlaneCayley upperHalfPlane := by - apply (differentiableOn_id.sub_const I).div (differentiableOn_id.add_const I) - exact fun z hz => halfPlaneCayley_den_ne (show 0 hz.2 (initialSubarc_range_subset p t h)⟩ + +lemma initialForwardChart_differentiableOn (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) : + DifferentiableOn ℂ (initialForwardChart p hi h0 hH t) (upperHalfPlane \ range p) := by + by_cases ht : t=0 + · simp only [ht,initialForwardChart_zero] + exact differentiableOn_id + · rw [initialForwardChart,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).forward_differentiable.mono + (fun _ hz => ⟨hz.1,fun h => hz.2 (initialSubarc_range_subset p t h)⟩) + +lemma initialInverse_forward_on_final (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : + initialInverseChart p hi h0 hH t (initialForwardChart p hi h0 hH t z)=z := by + by_cases ht : t=0 + · simp [ht] + · rw [initialInverseChart,dite_eq_right ht,initialForwardChart,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).left_inverse _ + ⟨hz.1,fun h => hz.2 (initialSubarc_range_subset p t h)⟩ + +lemma initialForward_inverse (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane) : + initialForwardChart p hi h0 hH t (initialInverseChart p hi h0 hH t z)=z := by + by_cases ht : t=0 + · simp [ht] + · rw [initialForwardChart,dite_eq_right ht,initialInverseChart,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).right_inverse _ hz + +lemma halfPlaneCayley_differentiableOn : DifferentiableOn ℂ halfPlaneCayley upperHalfPlane := by + apply (differentiableOn_id.sub_const I).div (differentiableOn_id.add_const I) + exact fun z hz => halfPlaneCayley_den_ne (show 0 initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (hs : s≠0) (ht : Tendsto t atTop (𝓝 s)) : - ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ z∈upperHalfPlane \ range p, - Tendsto (fun n => initialForwardChart p hi h0 hH (t (φ n)) z) - atTop (𝓝 ((initialSlitChart p hi h0 hH s hs).forward z)) := by - let P := initialSlitChart p hi h0 hH s hs - have heF := initialInverse_limit_eq_at p hi h0 hH hF hlim hs ht - have hPinv : EqOn F P.inverse upperHalfPlane := by - simpa only [initialInverseChart,dite_eq_right hs] using heF - obtain ⟨φ,hφ,hforward⟩ := initialInverse_limit_forward_subsequence p hi h0 hH hF hlim - refine ⟨φ,hφ,fun z hz => ?_⟩ - have hzP : z∈upperHalfPlane \ range (initialSubarc p s) := - ⟨hz.1,fun hh => hz.2 (initialSubarc_range_subset p s hh)⟩ - have hcov : SurjOn F upperHalfPlane (upperHalfPlane \ range p) := - initialInverse_limit_covers_final_domain p hi h0 hH hF hlim - have he : invFunOn F upperHalfPlane z=P.forward z := by - apply initialInverse_limit_injOn p hi h0 hH hlim - (hcov.mapsTo_invFunOn hz) (P.bijOn.mapsTo hzP) - rw [hcov.rightInvOn_invFunOn hz,hPinv (P.bijOn.mapsTo hzP),P.left_inverse z hzP] - rw [← he] - exact hforward z hz - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseLimitAt + +namespace OAI + +/-! The complementary hydrodynamic mass, measured from the common final +domain to each initial domain. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma initialForward_comparison_to_final (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (Q : SimpleSlitChart p) (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane) : + ‖initialForwardChart p hi h0 hH t (Q.inverse z)-z‖≤ + (Q.capacity-initialCapacity p hi h0 hH t)/z.im := by + by_cases ht : t=0 + · simpa only [ht,initialForwardChart_zero,initialCapacity_zero,id_eq,sub_zero] using + Q.inverse_displacement hz + · rw [initialForwardChart,dite_eq_right ht,initialCapacity,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).inverse_comparison Q + (initialSubarc_range_subset p t) hz + +lemma initialInverse_limit_forward_on_final (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (hs : s≠0) (ht : Tendsto t atTop (𝓝 s)) : + ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ z∈upperHalfPlane \ range p, + Tendsto (fun n => initialForwardChart p hi h0 hH (t (φ n)) z) + atTop (𝓝 ((initialSlitChart p hi h0 hH s hs).forward z)) := by + let P := initialSlitChart p hi h0 hH s hs + have heF := initialInverse_limit_eq_at p hi h0 hH hF hlim hs ht + have hPinv : EqOn F P.inverse upperHalfPlane := by + simpa only [initialInverseChart,dite_eq_right hs] using heF + obtain ⟨φ,hφ,hforward⟩ := initialInverse_limit_forward_subsequence p hi h0 hH hF hlim + refine ⟨φ,hφ,fun z hz => ?_⟩ + have hzP : z∈upperHalfPlane \ range (initialSubarc p s) := + ⟨hz.1,fun hh => hz.2 (initialSubarc_range_subset p s hh)⟩ + have hcov : SurjOn F upperHalfPlane (upperHalfPlane \ range p) := + initialInverse_limit_covers_final_domain p hi h0 hH hF hlim + have he : invFunOn F upperHalfPlane z=P.forward z := by + apply initialInverse_limit_injOn p hi h0 hH hlim + (hcov.mapsTo_invFunOn hz) (P.bijOn.mapsTo hzP) + rw [hcov.rightInvOn_invFunOn hz,hPinv (P.bijOn.mapsTo hzP),P.left_inverse z hzP] + rw [← he] + exact hforward z hz + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardIncrement.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardIncrement.lean index 189f296fa436ca42cdb60b439360de0172d1fc38..839f16d1cc37caa5cf951b986d7c3803837df10e 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardIncrement.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardIncrement.lean @@ -1,59 +1,59 @@ -import OAI.Probability.SLE.Loewner.FutureCapacityAdditive - -namespace OAI - -/-! Exact capacity increments control forward-map motion on the common -surviving domain. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem initialForward_increment_bound (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s (a.1,p (a.1*a.2))) := - continuous_fst.prodMk (p.continuous.comp (continuous_fst.mul continuous_snd)) - have hr : {a : unitInterval × ℂ | a.2∈range (initialSubarc p a.1)}= - range (fun a : unitInterval × unitInterval => (a.1,p (a.1*a.2))) := by - ext a - constructor - · rintro ⟨u,hu⟩ - exact ⟨(a.1,u),Prod.ext rfl hu⟩ - · rintro ⟨⟨s,u⟩,he⟩ - obtain ⟨hs,hz⟩ := Prod.mk.inj he - dsimp only at hs hz - subst s - exact ⟨u,hz⟩ - have hclosed : IsClosed {a : unitInterval × ℂ | a.2∈range (initialSubarc p a.1)} := by - rw [hr] - exact (isCompact_range hc).isClosed - exact (isOpen_upperHalfPlane.preimage continuous_snd).sdiff hclosed - -theorem initialForwardChart_joint_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - Continuous (fun a : {a : unitInterval × ℂ | - a.2∈upperHalfPlane \ range (initialSubarc p a.1)} => - initialForwardChart p hi h0 hH a.1.1 a.1.2) := by - apply SeqContinuous.continuous - intro v a hv - have ht := ((continuous_fst.comp continuous_subtype_val).tendsto a).comp hv - have hz := ((continuous_snd.comp continuous_subtype_val).tendsto a).comp hv - exact initialForwardChart_tendsto p hi h0 hH ht hz a.property (fun n => (v n).property) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialForwardTimeContinuity + +namespace OAI + +/-! The moving slit domain is open, and its actual forward coordinates +depend continuously on both time and the surviving point. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma isOpen_initialSlitDomain (p : C(unitInterval,ℂ)) : + IsOpen {a : unitInterval × ℂ | a.2∈upperHalfPlane \ range (initialSubarc p a.1)} := by + have hc : Continuous (fun a : unitInterval × unitInterval => (a.1,p (a.1*a.2))) := + continuous_fst.prodMk (p.continuous.comp (continuous_fst.mul continuous_snd)) + have hr : {a : unitInterval × ℂ | a.2∈range (initialSubarc p a.1)}= + range (fun a : unitInterval × unitInterval => (a.1,p (a.1*a.2))) := by + ext a + constructor + · rintro ⟨u,hu⟩ + exact ⟨(a.1,u),Prod.ext rfl hu⟩ + · rintro ⟨⟨s,u⟩,he⟩ + obtain ⟨hs,hz⟩ := Prod.mk.inj he + dsimp only at hs hz + subst s + exact ⟨u,hz⟩ + have hclosed : IsClosed {a : unitInterval × ℂ | a.2∈range (initialSubarc p a.1)} := by + rw [hr] + exact (isCompact_range hc).isClosed + exact (isOpen_upperHalfPlane.preimage continuous_snd).sdiff hclosed + +theorem initialForwardChart_joint_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + Continuous (fun a : {a : unitInterval × ℂ | + a.2∈upperHalfPlane \ range (initialSubarc p a.1)} => + initialForwardChart p hi h0 hH a.1.1 a.1.2) := by + apply SeqContinuous.continuous + intro v a hv + have ht := ((continuous_fst.comp continuous_subtype_val).tendsto a).comp hv + have hz := ((continuous_snd.comp continuous_subtype_val).tendsto a).comp hv + exact initialForwardChart_tendsto p hi h0 hH ht hz a.property (fun n => (v n).property) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardKernel.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardKernel.lean index 894f2152069d4419b98f495048163e9c79f01d89..171c839e79a9959eba22f2666e64cbf95c21ad7c 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardKernel.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardKernel.lean @@ -1,50 +1,50 @@ -import OAI.Probability.SLE.Loewner.SlitCapacityQuotient -import OAI.Probability.SLE.Loewner.InitialForwardIncrement -import OAI.Probability.SLE.Loewner.FutureSlitRadius - -namespace OAI - -/-! The exact chart composition turns the static small-slit estimate -into an estimate for the evolving map's capacity difference quotient. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -theorem initialForward_quotient_kernel (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s centeredFutureArc_upper p hi h0 hH hst ht hu) hR hp hw.1 hn - have he : (U:ℂ)+Q.forward w=initialForwardChart p hi h0 hH t z := - futureComposedForward_eq p hi h0 hH hst ht hz - have hd : Q.forward w-w= - initialForwardChart p hi h0 hH t z-initialForwardChart p hi h0 hH s z := by - dsimp only [w] at he ⊢ - linear_combination he - rwa [hd,show Q.capacity=_ from futureSlitChart_capacity_eq p hi h0 hH hst ht] at hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitCapacityQuotient +import OAI.Probability.SLE.Loewner.InitialForwardIncrement +import OAI.Probability.SLE.Loewner.FutureSlitRadius + +namespace OAI + +/-! The exact chart composition turns the static small-slit estimate +into an estimate for the evolving map's capacity difference quotient. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +theorem initialForward_quotient_kernel (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s centeredFutureArc_upper p hi h0 hH hst ht hu) hR hp hw.1 hn + have he : (U:ℂ)+Q.forward w=initialForwardChart p hi h0 hH t z := + futureComposedForward_eq p hi h0 hH hst ht hz + have hd : Q.forward w-w= + initialForwardChart p hi h0 hH t z-initialForwardChart p hi h0 hH s z := by + dsimp only [w] at he ⊢ + linear_combination he + rwa [hd,show Q.capacity=_ from futureSlitChart_capacity_eq p hi h0 hH hst ht] at hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardKernelLimit.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardKernelLimit.lean index d81e5ab0e1f5e5a91d8785757bbe311cd6580197..354747277897cbcd70796dc6e4312bbfb925b6e4 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardKernelLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardKernelLimit.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.Loewner.InitialForwardKernel -import OAI.Probability.SLE.Loewner.CapacityForwardDerivative - -namespace OAI - -/-! The exact capacity difference quotient tends to the Loewner -kernel as the future slit shrinks to its driving point. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -theorem initialForward_quotient_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s T : unitInterval} (hsT : s - (initialForwardChart p hi h0 hH t z-initialForwardChart p hi h0 hH s z)/ - ((initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s:ℝ):ℂ)) - (𝓝[Ioc s T] s) (𝓝 (1/(initialForwardChart p hi h0 hH s z- - (initialDrivingValue p hi h0 hH s (hsT.trans hT):ℂ)))) := by - let w := initialForwardChart p hi h0 hH s z- - (initialDrivingValue p hi h0 hH s (hsT.trans hT):ℂ) - have hzS : z∈upperHalfPlane \ range (initialSubarc p s) := - ⟨hz.1,fun hh => hz.2 (initialSubarc_range_mono p hsT.le hh)⟩ - have hwim : 0 (t:ℝ)-(s:ℝ)) (𝓝[Ioc s T] s) (𝓝 0) := by - have hcont : Continuous (fun t : unitInterval => (t:ℝ)-(s:ℝ)) := - continuous_subtype_val.sub continuous_const - have hh := (hcont.tendsto s).mono_left - (show 𝓝[Ioc s T] s≤𝓝 s from inf_le_left) - simpa only [sub_self] using hh - filter_upwards [self_mem_nhdsWithin,htime.eventually (gt_mem_nhds hδ)] with t ht htd - have htt : t<1 := ht.2.trans_lt hT - have hzt : z∈upperHalfPlane \ range (initialSubarc p t) := - ⟨hz.1,fun hh => hz.2 (initialSubarc_range_mono p ht.2 hh)⟩ - have hb := initialForward_quotient_kernel p hi h0 hH ht.1 htt hzt hR - (hsmall t ht.1 htt htd) hRn - rw [dist_eq_norm] - exact hb.trans_lt hRe - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialForwardKernel +import OAI.Probability.SLE.Loewner.CapacityForwardDerivative + +namespace OAI + +/-! The exact capacity difference quotient tends to the Loewner +kernel as the future slit shrinks to its driving point. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +theorem initialForward_quotient_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s T : unitInterval} (hsT : s + (initialForwardChart p hi h0 hH t z-initialForwardChart p hi h0 hH s z)/ + ((initialCapacity p hi h0 hH t-initialCapacity p hi h0 hH s:ℝ):ℂ)) + (𝓝[Ioc s T] s) (𝓝 (1/(initialForwardChart p hi h0 hH s z- + (initialDrivingValue p hi h0 hH s (hsT.trans hT):ℂ)))) := by + let w := initialForwardChart p hi h0 hH s z- + (initialDrivingValue p hi h0 hH s (hsT.trans hT):ℂ) + have hzS : z∈upperHalfPlane \ range (initialSubarc p s) := + ⟨hz.1,fun hh => hz.2 (initialSubarc_range_mono p hsT.le hh)⟩ + have hwim : 0 (t:ℝ)-(s:ℝ)) (𝓝[Ioc s T] s) (𝓝 0) := by + have hcont : Continuous (fun t : unitInterval => (t:ℝ)-(s:ℝ)) := + continuous_subtype_val.sub continuous_const + have hh := (hcont.tendsto s).mono_left + (show 𝓝[Ioc s T] s≤𝓝 s from inf_le_left) + simpa only [sub_self] using hh + filter_upwards [self_mem_nhdsWithin,htime.eventually (gt_mem_nhds hδ)] with t ht htd + have htt : t<1 := ht.2.trans_lt hT + have hzt : z∈upperHalfPlane \ range (initialSubarc p t) := + ⟨hz.1,fun hh => hz.2 (initialSubarc_range_mono p ht.2 hh)⟩ + have hb := initialForward_quotient_kernel p hi h0 hH ht.1 htt hzt hR + (hsmall t ht.1 htt htd) hRn + rw [dist_eq_norm] + exact hb.trans_lt hRe + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardLimit.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardLimit.lean index 2c128c93d6e710f4cd8b72ae050b035086ca432e..79e1acd7e3e41a9854b8ebb1fcdf0dea4fcac896 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardLimit.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Loewner.InitialForwardNormalFamily -import OAI.Probability.SLE.Loewner.InitialForwardBaseBounds -import OAI.Probability.SLE.Geometry.HolomorphicDiskInterior - -namespace OAI - -/-! Limits of the compactified forward charts stay in the open disk. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma initialForwardCayley_limit_interior (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {A : ℂ → ℂ} - (hA : DifferentiableOn ℂ A (upperHalfPlane \ range p)) - (hlim : TendstoLocallyUniformlyOn - (fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t n) z)) - A atTop (upperHalfPlane \ range p)) : - ∀ z∈upperHalfPlane \ range p, ‖A z‖<1 := by - obtain ⟨R0,hR0⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le - let R := max R0 1 - have hR : 0 by - rw [mem_closedBall,dist_zero_right] - exact (hbound n).2) - have hvim : 3*R/2≤v.im := ge_of_tendsto (continuous_im.continuousAt.tendsto.comp hv) - (Eventually.of_forall fun n => (hbound (φ n)).1) - have hvH : v∈upperHalfPlane := lt_of_lt_of_le (by positivity : 0<3*R/2) hvim - have he : A b=halfPlaneCayley v := by - apply tendsto_nhds_unique ((hlim.tendsto_at hb).comp hφ.tendsto_atTop) - exact (halfPlaneCayley_differentiableOn.continuousOn v hvH).continuousAt - (isOpen_upperHalfPlane.mem_nhds hvH) |>.tendsto.comp hv - have hAb : ‖A b‖<1 := by rw [he]; exact halfPlaneCayley_norm_lt hvH - have hnorm : ∀ z∈upperHalfPlane \ range p, ‖A z‖≤1 := by - intro z hz - apply le_of_tendsto (hlim.tendsto_at hz).norm - exact Eventually.of_forall fun n => (halfPlaneCayley_norm_lt - (initialForwardChart_mapsTo p hi h0 hH (t n) hz)).le - exact holomorphic_disk_strict_of_point - (isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed) - (simple_arc_halfPlane_complement_connected p hi h0 hH).isPreconnected hA hnorm hb hAb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialForwardNormalFamily +import OAI.Probability.SLE.Loewner.InitialForwardBaseBounds +import OAI.Probability.SLE.Geometry.HolomorphicDiskInterior + +namespace OAI + +/-! Limits of the compactified forward charts stay in the open disk. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +lemma initialForwardCayley_limit_interior (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {A : ℂ → ℂ} + (hA : DifferentiableOn ℂ A (upperHalfPlane \ range p)) + (hlim : TendstoLocallyUniformlyOn + (fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t n) z)) + A atTop (upperHalfPlane \ range p)) : + ∀ z∈upperHalfPlane \ range p, ‖A z‖<1 := by + obtain ⟨R0,hR0⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le + let R := max R0 1 + have hR : 0 by + rw [mem_closedBall,dist_zero_right] + exact (hbound n).2) + have hvim : 3*R/2≤v.im := ge_of_tendsto (continuous_im.continuousAt.tendsto.comp hv) + (Eventually.of_forall fun n => (hbound (φ n)).1) + have hvH : v∈upperHalfPlane := lt_of_lt_of_le (by positivity : 0<3*R/2) hvim + have he : A b=halfPlaneCayley v := by + apply tendsto_nhds_unique ((hlim.tendsto_at hb).comp hφ.tendsto_atTop) + exact (halfPlaneCayley_differentiableOn.continuousOn v hvH).continuousAt + (isOpen_upperHalfPlane.mem_nhds hvH) |>.tendsto.comp hv + have hAb : ‖A b‖<1 := by rw [he]; exact halfPlaneCayley_norm_lt hvH + have hnorm : ∀ z∈upperHalfPlane \ range p, ‖A z‖≤1 := by + intro z hz + apply le_of_tendsto (hlim.tendsto_at hz).norm + exact Eventually.of_forall fun n => (halfPlaneCayley_norm_lt + (initialForwardChart_mapsTo p hi h0 hH (t n) hz)).le + exact holomorphic_disk_strict_of_point + (isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed) + (simple_arc_halfPlane_complement_connected p hi h0 hH).isPreconnected hA hnorm hb hAb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardNormalFamily.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardNormalFamily.lean index e57dda6a8605ed7c9fa4047573e3b61b63ed2723..fa98aff38ba3c6ae17e6be9de4c95a3d8e10d2e6 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardNormalFamily.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardNormalFamily.lean @@ -1,37 +1,37 @@ -import OAI.Probability.SLE.Loewner.InitialForwardChart -import OAI.Probability.SLE.Loewner.InitialInverseNormalFamily - -namespace OAI - -/-! The compactified forward charts form a bounded normal family on the final domain. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -theorem initialForwardCayley_subsequence (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : ℕ → unitInterval) : - ∃ (φ : ℕ → ℕ) (F : ℂ → ℂ), StrictMono φ ∧ - DifferentiableOn ℂ F (upperHalfPlane \ range p) ∧ - TendstoLocallyUniformlyOn - (fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t (φ n)) z)) - F atTop (upperHalfPlane \ range p) := by - have hD : IsOpen (upperHalfPlane \ range p) := - isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed - apply TauCeti.montel (F := fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t n) z)) hD - · intro n - exact halfPlaneCayley_differentiableOn.comp - (initialForwardChart_differentiableOn p hi h0 hH (t n)) - (initialForwardChart_mapsTo p hi h0 hH (t n)) - · apply TauCeti.isLocallyBoundedOn_of_forall_norm_le (C := 1) - intro n z hz - exact (halfPlaneCayley_norm_lt - (initialForwardChart_mapsTo p hi h0 hH (t n) hz)).le - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialForwardChart +import OAI.Probability.SLE.Loewner.InitialInverseNormalFamily + +namespace OAI + +/-! The compactified forward charts form a bounded normal family on the final domain. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +theorem initialForwardCayley_subsequence (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : ℕ → unitInterval) : + ∃ (φ : ℕ → ℕ) (F : ℂ → ℂ), StrictMono φ ∧ + DifferentiableOn ℂ F (upperHalfPlane \ range p) ∧ + TendstoLocallyUniformlyOn + (fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t (φ n)) z)) + F atTop (upperHalfPlane \ range p) := by + have hD : IsOpen (upperHalfPlane \ range p) := + isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed + apply TauCeti.montel (F := fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t n) z)) hD + · intro n + exact halfPlaneCayley_differentiableOn.comp + (initialForwardChart_differentiableOn p hi h0 hH (t n)) + (initialForwardChart_mapsTo p hi h0 hH (t n)) + · apply TauCeti.isLocallyBoundedOn_of_forall_norm_le (C := 1) + intro n z hz + exact (halfPlaneCayley_norm_lt + (initialForwardChart_mapsTo p hi h0 hH (t n) hz)).le + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardRecovery.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardRecovery.lean index 5969a14734ca00de2ab51fd0c4ee4c716d20db43..4d1c79a8d85cd68b67e63cf15a2d0da0e154645a 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardRecovery.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardRecovery.lean @@ -1,41 +1,41 @@ -import OAI.Probability.SLE.Loewner.InitialForwardLimit - -namespace OAI - -/-! Recover actual forward coordinates from their compactified limits. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialForward_limit_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {A : ℂ → ℂ} - (hA : DifferentiableOn ℂ A (upperHalfPlane \ range p)) - (hlim : TendstoLocallyUniformlyOn - (fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t n) z)) - A atTop (upperHalfPlane \ range p)) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : - Tendsto (fun n => initialForwardChart p hi h0 hH (t n) z) - atTop (𝓝 (halfPlaneCayleyInv (A z))) := by - have hn := initialForwardCayley_limit_interior p hi h0 hH hA hlim z hz - have hne : A z≠1 := by - intro he - simp only [he,norm_one,lt_self_iff_false] at hn - have hc : ContinuousAt halfPlaneCayleyInv (A z) := by - apply (continuousAt_const.mul (continuousAt_const.add continuousAt_id)).div - (continuousAt_const.sub continuousAt_id) - exact sub_ne_zero.mpr hne.symm - have ht := hc.tendsto.comp (hlim.tendsto_at hz) - apply ht.congr - intro n - exact halfPlaneCayleyInv_cayley - (show 0≤(initialForwardChart p hi h0 hH (t n) z).im from - (show 0<(initialForwardChart p hi h0 hH (t n) z).im from - initialForwardChart_mapsTo p hi h0 hH (t n) hz).le) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialForwardLimit + +namespace OAI + +/-! Recover actual forward coordinates from their compactified limits. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma initialForward_limit_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {A : ℂ → ℂ} + (hA : DifferentiableOn ℂ A (upperHalfPlane \ range p)) + (hlim : TendstoLocallyUniformlyOn + (fun n z => halfPlaneCayley (initialForwardChart p hi h0 hH (t n) z)) + A atTop (upperHalfPlane \ range p)) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : + Tendsto (fun n => initialForwardChart p hi h0 hH (t n) z) + atTop (𝓝 (halfPlaneCayleyInv (A z))) := by + have hn := initialForwardCayley_limit_interior p hi h0 hH hA hlim z hz + have hne : A z≠1 := by + intro he + simp only [he,norm_one,lt_self_iff_false] at hn + have hc : ContinuousAt halfPlaneCayleyInv (A z) := by + apply (continuousAt_const.mul (continuousAt_const.add continuousAt_id)).div + (continuousAt_const.sub continuousAt_id) + exact sub_ne_zero.mpr hne.symm + have ht := hc.tendsto.comp (hlim.tendsto_at hz) + apply ht.congr + intro n + exact halfPlaneCayleyInv_cayley + (show 0≤(initialForwardChart p hi h0 hH (t n) z).im from + (show 0<(initialForwardChart p hi h0 hH (t n) z).im from + initialForwardChart_mapsTo p hi h0 hH (t n) hz).le) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialForwardTimeContinuity.lean b/lean/OAI/Probability/SLE/Loewner/InitialForwardTimeContinuity.lean index d48206791186c894ce2b43554079973ee11f140d..a07141b09999cf17392b8cd30b078043e3666ec7 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialForwardTimeContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialForwardTimeContinuity.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Loewner.InitialInverseTimeContinuity -import OAI.Probability.SLE.Loewner.InitialChartIdentities - -namespace OAI - -/-! Continuity of the forward coordinates along any sequence of surviving -points, using the actual inverse charts and Hurwitz's theorem. -/ -noncomputable section -open Set Filter Function Metric -open scoped Topology - -namespace SLEExactGauge - -theorem initialForwardChart_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) - {z : ℕ → ℂ} {a : ℂ} (hz : Tendsto z atTop (𝓝 a)) - (ha : a∈upperHalfPlane \ range (initialSubarc p s)) - (hzD : ∀ n, z n∈upperHalfPlane \ range (initialSubarc p (t n))) : - Tendsto (fun n => initialForwardChart p hi h0 hH (t n) (z n)) - atTop (𝓝 (initialForwardChart p hi h0 hH s a)) := by - have hbase := initialInverseChart_locallyUniform_continuous p hi h0 hH s - have hconv : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - (initialInverseChart p hi h0 hH s) atTop upperHalfPlane := by - intro U hU w hw - obtain ⟨V,hV,hN⟩ := hbase U hU w hw - exact ⟨V,hV,ht.eventually hN⟩ - have hconst := hz.tendstoUniformlyOn_const upperHalfPlane |>.tendstoLocallyUniformlyOn - have hdiff := hconv.sub hconst - have hnc : ¬ ∀ w∈upperHalfPlane, initialInverseChart p hi h0 hH s w-a=0 := by - intro h - exact initialInverse_limit_not_constant p hi h0 hH hconv a - (fun w hw => sub_eq_zero.mp (h w hw)) - apply Metric.tendsto_nhds.mpr - intro ε hε - have hevent := TauCeti.hurwitz_eventually_exists_eq_zero isOpen_upperHalfPlane - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected - (Eventually.of_forall fun n => (initialInverseChart_differentiableOn p hi h0 hH (t n)).sub_const (z n)) - hdiff (initialForwardChart_mem_upper p hi h0 hH s ha) - (sub_eq_zero.mpr (initialInverse_forward_at p hi h0 hH s ha)) hnc hε - filter_upwards [hevent] with n hn - obtain ⟨w,hw,hew⟩ := hn - have he : w=initialForwardChart p hi h0 hH (t n) (z n) := by - apply initialInverseChart_injOn p hi h0 hH (t n) hw.1 - (initialForwardChart_mem_upper p hi h0 hH (t n) (hzD n)) - exact (sub_eq_zero.mp hew).trans (initialInverse_forward_at p hi h0 hH (t n) (hzD n)).symm - simpa only [← he,mem_ball] using hw.2 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseTimeContinuity +import OAI.Probability.SLE.Loewner.InitialChartIdentities + +namespace OAI + +/-! Continuity of the forward coordinates along any sequence of surviving +points, using the actual inverse charts and Hurwitz's theorem. -/ +noncomputable section +open Set Filter Function Metric +open scoped Topology + +namespace SLEExactGauge + +theorem initialForwardChart_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) + {z : ℕ → ℂ} {a : ℂ} (hz : Tendsto z atTop (𝓝 a)) + (ha : a∈upperHalfPlane \ range (initialSubarc p s)) + (hzD : ∀ n, z n∈upperHalfPlane \ range (initialSubarc p (t n))) : + Tendsto (fun n => initialForwardChart p hi h0 hH (t n) (z n)) + atTop (𝓝 (initialForwardChart p hi h0 hH s a)) := by + have hbase := initialInverseChart_locallyUniform_continuous p hi h0 hH s + have hconv : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + (initialInverseChart p hi h0 hH s) atTop upperHalfPlane := by + intro U hU w hw + obtain ⟨V,hV,hN⟩ := hbase U hU w hw + exact ⟨V,hV,ht.eventually hN⟩ + have hconst := hz.tendstoUniformlyOn_const upperHalfPlane |>.tendstoLocallyUniformlyOn + have hdiff := hconv.sub hconst + have hnc : ¬ ∀ w∈upperHalfPlane, initialInverseChart p hi h0 hH s w-a=0 := by + intro h + exact initialInverse_limit_not_constant p hi h0 hH hconv a + (fun w hw => sub_eq_zero.mp (h w hw)) + apply Metric.tendsto_nhds.mpr + intro ε hε + have hevent := TauCeti.hurwitz_eventually_exists_eq_zero isOpen_upperHalfPlane + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected + (Eventually.of_forall fun n => (initialInverseChart_differentiableOn p hi h0 hH (t n)).sub_const (z n)) + hdiff (initialForwardChart_mem_upper p hi h0 hH s ha) + (sub_eq_zero.mpr (initialInverse_forward_at p hi h0 hH s ha)) hnc hε + filter_upwards [hevent] with n hn + obtain ⟨w,hw,hew⟩ := hn + have he : w=initialForwardChart p hi h0 hH (t n) (z n) := by + apply initialInverseChart_injOn p hi h0 hH (t n) hw.1 + (initialForwardChart_mem_upper p hi h0 hH (t n) (hzD n)) + exact (sub_eq_zero.mp hew).trans (initialInverse_forward_at p hi h0 hH (t n) (hzD n)).symm + simpa only [← he,mem_ball] using hw.2 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialHeightCapacityIncrement.lean b/lean/OAI/Probability/SLE/Loewner/InitialHeightCapacityIncrement.lean index 7353d7bd2fc7effbecb08a2d2f233eab1dfc8094..eb0bb4c28a4c22d766a5f7dfca09d8df5e048efd 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialHeightCapacityIncrement.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialHeightCapacityIncrement.lean @@ -1,41 +1,41 @@ -import OAI.Probability.SLE.Loewner.ForwardHeightSquareIncrement - -namespace OAI - -/-! Height loss on a surviving initial-slit prefix is bounded by twice -the exact capacity increment. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem initialForward_height_capacity_increment (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s r T : unitInterval} (hsr : s≤r) (hrT : r≤T) (hT : T<1) - {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p T)) : - (initialForwardChart p hi h0 hH s z).im^2≤ - (initialForwardChart p hi h0 hH r z).im^2+ - 2*(initialCapacity p hi h0 hH r-initialCapacity p hi h0 hH s) := by - have ha := initialCapacity_nonneg p hi h0 hH s - have hsr' := initialCapacity_monotone p hi h0 hH hsr - have hrT' := initialCapacity_monotone p hi h0 hH hrT - have hc : ContinuousOn (capacityForwardOnPrefix p hi h0 hH T z) - (Icc (initialCapacity p hi h0 hH s) (initialCapacity p hi h0 hH r)) := - (capacityForwardOnPrefix_lipschitz p hi h0 hH hT hz).continuous.continuousOn - have hd (c : ℝ) (hcc : c∈Ico (initialCapacity p hi h0 hH s) - (initialCapacity p hi h0 hH r)) : - HasDerivWithinAt (capacityForwardOnPrefix p hi h0 hH T z) - (1/(capacityForwardOnPrefix p hi h0 hH T z c- - (capacityDriverOnPrefix p hi h0 hH T hT c:ℂ))) (Ici c) c := - capacityForward_hasDerivWithinAt_Ici p hi h0 hH hT hz - ⟨ha.trans hcc.1,hcc.2.trans_le hrT'⟩ - have hh := height_square_increment hsr' hc hd - simpa only [capacityForwardOnPrefix_at_initialCapacity p hi h0 hH T z (hsr.trans hrT), - capacityForwardOnPrefix_at_initialCapacity p hi h0 hH T z hrT] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ForwardHeightSquareIncrement + +namespace OAI + +/-! Height loss on a surviving initial-slit prefix is bounded by twice +the exact capacity increment. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +theorem initialForward_height_capacity_increment (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s r T : unitInterval} (hsr : s≤r) (hrT : r≤T) (hT : T<1) + {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p T)) : + (initialForwardChart p hi h0 hH s z).im^2≤ + (initialForwardChart p hi h0 hH r z).im^2+ + 2*(initialCapacity p hi h0 hH r-initialCapacity p hi h0 hH s) := by + have ha := initialCapacity_nonneg p hi h0 hH s + have hsr' := initialCapacity_monotone p hi h0 hH hsr + have hrT' := initialCapacity_monotone p hi h0 hH hrT + have hc : ContinuousOn (capacityForwardOnPrefix p hi h0 hH T z) + (Icc (initialCapacity p hi h0 hH s) (initialCapacity p hi h0 hH r)) := + (capacityForwardOnPrefix_lipschitz p hi h0 hH hT hz).continuous.continuousOn + have hd (c : ℝ) (hcc : c∈Ico (initialCapacity p hi h0 hH s) + (initialCapacity p hi h0 hH r)) : + HasDerivWithinAt (capacityForwardOnPrefix p hi h0 hH T z) + (1/(capacityForwardOnPrefix p hi h0 hH T z c- + (capacityDriverOnPrefix p hi h0 hH T hT c:ℂ))) (Ici c) c := + capacityForward_hasDerivWithinAt_Ici p hi h0 hH hT hz + ⟨ha.trans hcc.1,hcc.2.trans_le hrT'⟩ + have hh := height_square_increment hsr' hc hd + simpa only [capacityForwardOnPrefix_at_initialCapacity p hi h0 hH T z (hsr.trans hrT), + capacityForwardOnPrefix_at_initialCapacity p hi h0 hH T z hrT] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseChart.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseChart.lean index 3d45701417dbeaac546c9e833922262973399f3f..fe3f79cb0920c729727046f6dd9be7be92ec5884 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseChart.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseChart.lean @@ -1,71 +1,71 @@ -import OAI.Probability.SLE.Loewner.SlitChartDisplacement -import OAI.Probability.SLE.Loewner.InitialCapacityAtZero - -namespace OAI - -/-! The actual normalized inverse charts of initial subarcs start at the identity. -/ -noncomputable section -open Set Filter Function Metric -open scoped Topology - -namespace SLEExactGauge - -noncomputable def initialInverseChart (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) : ℂ → ℂ := - if ht : t=0 then id else (initialSlitChart p hi h0 hH t ht).inverse - -@[simp] lemma initialInverseChart_zero (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - initialInverseChart p hi h0 hH 0=id := by simp [initialInverseChart] - -lemma initialInverseChart_differentiableOn (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) : DifferentiableOn ℂ (initialInverseChart p hi h0 hH t) upperHalfPlane := by - by_cases ht : t=0 - · simp only [ht,initialInverseChart_zero] - exact differentiableOn_id - · rw [initialInverseChart,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).inverse_differentiable - -lemma initialInverseChart_displacement (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane) : - ‖initialInverseChart p hi h0 hH t z-z‖≤ initialCapacity p hi h0 hH t/z.im := by - by_cases ht : t=0 - · simp [ht] - · rw [initialInverseChart,dite_eq_right ht,initialCapacity,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).inverse_displacement hz - -lemma compact_upperHalfPlane_min_height {K : Set ℂ} (hK : IsCompact K) - (hKH : K ⊆ upperHalfPlane) (hne : K.Nonempty) : - ∃ m : ℝ, 0< m ∧ ∀ z∈K, m≤ z.im := by - obtain ⟨z,hz,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn - exact ⟨z.im,hKH hz,fun w hw => hmin hw⟩ - -theorem initialInverseChart_tendsto_identity (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - TendstoLocallyUniformlyOn (initialInverseChart p hi h0 hH) id (𝓝 0) upperHalfPlane := by - rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane] - intro K hKH hK - by_cases hne : K.Nonempty - · obtain ⟨m,hm,hmin⟩ := compact_upperHalfPlane_min_height hK hKH hne - have hcap : Tendsto (initialCapacity p hi h0 hH) (𝓝 0) (𝓝 0) := by - simpa only [initialCapacity_zero] using (initialCapacity_continuousAt_zero p hi h0 hH).tendsto - have hsmall : Tendsto (fun t => initialCapacity p hi h0 hH t/m) (𝓝 0) (𝓝 0) := by - simpa only [zero_div] using hcap.div_const m - rw [Metric.tendstoUniformlyOn_iff] - intro ε hε - filter_upwards [hsmall.eventually (gt_mem_nhds hε)] with t ht - intro z hz - rw [dist_eq_norm,show id z=z from rfl,norm_sub_rev] - exact ((initialInverseChart_displacement p hi h0 hH t (hKH hz)).trans - (div_le_div_of_nonneg_left (initialCapacity_nonneg p hi h0 hH t) hm (hmin z hz))).trans_lt ht - · simpa only [not_nonempty_iff_eq_empty.mp hne] using - (tendstoUniformlyOn_empty : TendstoUniformlyOn (initialInverseChart p hi h0 hH) id (𝓝 0) ∅) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartDisplacement +import OAI.Probability.SLE.Loewner.InitialCapacityAtZero + +namespace OAI + +/-! The actual normalized inverse charts of initial subarcs start at the identity. -/ +noncomputable section +open Set Filter Function Metric +open scoped Topology + +namespace SLEExactGauge + +noncomputable def initialInverseChart (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) : ℂ → ℂ := + if ht : t=0 then id else (initialSlitChart p hi h0 hH t ht).inverse + +@[simp] lemma initialInverseChart_zero (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + initialInverseChart p hi h0 hH 0=id := by simp [initialInverseChart] + +lemma initialInverseChart_differentiableOn (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) : DifferentiableOn ℂ (initialInverseChart p hi h0 hH t) upperHalfPlane := by + by_cases ht : t=0 + · simp only [ht,initialInverseChart_zero] + exact differentiableOn_id + · rw [initialInverseChart,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).inverse_differentiable + +lemma initialInverseChart_displacement (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane) : + ‖initialInverseChart p hi h0 hH t z-z‖≤ initialCapacity p hi h0 hH t/z.im := by + by_cases ht : t=0 + · simp [ht] + · rw [initialInverseChart,dite_eq_right ht,initialCapacity,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).inverse_displacement hz + +lemma compact_upperHalfPlane_min_height {K : Set ℂ} (hK : IsCompact K) + (hKH : K ⊆ upperHalfPlane) (hne : K.Nonempty) : + ∃ m : ℝ, 0< m ∧ ∀ z∈K, m≤ z.im := by + obtain ⟨z,hz,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn + exact ⟨z.im,hKH hz,fun w hw => hmin hw⟩ + +theorem initialInverseChart_tendsto_identity (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + TendstoLocallyUniformlyOn (initialInverseChart p hi h0 hH) id (𝓝 0) upperHalfPlane := by + rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane] + intro K hKH hK + by_cases hne : K.Nonempty + · obtain ⟨m,hm,hmin⟩ := compact_upperHalfPlane_min_height hK hKH hne + have hcap : Tendsto (initialCapacity p hi h0 hH) (𝓝 0) (𝓝 0) := by + simpa only [initialCapacity_zero] using (initialCapacity_continuousAt_zero p hi h0 hH).tendsto + have hsmall : Tendsto (fun t => initialCapacity p hi h0 hH t/m) (𝓝 0) (𝓝 0) := by + simpa only [zero_div] using hcap.div_const m + rw [Metric.tendstoUniformlyOn_iff] + intro ε hε + filter_upwards [hsmall.eventually (gt_mem_nhds hε)] with t ht + intro z hz + rw [dist_eq_norm,show id z=z from rfl,norm_sub_rev] + exact ((initialInverseChart_displacement p hi h0 hH t (hKH hz)).trans + (div_le_div_of_nonneg_left (initialCapacity_nonneg p hi h0 hH t) hm (hmin z hz))).trans_lt ht + · simpa only [not_nonempty_iff_eq_empty.mp hne] using + (tendstoUniformlyOn_empty : TendstoUniformlyOn (initialInverseChart p hi h0 hH) id (𝓝 0) ∅) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseCommonDomain.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseCommonDomain.lean index 494167874ae6288514d89c2a7ce2ff03c202c9a9..8797a2cd7ff690c4938de23c9d08efd37760b9d1 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseCommonDomain.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseCommonDomain.lean @@ -1,45 +1,45 @@ -import OAI.Probability.SLE.Loewner.InitialForwardRecovery -import OAI.Probability.SLE.Loewner.InitialInverseLimit -import OAI.Probability.SLE.Geometry.LocallyUniformEvaluation - -namespace OAI - -/-! Every inverse-chart limit covers the common final slit domain. -No convergence of the times, nor a boundary driving value, is assumed here. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem initialInverse_limit_covers_final_domain (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) : - ∀ z∈upperHalfPlane \ range p, ∃ w∈upperHalfPlane, F w=z := by - obtain ⟨φ,A,hφ,hA,hAlim⟩ := initialForwardCayley_subsequence p hi h0 hH t - have hFsub := locallyUniform_reindex hlim hφ.tendsto_atTop - intro z hz - have hAz := initialForwardCayley_limit_interior p hi h0 hH hA hAlim z hz - have hw : halfPlaneCayleyInv (A z)∈upperHalfPlane := halfPlaneCayleyInv_open hAz - refine ⟨halfPlaneCayleyInv (A z),hw,?_⟩ - have hforward := initialForward_limit_tendsto p hi h0 hH hA hAlim hz - have heval := locallyUniform_evaluation_tendsto isOpen_upperHalfPlane - hF.continuousOn hFsub hw hforward - have hconstant : Tendsto (fun n => initialInverseChart p hi h0 hH (t (φ n)) - (initialForwardChart p hi h0 hH (t (φ n)) z)) atTop (𝓝 z) := by - have he : (fun n => initialInverseChart p hi h0 hH (t (φ n)) - (initialForwardChart p hi h0 hH (t (φ n)) z)) = (fun _ : ℕ => z) := by - funext n - exact initialInverse_forward_on_final p hi h0 hH (t (φ n)) hz - rw [he] - exact tendsto_const_nhds - exact tendsto_nhds_unique heval hconstant - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialForwardRecovery +import OAI.Probability.SLE.Loewner.InitialInverseLimit +import OAI.Probability.SLE.Geometry.LocallyUniformEvaluation + +namespace OAI + +/-! Every inverse-chart limit covers the common final slit domain. +No convergence of the times, nor a boundary driving value, is assumed here. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +theorem initialInverse_limit_covers_final_domain (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) : + ∀ z∈upperHalfPlane \ range p, ∃ w∈upperHalfPlane, F w=z := by + obtain ⟨φ,A,hφ,hA,hAlim⟩ := initialForwardCayley_subsequence p hi h0 hH t + have hFsub := locallyUniform_reindex hlim hφ.tendsto_atTop + intro z hz + have hAz := initialForwardCayley_limit_interior p hi h0 hH hA hAlim z hz + have hw : halfPlaneCayleyInv (A z)∈upperHalfPlane := halfPlaneCayleyInv_open hAz + refine ⟨halfPlaneCayleyInv (A z),hw,?_⟩ + have hforward := initialForward_limit_tendsto p hi h0 hH hA hAlim hz + have heval := locallyUniform_evaluation_tendsto isOpen_upperHalfPlane + hF.continuousOn hFsub hw hforward + have hconstant : Tendsto (fun n => initialInverseChart p hi h0 hH (t (φ n)) + (initialForwardChart p hi h0 hH (t (φ n)) z)) atTop (𝓝 z) := by + have he : (fun n => initialInverseChart p hi h0 hH (t (φ n)) + (initialForwardChart p hi h0 hH (t (φ n)) z)) = (fun _ : ℕ => z) := by + funext n + exact initialInverse_forward_on_final p hi h0 hH (t (φ n)) hz + rw [he] + exact tendsto_const_nhds + exact tendsto_nhds_unique heval hconstant + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseEnergyBound.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseEnergyBound.lean index 05e51f45f005aebeedd0447551f3259c776a8fb3..27528aad5a4470dd6ee6f3d4011c20cf6806498a 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseEnergyBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseEnergyBound.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Loewner.SlitUniformInverseBound -import OAI.Probability.SLE.Loewner.InverseDirichletEnergy - -namespace OAI - -/-! A common finite Dirichlet-energy bound for the whole initial inverse -family on a fixed upper half-disk. -/ -noncomputable section -open Set Filter Function Complex Metric MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -theorem initialInverse_dirichlet_bound (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hinj - simpa only [complex_derivative_real_det,abs_pow,abs_norm] using hh - change (∫⁻ z in S, ENNReal.ofReal (‖deriv F z‖^2))≤_ - rw [he] - apply measure_mono - rintro _ ⟨z,hz,rfl⟩ - rw [mem_closedBall_zero_iff] - have hb := initialInverse_uniform_bound p hi h0 hH hR hp s hz.1 - have hzA : ‖z‖ ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hinj + simpa only [complex_derivative_real_det,abs_pow,abs_norm] using hh + change (∫⁻ z in S, ENNReal.ofReal (‖deriv F z‖^2))≤_ + rw [he] + apply measure_mono + rintro _ ⟨z,hz,rfl⟩ + rw [mem_closedBall_zero_iff] + have hb := initialInverse_uniform_bound p hi h0 hH hR hp s hz.1 + have hzA : ‖z‖ ‖deriv (initialInverseChart p hi h0 hH s) z‖^2) - (upperHalfPlane ∩ ball (0:ℂ) A) := by - have hm : Measurable (fun z => ‖deriv (initialInverseChart p hi h0 hH s) z‖^2) := - (measurable_deriv _).norm.pow_const 2 - refine ⟨hm.aestronglyMeasurable,?_⟩ - rw [hasFiniteIntegral_iff_ofReal (ae_of_all _ (fun z => sq_nonneg _))] - exact (initialInverse_dirichlet_bound p hi h0 hH hR hp s A).trans_lt - (isCompact_closedBall (0:ℂ) (A+3*R)).measure_lt_top - -lemma initialInverse_energy_integral_le (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 sq_nonneg _)) - ((measurable_deriv _).norm.pow_const 2).aestronglyMeasurable] - exact ENNReal.toReal_mono (isCompact_closedBall (0:ℂ) (A+3*R)).measure_lt_top.ne - (initialInverse_dirichlet_bound p hi h0 hH hR hp s A) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseEnergyBound + +namespace OAI + +/-! Real, integrable versions of the common inverse Dirichlet bound. -/ +noncomputable section +open Set Filter Function Complex Metric MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma initialInverse_energy_integrable (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 ‖deriv (initialInverseChart p hi h0 hH s) z‖^2) + (upperHalfPlane ∩ ball (0:ℂ) A) := by + have hm : Measurable (fun z => ‖deriv (initialInverseChart p hi h0 hH s) z‖^2) := + (measurable_deriv _).norm.pow_const 2 + refine ⟨hm.aestronglyMeasurable,?_⟩ + rw [hasFiniteIntegral_iff_ofReal (ae_of_all _ (fun z => sq_nonneg _))] + exact (initialInverse_dirichlet_bound p hi h0 hH hR hp s A).trans_lt + (isCompact_closedBall (0:ℂ) (A+3*R)).measure_lt_top + +lemma initialInverse_energy_integral_le (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 sq_nonneg _)) + ((measurable_deriv _).norm.pow_const 2).aestronglyMeasurable] + exact ENNReal.toReal_mono (isCompact_closedBall (0:ℂ) (A+3*R)).measure_lt_top.ne + (initialInverse_dirichlet_bound p hi h0 hH hR hp s A) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseJointContinuity.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseJointContinuity.lean index 18ca51df5e3107fba0684ee1fcff8fbbfbfaba8b..ebd2486fc30b7ee03f1a912818ff0a04085b7afe 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseJointContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseJointContinuity.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.Loewner.InitialInverseLimitAt - -namespace OAI - -/-! Joint continuity of the exact inverse charts in time and an interior -half-plane coordinate. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -lemma initialInverse_limit_eq (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : - EqOn F (initialInverseChart p hi h0 hH s) upperHalfPlane := by - by_cases hs : s=0 - · subst s - intro z hz - have hzero := ((initialInverseChart_tendsto_identity p hi h0 hH).tendsto_at hz).comp ht - have he : F z=id z := tendsto_nhds_unique (hlim.tendsto_at hz) hzero - simpa only [initialInverseChart_zero] using he - · exact initialInverse_limit_eq_at p hi h0 hH hF hlim hs ht - -theorem initialInverseChart_joint_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - Continuous (fun a : unitInterval × upperHalfPlane => - initialInverseChart p hi h0 hH a.1 a.2) := by - rw [continuous_iff_continuousAt] - intro a - apply tendsto_of_subseq_tendsto - intro v hv - have ht : Tendsto (fun n => (v n).1) atTop (𝓝 a.1) := - (continuous_fst.tendsto a).comp hv - have hz : Tendsto (fun n => ((v n).2:ℂ)) atTop (𝓝 (a.2:ℂ)) := - ((continuous_subtype_val.comp continuous_snd).tendsto a).comp hv - obtain ⟨φ,F,hφ,hF,hlim⟩ := initialInverseChart_subsequence p hi h0 hH (fun n => (v n).1) - have he := initialInverse_limit_eq p hi h0 hH hF hlim (ht.comp hφ.tendsto_atTop) - have heval := locallyUniform_evaluation_tendsto isOpen_upperHalfPlane hF.continuousOn - hlim a.2.property (hz.comp hφ.tendsto_atTop) - rw [he a.2.property] at heval - refine ⟨φ,?_⟩ - exact heval - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseLimitAt + +namespace OAI + +/-! Joint continuity of the exact inverse charts in time and an interior +half-plane coordinate. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma initialInverse_limit_eq (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : + EqOn F (initialInverseChart p hi h0 hH s) upperHalfPlane := by + by_cases hs : s=0 + · subst s + intro z hz + have hzero := ((initialInverseChart_tendsto_identity p hi h0 hH).tendsto_at hz).comp ht + have he : F z=id z := tendsto_nhds_unique (hlim.tendsto_at hz) hzero + simpa only [initialInverseChart_zero] using he + · exact initialInverse_limit_eq_at p hi h0 hH hF hlim hs ht + +theorem initialInverseChart_joint_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + Continuous (fun a : unitInterval × upperHalfPlane => + initialInverseChart p hi h0 hH a.1 a.2) := by + rw [continuous_iff_continuousAt] + intro a + apply tendsto_of_subseq_tendsto + intro v hv + have ht : Tendsto (fun n => (v n).1) atTop (𝓝 a.1) := + (continuous_fst.tendsto a).comp hv + have hz : Tendsto (fun n => ((v n).2:ℂ)) atTop (𝓝 (a.2:ℂ)) := + ((continuous_subtype_val.comp continuous_snd).tendsto a).comp hv + obtain ⟨φ,F,hφ,hF,hlim⟩ := initialInverseChart_subsequence p hi h0 hH (fun n => (v n).1) + have he := initialInverse_limit_eq p hi h0 hH hF hlim (ht.comp hφ.tendsto_atTop) + have heval := locallyUniform_evaluation_tendsto isOpen_upperHalfPlane hF.continuousOn + hlim a.2.property (hz.comp hφ.tendsto_atTop) + rw [he a.2.property] at heval + refine ⟨φ,?_⟩ + exact heval + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseLimit.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseLimit.lean index 93f540d5bec5a81ec37a89b1ecfbfb0bbe5b5f72..7fa2b9a9dee8fa5fb37d0c3100bf3f75bbb13985 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseLimit.lean @@ -1,72 +1,72 @@ -import OAI.Probability.SLE.Loewner.InitialInverseNormalFamily -import TauCeti.Analysis.Complex.Conformal.Hurwitz - -namespace OAI - -/-! Limits of initial inverse slit charts remain nondegenerate and univalent. -/ -noncomputable section -open Set Filter Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma initialInverseChart_im_ge (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane) : - z.im≤(initialInverseChart p hi h0 hH t z).im := by - by_cases ht : t=0 - · simp only [ht,initialInverseChart_zero,id_eq,le_refl] - · rw [initialInverseChart,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).inverse_im_ge hz - -lemma initialInverseChart_injOn (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : unitInterval) : InjOn (initialInverseChart p hi h0 hH t) upperHalfPlane := by - by_cases ht : t=0 - · simp only [ht,initialInverseChart_zero] - exact injective_id.injOn - · rw [initialInverseChart,dite_eq_right ht] - exact (initialSlitChart p hi h0 hH t ht).inverse_injOn - -lemma initialInverse_limit_im_ge (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {z : ℂ} (hz : z∈upperHalfPlane) : z.im≤(F z).im := - ge_of_tendsto (Complex.continuous_im.continuousAt.tendsto.comp (hlim.tendsto_at hz)) - (Eventually.of_forall fun n => initialInverseChart_im_ge p hi h0 hH (t n) hz) - -lemma initialInverse_limit_injOn (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) : InjOn F upperHalfPlane := by - rcases TauCeti.hurwitz_injOn isOpen_upperHalfPlane - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected - (Eventually.of_forall fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) hlim - (Eventually.of_forall fun n => initialInverseChart_injOn p hi h0 hH (t n)) with hh | ⟨c,hc⟩ - · exact hh - · let y := max c.im 0+1 - have hy : 0 initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) - (hcap : Tendsto (fun n => initialCapacity p hi h0 hH (t n)) atTop (𝓝 c)) - {z : ℂ} (hz : z∈upperHalfPlane) : ‖F z-z‖≤c/z.im := - le_of_tendsto_of_tendsto ((hlim.tendsto_at hz).sub_const z).norm (hcap.div_const z.im) - (Eventually.of_forall fun n => initialInverseChart_displacement p hi h0 hH (t n) hz) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseNormalFamily +import TauCeti.Analysis.Complex.Conformal.Hurwitz + +namespace OAI + +/-! Limits of initial inverse slit charts remain nondegenerate and univalent. -/ +noncomputable section +open Set Filter Function Metric +open scoped Topology + +namespace SLEExactGauge + +lemma initialInverseChart_im_ge (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane) : + z.im≤(initialInverseChart p hi h0 hH t z).im := by + by_cases ht : t=0 + · simp only [ht,initialInverseChart_zero,id_eq,le_refl] + · rw [initialInverseChart,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).inverse_im_ge hz + +lemma initialInverseChart_injOn (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : unitInterval) : InjOn (initialInverseChart p hi h0 hH t) upperHalfPlane := by + by_cases ht : t=0 + · simp only [ht,initialInverseChart_zero] + exact injective_id.injOn + · rw [initialInverseChart,dite_eq_right ht] + exact (initialSlitChart p hi h0 hH t ht).inverse_injOn + +lemma initialInverse_limit_im_ge (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {z : ℂ} (hz : z∈upperHalfPlane) : z.im≤(F z).im := + ge_of_tendsto (Complex.continuous_im.continuousAt.tendsto.comp (hlim.tendsto_at hz)) + (Eventually.of_forall fun n => initialInverseChart_im_ge p hi h0 hH (t n) hz) + +lemma initialInverse_limit_injOn (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) : InjOn F upperHalfPlane := by + rcases TauCeti.hurwitz_injOn isOpen_upperHalfPlane + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected + (Eventually.of_forall fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) hlim + (Eventually.of_forall fun n => initialInverseChart_injOn p hi h0 hH (t n)) with hh | ⟨c,hc⟩ + · exact hh + · let y := max c.im 0+1 + have hy : 0 initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) + (hcap : Tendsto (fun n => initialCapacity p hi h0 hH (t n)) atTop (𝓝 c)) + {z : ℂ} (hz : z∈upperHalfPlane) : ‖F z-z‖≤c/z.im := + le_of_tendsto_of_tendsto ((hlim.tendsto_at hz).sub_const z).norm (hcap.div_const z.im) + (Eventually.of_forall fun n => initialInverseChart_displacement p hi h0 hH (t n) hz) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseLimitAt.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseLimitAt.lean index 9436092d50ed4cef5b264f3a8bcbb52c01ebfd7f..964f8c681861e3335b2022b2c1feeda3c41b8cc9 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseLimitAt.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseLimitAt.lean @@ -1,35 +1,35 @@ -import OAI.Probability.SLE.Loewner.InitialInverseRangeAt -import OAI.Probability.SLE.Loewner.SlitChartLimitUniqueness - -namespace OAI - -/-! Both-sided identification of the actual initial inverse-chart limits. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -theorem initialInverse_limit_eq_at (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (hs : s≠0) (ht : Tendsto t atTop (𝓝 s)) : - EqOn F (initialInverseChart p hi h0 hH s) upperHalfPlane := by - obtain ⟨R0,hR0⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le - let R := max R0 1 - have hR : 0 initialInverse_limit_im_ge p hi h0 hH hlim hz) - hR (fun u => hp (s*u)) - (fun _ hz => initialInverse_limit_uniform_displacement p hi h0 hH hlim hz) - (fun _ hz => initialInverse_limit_inverse_far_height p hi h0 hH hF hlim hR hp hz) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseRangeAt +import OAI.Probability.SLE.Loewner.SlitChartLimitUniqueness + +namespace OAI + +/-! Both-sided identification of the actual initial inverse-chart limits. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +theorem initialInverse_limit_eq_at (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (hs : s≠0) (ht : Tendsto t atTop (𝓝 s)) : + EqOn F (initialInverseChart p hi h0 hH s) upperHalfPlane := by + obtain ⟨R0,hR0⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le + let R := max R0 1 + have hR : 0 initialInverse_limit_im_ge p hi h0 hH hlim hz) + hR (fun u => hp (s*u)) + (fun _ hz => initialInverse_limit_uniform_displacement p hi h0 hH hlim hz) + (fun _ hz => initialInverse_limit_inverse_far_height p hi h0 hH hF hlim hR hp hz) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseNormalFamily.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseNormalFamily.lean index d8b297a41a3830dc6a8d90a8677b99196129eb6b..f3319680526983fc148428dea85262744216b3c5 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseNormalFamily.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseNormalFamily.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Loewner.InitialInverseChart -import TauCeti.Analysis.Complex.Conformal.Montel.Basic - -namespace OAI - -/-! Compactness of the actual inverse charts of all initial subarcs. -The uniform bound is supplied by their common final capacity. -/ -noncomputable section -open Set Filter Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma initialInverseChart_locallyBounded (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - TauCeti.IsLocallyBoundedOn (initialInverseChart p hi h0 hH) upperHalfPlane := by - intro K hKH hK - by_cases hne : K.Nonempty - · obtain ⟨m,hm,hmin⟩ := compact_upperHalfPlane_min_height hK hKH hne - obtain ⟨R,hR⟩ := hK.isBounded.exists_norm_le - let C := initialCapacity p hi h0 hH 1 - have hC : 0≤C := initialCapacity_nonneg p hi h0 hH 1 - refine ⟨R+C/m,fun t z hz => ?_⟩ - have hcap : initialCapacity p hi h0 hH t≤C := - initialCapacity_monotone p hi h0 hH (show t≤1 from le_top) - have him : 0 False.elim (hne ⟨_,hz⟩)⟩ - -theorem initialInverseChart_subsequence (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t : ℕ → unitInterval) : - ∃ (φ : ℕ → ℕ) (F : ℂ → ℂ), StrictMono φ ∧ DifferentiableOn ℂ F upperHalfPlane ∧ - TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t (φ n))) - F atTop upperHalfPlane := - TauCeti.montel isOpen_upperHalfPlane - (fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) - ((initialInverseChart_locallyBounded p hi h0 hH).comp t) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseChart +import TauCeti.Analysis.Complex.Conformal.Montel.Basic + +namespace OAI + +/-! Compactness of the actual inverse charts of all initial subarcs. +The uniform bound is supplied by their common final capacity. -/ +noncomputable section +open Set Filter Function Metric +open scoped Topology + +namespace SLEExactGauge + +lemma initialInverseChart_locallyBounded (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + TauCeti.IsLocallyBoundedOn (initialInverseChart p hi h0 hH) upperHalfPlane := by + intro K hKH hK + by_cases hne : K.Nonempty + · obtain ⟨m,hm,hmin⟩ := compact_upperHalfPlane_min_height hK hKH hne + obtain ⟨R,hR⟩ := hK.isBounded.exists_norm_le + let C := initialCapacity p hi h0 hH 1 + have hC : 0≤C := initialCapacity_nonneg p hi h0 hH 1 + refine ⟨R+C/m,fun t z hz => ?_⟩ + have hcap : initialCapacity p hi h0 hH t≤C := + initialCapacity_monotone p hi h0 hH (show t≤1 from le_top) + have him : 0 False.elim (hne ⟨_,hz⟩)⟩ + +theorem initialInverseChart_subsequence (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t : ℕ → unitInterval) : + ∃ (φ : ℕ → ℕ) (F : ℂ → ℂ), StrictMono φ ∧ DifferentiableOn ℂ F upperHalfPlane ∧ + TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t (φ n))) + F atTop upperHalfPlane := + TauCeti.montel isOpen_upperHalfPlane + (fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) + ((initialInverseChart_locallyBounded p hi h0 hH).comp t) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInversePartialDomain.lean b/lean/OAI/Probability/SLE/Loewner/InitialInversePartialDomain.lean index 6205cac12778fcc400bb44efb2261ff6415c7914..a3a4f48d567745c5ea262035c55dd0c31731320d 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInversePartialDomain.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInversePartialDomain.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Loewner.InitialLimitInverse -import OAI.Probability.SLE.Geometry.SubarcParameter - -namespace OAI - -/-! The inverse limit covers any fixed later subarc complement on which -the approximating charts are eventually defined. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -lemma initialInverse_limit_covers_subarc (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) (b : unitInterval) (hb : b≠0) - (hbt : ∀ᶠ n in atTop, t n≤b) : - SurjOn F upperHalfPlane (upperHalfPlane \ range (initialSubarc p b)) := by - obtain ⟨N,hN⟩ := eventually_atTop.mp hbt - let hib := initialSubarc_injective hi hb - have h0b : initialSubarc p b 0=0 := by simpa only [initialSubarc_zero] using h0 - let hHb := initialSubarc_upper hH hb - let u : ℕ → unitInterval := fun n => subarcParameter b hb (t (n+N)) - have he (n : ℕ) : b*u n=t (n+N) := - subarcParameter_mul_of_le b hb (hN (n+N) (by omega)) - have hshift := locallyUniform_reindex hlim (tendsto_add_atTop_nat N) - have hq : TendstoLocallyUniformlyOn - (fun n => initialInverseChart (initialSubarc p b) hib h0b hHb (u n)) F atTop upperHalfPlane := by - apply hshift.congr - intro n z hz - have hh := initialInverseChart_subarc p hi h0 hH b hb hib h0b hHb (u n) hz - rw [he n] at hh - exact hh.symm - exact initialInverse_limit_covers_final_domain (initialSubarc p b) hib h0b hHb hF hq - -lemma exists_later_subarc_domain (p : C(unitInterval,ℂ)) (hi : Injective p) - {s : unitInterval} (hs : s<1) {z : ℂ} - (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : - ∃ b : unitInterval, s hzp (initialSubarc_range_subset p 1 hh) - -lemma initialInverse_limit_covers_at (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : - SurjOn F upperHalfPlane (upperHalfPlane \ range (initialSubarc p s)) := by - intro z hz - by_cases hs : s=1 - · subst s - apply initialInverse_limit_covers_final_domain p hi h0 hH hF hlim z - simpa only [Set.mem_sdiff,mem_range,initialSubarc_apply,one_mul] using hz - · obtain ⟨b,hsb,hzb⟩ := exists_later_subarc_domain p hi (lt_of_le_of_ne le_top hs) hz - have hb : b≠0 := ne_of_gt (lt_of_le_of_lt bot_le hsb) - exact initialInverse_limit_covers_subarc p hi h0 hH hF hlim b hb - (ht.eventually (Iio_mem_nhds hsb) |>.mono fun _ hh => hh.le) hzb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialLimitInverse +import OAI.Probability.SLE.Geometry.SubarcParameter + +namespace OAI + +/-! The inverse limit covers any fixed later subarc complement on which +the approximating charts are eventually defined. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma initialInverse_limit_covers_subarc (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) (b : unitInterval) (hb : b≠0) + (hbt : ∀ᶠ n in atTop, t n≤b) : + SurjOn F upperHalfPlane (upperHalfPlane \ range (initialSubarc p b)) := by + obtain ⟨N,hN⟩ := eventually_atTop.mp hbt + let hib := initialSubarc_injective hi hb + have h0b : initialSubarc p b 0=0 := by simpa only [initialSubarc_zero] using h0 + let hHb := initialSubarc_upper hH hb + let u : ℕ → unitInterval := fun n => subarcParameter b hb (t (n+N)) + have he (n : ℕ) : b*u n=t (n+N) := + subarcParameter_mul_of_le b hb (hN (n+N) (by omega)) + have hshift := locallyUniform_reindex hlim (tendsto_add_atTop_nat N) + have hq : TendstoLocallyUniformlyOn + (fun n => initialInverseChart (initialSubarc p b) hib h0b hHb (u n)) F atTop upperHalfPlane := by + apply hshift.congr + intro n z hz + have hh := initialInverseChart_subarc p hi h0 hH b hb hib h0b hHb (u n) hz + rw [he n] at hh + exact hh.symm + exact initialInverse_limit_covers_final_domain (initialSubarc p b) hib h0b hHb hF hq + +lemma exists_later_subarc_domain (p : C(unitInterval,ℂ)) (hi : Injective p) + {s : unitInterval} (hs : s<1) {z : ℂ} + (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : + ∃ b : unitInterval, s hzp (initialSubarc_range_subset p 1 hh) + +lemma initialInverse_limit_covers_at (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : + SurjOn F upperHalfPlane (upperHalfPlane \ range (initialSubarc p s)) := by + intro z hz + by_cases hs : s=1 + · subst s + apply initialInverse_limit_covers_final_domain p hi h0 hH hF hlim z + simpa only [Set.mem_sdiff,mem_range,initialSubarc_apply,one_mul] using hz + · obtain ⟨b,hsb,hzb⟩ := exists_later_subarc_domain p hi (lt_of_le_of_ne le_top hs) hz + have hb : b≠0 := ne_of_gt (lt_of_le_of_lt bot_le hsb) + exact initialInverse_limit_covers_subarc p hi h0 hH hF hlim b hb + (ht.eventually (Iio_mem_nhds hsb) |>.mono fun _ hh => hh.le) hzb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseRange.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseRange.lean index 00be8e952acf9b7768807800fe4e6efa69aad7c9..0ecefd34e4cb6a8f7cd0b22ddf1f155cd27557e1 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseRange.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseRange.lean @@ -1,102 +1,102 @@ -import OAI.Probability.SLE.Loewner.InitialInverseCommonDomain -import Mathlib.Topology.Order.DenselyOrdered - -namespace OAI - -/-! At the terminal initial-arc time, the inverse limits have exactly the -prescribed slit domain as their image. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialInverseChart_avoids_prefix (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (t u : unitInterval) (hut : u≤t) {z : ℂ} (hz : z∈upperHalfPlane) : - initialInverseChart p hi h0 hH t z≠p u := by - by_cases ht : t=0 - · have hu : u=0 := le_antisymm (ht ▸ hut) bot_le - simp only [ht,initialInverseChart_zero,id_eq,hu,h0] - intro he - have hh : 0 initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) (c : ℂ) : ¬ ∀ z∈upperHalfPlane, F z=c := by - intro hc - have hinj := initialInverse_limit_injOn p hi h0 hH hlim - have h1 : I∈upperHalfPlane := by simp [upperHalfPlane] - have h2 : 2*I∈upperHalfPlane := by simp [upperHalfPlane] - have he := hinj h1 h2 ((hc I h1).trans (hc (2*I) h2).symm) - have hh := congrArg Complex.im he - norm_num at hh - -lemma initialInverse_limit_avoids_before_one (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) - (u : unitInterval) (hu : u<1) : ∀ z∈upperHalfPlane, F z≠p u := by - have hne : ∀ᶠ n in atTop, ∀ z∈upperHalfPlane, - initialInverseChart p hi h0 hH (t n) z≠p u := by - filter_upwards [ht.eventually (Ioi_mem_nhds hu)] with n hn - exact fun _ hz => initialInverseChart_avoids_prefix p hi h0 hH (t n) u hn.le hz - exact TauCeti.hurwitz_forall_ne isOpen_upperHalfPlane - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected - (Eventually.of_forall fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) - hlim hne (initialInverse_limit_not_constant p hi h0 hH hlim (p u)) - -lemma initialInverse_limit_mapsTo_final (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) : - MapsTo F upperHalfPlane (upperHalfPlane \ range p) := by - have hopen : IsOpen (F '' upperHalfPlane) := by - rcases (hF.analyticOnNhd isOpen_upperHalfPlane).is_constant_or_isOpen - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected with ⟨c,hc⟩ | ho - · exact False.elim (initialInverse_limit_not_constant p hi h0 hH hlim c hc) - · exact ho _ Subset.rfl isOpen_upperHalfPlane - intro z hz - refine ⟨lt_of_lt_of_le hz (initialInverse_limit_im_ge p hi h0 hH hlim hz),?_⟩ - rintro ⟨u,hu⟩ - have hbefore := initialInverse_limit_avoids_before_one p hi h0 hH hlim ht - by_cases hu1 : u<1 - · exact hbefore u hu1 z hz hu.symm - have heu : u=1 := le_antisymm le_top (le_of_not_gt hu1) - subst u - have hp1 : p 1∈F '' upperHalfPlane := ⟨z,hz,hu.symm⟩ - have hpre : p ⁻¹' (F '' upperHalfPlane)∈𝓝 (1:unitInterval) := - p.continuous.continuousAt.preimage_mem_nhds (hopen.mem_nhds hp1) - have hcl : (1:unitInterval)∈closure (Iio (1:unitInterval)) := by - rw [closure_Iio' (show (Iio (1:unitInterval)).Nonempty from ⟨0,by norm_num⟩)] - exact mem_Iic.mpr (le_refl (1:unitInterval)) - obtain ⟨v,hvp,hvlt⟩ := mem_closure_iff_nhds.mp hcl _ hpre - obtain ⟨w,hw,hew⟩ := hvp - exact hbefore v hvlt w hw hew - -theorem initialInverse_limit_bijOn_final (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) : - BijOn F upperHalfPlane (upperHalfPlane \ range p) := - ⟨initialInverse_limit_mapsTo_final p hi h0 hH hF hlim ht, - initialInverse_limit_injOn p hi h0 hH hlim, - initialInverse_limit_covers_final_domain p hi h0 hH hF hlim⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseCommonDomain +import Mathlib.Topology.Order.DenselyOrdered + +namespace OAI + +/-! At the terminal initial-arc time, the inverse limits have exactly the +prescribed slit domain as their image. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma initialInverseChart_avoids_prefix (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (t u : unitInterval) (hut : u≤t) {z : ℂ} (hz : z∈upperHalfPlane) : + initialInverseChart p hi h0 hH t z≠p u := by + by_cases ht : t=0 + · have hu : u=0 := le_antisymm (ht ▸ hut) bot_le + simp only [ht,initialInverseChart_zero,id_eq,hu,h0] + intro he + have hh : 0 initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) (c : ℂ) : ¬ ∀ z∈upperHalfPlane, F z=c := by + intro hc + have hinj := initialInverse_limit_injOn p hi h0 hH hlim + have h1 : I∈upperHalfPlane := by simp [upperHalfPlane] + have h2 : 2*I∈upperHalfPlane := by simp [upperHalfPlane] + have he := hinj h1 h2 ((hc I h1).trans (hc (2*I) h2).symm) + have hh := congrArg Complex.im he + norm_num at hh + +lemma initialInverse_limit_avoids_before_one (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) + (u : unitInterval) (hu : u<1) : ∀ z∈upperHalfPlane, F z≠p u := by + have hne : ∀ᶠ n in atTop, ∀ z∈upperHalfPlane, + initialInverseChart p hi h0 hH (t n) z≠p u := by + filter_upwards [ht.eventually (Ioi_mem_nhds hu)] with n hn + exact fun _ hz => initialInverseChart_avoids_prefix p hi h0 hH (t n) u hn.le hz + exact TauCeti.hurwitz_forall_ne isOpen_upperHalfPlane + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected + (Eventually.of_forall fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) + hlim hne (initialInverse_limit_not_constant p hi h0 hH hlim (p u)) + +lemma initialInverse_limit_mapsTo_final (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) : + MapsTo F upperHalfPlane (upperHalfPlane \ range p) := by + have hopen : IsOpen (F '' upperHalfPlane) := by + rcases (hF.analyticOnNhd isOpen_upperHalfPlane).is_constant_or_isOpen + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected with ⟨c,hc⟩ | ho + · exact False.elim (initialInverse_limit_not_constant p hi h0 hH hlim c hc) + · exact ho _ Subset.rfl isOpen_upperHalfPlane + intro z hz + refine ⟨lt_of_lt_of_le hz (initialInverse_limit_im_ge p hi h0 hH hlim hz),?_⟩ + rintro ⟨u,hu⟩ + have hbefore := initialInverse_limit_avoids_before_one p hi h0 hH hlim ht + by_cases hu1 : u<1 + · exact hbefore u hu1 z hz hu.symm + have heu : u=1 := le_antisymm le_top (le_of_not_gt hu1) + subst u + have hp1 : p 1∈F '' upperHalfPlane := ⟨z,hz,hu.symm⟩ + have hpre : p ⁻¹' (F '' upperHalfPlane)∈𝓝 (1:unitInterval) := + p.continuous.continuousAt.preimage_mem_nhds (hopen.mem_nhds hp1) + have hcl : (1:unitInterval)∈closure (Iio (1:unitInterval)) := by + rw [closure_Iio' (show (Iio (1:unitInterval)).Nonempty from ⟨0,by norm_num⟩)] + exact mem_Iic.mpr (le_refl (1:unitInterval)) + obtain ⟨v,hvp,hvlt⟩ := mem_closure_iff_nhds.mp hcl _ hpre + obtain ⟨w,hw,hew⟩ := hvp + exact hbefore v hvlt w hw hew + +theorem initialInverse_limit_bijOn_final (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) : + BijOn F upperHalfPlane (upperHalfPlane \ range p) := + ⟨initialInverse_limit_mapsTo_final p hi h0 hH hF hlim ht, + initialInverse_limit_injOn p hi h0 hH hlim, + initialInverse_limit_covers_final_domain p hi h0 hH hF hlim⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseRangeAt.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseRangeAt.lean index d9321bfa8af0f69a9c0f7af5d4b6523225c1df06..8ce1d1d95e46f6f539e1d46f38b498a21a41b8bf 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseRangeAt.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseRangeAt.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Loewner.InitialInversePartialDomain - -namespace OAI - -/-! Exact image of an inverse-chart limit at an arbitrary initial time, -from either side of that time. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialInverse_limit_avoids_before_time (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) - (u : unitInterval) (hu : u initialInverseChart_avoids_prefix p hi h0 hH (t n) u hn.le hz - exact TauCeti.hurwitz_forall_ne isOpen_upperHalfPlane - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected - (Eventually.of_forall fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) - hlim hne (initialInverse_limit_not_constant p hi h0 hH hlim (p u)) - -lemma initialInverse_limit_mapsTo_at (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : - MapsTo F upperHalfPlane (upperHalfPlane \ range (initialSubarc p s)) := by - have hopen : IsOpen (F '' upperHalfPlane) := by - rcases (hF.analyticOnNhd isOpen_upperHalfPlane).is_constant_or_isOpen - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected with ⟨c,hc⟩ | ho - · exact False.elim (initialInverse_limit_not_constant p hi h0 hH hlim c hc) - · exact ho _ Subset.rfl isOpen_upperHalfPlane - intro z hz - have hpos : 0<(F z).im := lt_of_lt_of_le hz (initialInverse_limit_im_ge p hi h0 hH hlim hz) - refine ⟨hpos,?_⟩ - rw [initialSubarc_range] - rintro ⟨u,hu,heu⟩ - have hbefore := initialInverse_limit_avoids_before_time p hi h0 hH hlim ht - by_cases hus : u initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : - BijOn F upperHalfPlane (upperHalfPlane \ range (initialSubarc p s)) := - ⟨initialInverse_limit_mapsTo_at p hi h0 hH hF hlim ht, - initialInverse_limit_injOn p hi h0 hH hlim, - initialInverse_limit_covers_at p hi h0 hH hF hlim ht⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInversePartialDomain + +namespace OAI + +/-! Exact image of an inverse-chart limit at an arbitrary initial time, +from either side of that time. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma initialInverse_limit_avoids_before_time (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) + (u : unitInterval) (hu : u initialInverseChart_avoids_prefix p hi h0 hH (t n) u hn.le hz + exact TauCeti.hurwitz_forall_ne isOpen_upperHalfPlane + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected + (Eventually.of_forall fun n => initialInverseChart_differentiableOn p hi h0 hH (t n)) + hlim hne (initialInverse_limit_not_constant p hi h0 hH hlim (p u)) + +lemma initialInverse_limit_mapsTo_at (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : + MapsTo F upperHalfPlane (upperHalfPlane \ range (initialSubarc p s)) := by + have hopen : IsOpen (F '' upperHalfPlane) := by + rcases (hF.analyticOnNhd isOpen_upperHalfPlane).is_constant_or_isOpen + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected with ⟨c,hc⟩ | ho + · exact False.elim (initialInverse_limit_not_constant p hi h0 hH hlim c hc) + · exact ho _ Subset.rfl isOpen_upperHalfPlane + intro z hz + have hpos : 0<(F z).im := lt_of_lt_of_le hz (initialInverse_limit_im_ge p hi h0 hH hlim hz) + refine ⟨hpos,?_⟩ + rw [initialSubarc_range] + rintro ⟨u,hu,heu⟩ + have hbefore := initialInverse_limit_avoids_before_time p hi h0 hH hlim ht + by_cases hus : u initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {s : unitInterval} (ht : Tendsto t atTop (𝓝 s)) : + BijOn F upperHalfPlane (upperHalfPlane \ range (initialSubarc p s)) := + ⟨initialInverse_limit_mapsTo_at p hi h0 hH hF hlim ht, + initialInverse_limit_injOn p hi h0 hH hlim, + initialInverse_limit_covers_at p hi h0 hH hF hlim ht⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialInverseTimeContinuity.lean b/lean/OAI/Probability/SLE/Loewner/InitialInverseTimeContinuity.lean index 1335e68fbf253ae5a6932bf1b161c23b751bce7f..c2e40e15de74c273703f632debb0a216e226665a 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialInverseTimeContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialInverseTimeContinuity.lean @@ -1,34 +1,34 @@ -import OAI.Probability.SLE.Loewner.InitialInverseJointContinuity -import Mathlib.Topology.UniformSpace.HeineCantor - -namespace OAI - -/-! Local uniform time continuity of the exact inverse slit charts. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -theorem initialInverseChart_locallyUniform_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (s : unitInterval) : - TendstoLocallyUniformlyOn (initialInverseChart p hi h0 hH) - (initialInverseChart p hi h0 hH s) (𝓝 s) upperHalfPlane := by - rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane] - intro K hKH hK - let : CompactSpace K := isCompact_iff_compactSpace.mp hK - have hc : Continuous (fun a : unitInterval × K => - initialInverseChart p hi h0 hH a.1 a.2) := by - exact (initialInverseChart_joint_continuous p hi h0 hH).comp - (continuous_fst.prodMk (Continuous.subtype_mk - (continuous_subtype_val.comp continuous_snd) (fun a => hKH a.2.property))) - have hu := Continuous.tendstoUniformly - (fun t (z : K) => initialInverseChart p hi h0 hH t z) hc s - exact tendstoUniformlyOn_iff_tendstoUniformly_comp_coe.mpr hu - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseJointContinuity +import Mathlib.Topology.UniformSpace.HeineCantor + +namespace OAI + +/-! Local uniform time continuity of the exact inverse slit charts. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +theorem initialInverseChart_locallyUniform_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (s : unitInterval) : + TendstoLocallyUniformlyOn (initialInverseChart p hi h0 hH) + (initialInverseChart p hi h0 hH s) (𝓝 s) upperHalfPlane := by + rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane] + intro K hKH hK + let : CompactSpace K := isCompact_iff_compactSpace.mp hK + have hc : Continuous (fun a : unitInterval × K => + initialInverseChart p hi h0 hH a.1 a.2) := by + exact (initialInverseChart_joint_continuous p hi h0 hH).comp + (continuous_fst.prodMk (Continuous.subtype_mk + (continuous_subtype_val.comp continuous_snd) (fun a => hKH a.2.property))) + have hu := Continuous.tendstoUniformly + (fun t (z : K) => initialInverseChart p hi h0 hH t z) hc s + exact tendstoUniformlyOn_iff_tendstoUniformly_comp_coe.mpr hu + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialLimitInverse.lean b/lean/OAI/Probability/SLE/Loewner/InitialLimitInverse.lean index d362811b59d5214af78c7f3d8ae4d05d02e2d994..dd212dc5783230d32e45330a23ee5179c77a9ae3 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialLimitInverse.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialLimitInverse.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Loewner.InitialInverseRange -import OAI.Probability.SLE.Loewner.SlitChartFarHeight -import TauCeti.Analysis.Complex.Conformal.Inverse.Function - -namespace OAI - -/-! The inverse of a terminal inverse-chart limit is holomorphic and retains -the common far-height bound. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialForwardChart_far_height (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {R : ℝ} (hR : 0 hp (t*u)) hz - -lemma initialInverse_limit_forward_subsequence (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) : - ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ z∈upperHalfPlane \ range p, - Tendsto (fun n => initialForwardChart p hi h0 hH (t (φ n)) z) - atTop (𝓝 (invFunOn F upperHalfPlane z)) := by - have hcov : SurjOn F upperHalfPlane (upperHalfPlane \ range p) := - initialInverse_limit_covers_final_domain p hi h0 hH hF hlim - have hinj := initialInverse_limit_injOn p hi h0 hH hlim - obtain ⟨φ,A,hφ,hA,hAlim⟩ := initialForwardCayley_subsequence p hi h0 hH t - refine ⟨φ,hφ,fun z hz => ?_⟩ - have hw := halfPlaneCayleyInv_open - (initialForwardCayley_limit_interior p hi h0 hH hA hAlim z hz) - have hfwd := initialForward_limit_tendsto p hi h0 hH hA hAlim hz - have heval := locallyUniform_evaluation_tendsto isOpen_upperHalfPlane - hF.continuousOn (locallyUniform_reindex hlim hφ.tendsto_atTop) hw hfwd - have he : F (halfPlaneCayleyInv (A z))=z := by - apply tendsto_nhds_unique heval - simpa only [initialInverse_forward_on_final p hi h0 hH _ hz] using - (tendsto_const_nhds : Tendsto (fun _ : ℕ => z) atTop (𝓝 z)) - have hwG : halfPlaneCayleyInv (A z)=invFunOn F upperHalfPlane z := by - apply hinj hw (hcov.mapsTo_invFunOn hz) - exact he.trans (hcov.rightInvOn_invFunOn hz).symm - rwa [hwG] at hfwd - -lemma initialInverse_limit_inverse_far_height (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) - {R : ℝ} (hR : 0 initialForwardChart_far_height p hi h0 hH hR hp _ hz) - -lemma initialInverse_limit_uniform_displacement (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {t : ℕ → unitInterval} {F : ℂ → ℂ} - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) {z : ℂ} (hz : z∈upperHalfPlane) : - ‖F z-z‖ ≤ initialCapacity p hi h0 hH 1/z.im := by - apply le_of_tendsto ((hlim.tendsto_at hz).sub_const z).norm - exact Eventually.of_forall fun n => (initialInverseChart_displacement p hi h0 hH (t n) hz).trans - (div_le_div_of_nonneg_right (initialCapacity_monotone p hi h0 hH (show t n≤1 from le_top)) - (show 0 hp (t*u)) hz + +lemma initialInverse_limit_forward_subsequence (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) : + ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ z∈upperHalfPlane \ range p, + Tendsto (fun n => initialForwardChart p hi h0 hH (t (φ n)) z) + atTop (𝓝 (invFunOn F upperHalfPlane z)) := by + have hcov : SurjOn F upperHalfPlane (upperHalfPlane \ range p) := + initialInverse_limit_covers_final_domain p hi h0 hH hF hlim + have hinj := initialInverse_limit_injOn p hi h0 hH hlim + obtain ⟨φ,A,hφ,hA,hAlim⟩ := initialForwardCayley_subsequence p hi h0 hH t + refine ⟨φ,hφ,fun z hz => ?_⟩ + have hw := halfPlaneCayleyInv_open + (initialForwardCayley_limit_interior p hi h0 hH hA hAlim z hz) + have hfwd := initialForward_limit_tendsto p hi h0 hH hA hAlim hz + have heval := locallyUniform_evaluation_tendsto isOpen_upperHalfPlane + hF.continuousOn (locallyUniform_reindex hlim hφ.tendsto_atTop) hw hfwd + have he : F (halfPlaneCayleyInv (A z))=z := by + apply tendsto_nhds_unique heval + simpa only [initialInverse_forward_on_final p hi h0 hH _ hz] using + (tendsto_const_nhds : Tendsto (fun _ : ℕ => z) atTop (𝓝 z)) + have hwG : halfPlaneCayleyInv (A z)=invFunOn F upperHalfPlane z := by + apply hinj hw (hcov.mapsTo_invFunOn hz) + exact he.trans (hcov.rightInvOn_invFunOn hz).symm + rwa [hwG] at hfwd + +lemma initialInverse_limit_inverse_far_height (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) + {R : ℝ} (hR : 0 initialForwardChart_far_height p hi h0 hH hR hp _ hz) + +lemma initialInverse_limit_uniform_displacement (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {t : ℕ → unitInterval} {F : ℂ → ℂ} + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) {z : ℂ} (hz : z∈upperHalfPlane) : + ‖F z-z‖ ≤ initialCapacity p hi h0 hH 1/z.im := by + apply le_of_tendsto ((hlim.tendsto_at hz).sub_const z).norm + exact Eventually.of_forall fun n => (initialInverseChart_displacement p hi h0 hH (t n) hz).trans + (div_le_div_of_nonneg_right (initialCapacity_monotone p hi h0 hH (show t n≤1 from le_top)) + (show 0.symm - -lemma initialReflectedForward_differentiable (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) : - DifferentiableOn ℂ (initialReflectedForward p hi h0 hH s) (reflectedArcSet p)ᶜ := by - by_cases hs : s=0 - · simp only [initialReflectedForward,dite_eq_left hs] - exact differentiableOn_id - · rw [initialReflectedForward,dite_eq_right hs] - exact ((initialReflectedChart p hi h0 hH s hs).forward_differentiable - (by simpa only [initialSubarc_zero] using h0)).mono - (compl_subset_compl.mpr (reflected_initial_subset p s)) - -lemma real_not_reflected_arc (p : C(unitInterval,ℂ)) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {x : ℝ} (hx : x≠0) : (x:ℂ)∉reflectedArcSet p := by - have hreal : (x:ℂ)∉range p := by - rintro ⟨s,hs⟩ - by_cases hs0 : s=0 - · have he : (x:ℂ)=0 := by simpa only [hs0,h0] using hs.symm - exact hx (ofReal_eq_zero.mp he) - · have hh := hH s hs0 - rw [hs] at hh - simp [upperHalfPlane] at hh - rintro (hz | ⟨w,hw,he⟩) - · exact hreal hz - · apply hreal - have hwx : w=(x:ℂ) := by - have hh := congrArg conj he - simpa only [conj_conj,conj_ofReal] using hh - rwa [hwx] at hw - -lemma initialReflectedForward_real (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) {x : ℝ} (hx : x≠0) : - (initialReflectedForward p hi h0 hH s (x:ℂ)).im=0 := by - by_cases hs : s=0 - · simp [initialReflectedForward,hs] - · rw [initialReflectedForward,dite_eq_right hs] - have hh := (initialReflectedChart p hi h0 hH s hs).forward_reflection - (fun hz => real_not_reflected_arc p h0 hH hx (reflected_initial_subset p s hz)) - rw [conj_ofReal] at hh - have he := congrArg Complex.im hh - simp only [conj_im] at he - linarith - -lemma initialReflectedForward_uniform_bound (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 hp (s*t)) - (fun he => hz (reflected_initial_subset p s he)) hzR - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitUniformForwardBound +import OAI.Probability.SLE.Loewner.ReflectedExteriorHeight +import OAI.Probability.SLE.Loewner.InitialForwardJointContinuity + +namespace OAI + +/-! The normalized initial forward maps continued across the real axis +away from the attached slit. These are the same maps on the upper side. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +noncomputable def initialReflectedChart (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) (hs : s≠0) : ReflectedCapacityChart (initialSubarc p s) := + Classical.choice (exists_reflectedCapacityChart (initialSubarc p s) + (initialSubarc_injective hi hs) (by simpa only [initialSubarc_zero] using h0) + (initialSubarc_upper hH hs)) + +noncomputable def initialReflectedForward (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) : ℂ → ℂ := + if hs : s=0 then id else (initialReflectedChart p hi h0 hH s hs).chart.forward + +lemma reflected_initial_subset (p : C(unitInterval,ℂ)) (s : unitInterval) : + reflectedArcSet (initialSubarc p s) ⊆ reflectedArcSet p := by + intro z hz + rcases hz with hz | ⟨w,hw,rfl⟩ + · exact Or.inl (initialSubarc_range_subset p s hz) + · exact Or.inr ⟨w,initialSubarc_range_subset p s hw,rfl⟩ + +lemma initialReflectedForward_eq (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : + initialReflectedForward p hi h0 hH s z=initialForwardChart p hi h0 hH s z := by + by_cases hs : s=0 + · subst s + simp [initialReflectedForward] + · simp only [initialReflectedForward,initialForwardChart,dite_eq_right hs] + exact ((initialSlitChart p hi h0 hH s hs).unique_on_domain + (initialReflectedChart p hi h0 hH s hs).chart).1 hz |>.symm + +lemma initialReflectedForward_differentiable (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) : + DifferentiableOn ℂ (initialReflectedForward p hi h0 hH s) (reflectedArcSet p)ᶜ := by + by_cases hs : s=0 + · simp only [initialReflectedForward,dite_eq_left hs] + exact differentiableOn_id + · rw [initialReflectedForward,dite_eq_right hs] + exact ((initialReflectedChart p hi h0 hH s hs).forward_differentiable + (by simpa only [initialSubarc_zero] using h0)).mono + (compl_subset_compl.mpr (reflected_initial_subset p s)) + +lemma real_not_reflected_arc (p : C(unitInterval,ℂ)) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {x : ℝ} (hx : x≠0) : (x:ℂ)∉reflectedArcSet p := by + have hreal : (x:ℂ)∉range p := by + rintro ⟨s,hs⟩ + by_cases hs0 : s=0 + · have he : (x:ℂ)=0 := by simpa only [hs0,h0] using hs.symm + exact hx (ofReal_eq_zero.mp he) + · have hh := hH s hs0 + rw [hs] at hh + simp [upperHalfPlane] at hh + rintro (hz | ⟨w,hw,he⟩) + · exact hreal hz + · apply hreal + have hwx : w=(x:ℂ) := by + have hh := congrArg conj he + simpa only [conj_conj,conj_ofReal] using hh + rwa [hwx] at hw + +lemma initialReflectedForward_real (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (s : unitInterval) {x : ℝ} (hx : x≠0) : + (initialReflectedForward p hi h0 hH s (x:ℂ)).im=0 := by + by_cases hs : s=0 + · simp [initialReflectedForward,hs] + · rw [initialReflectedForward,dite_eq_right hs] + have hh := (initialReflectedChart p hi h0 hH s hs).forward_reflection + (fun hz => real_not_reflected_arc p h0 hH hx (reflected_initial_subset p s hz)) + rw [conj_ofReal] at hh + have he := congrArg Complex.im hh + simp only [conj_im] at he + linarith + +lemma initialReflectedForward_uniform_bound (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 hp (s*t)) + (fun he => hz (reflected_initial_subset p s he)) hzR + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialSlitFuture.lean b/lean/OAI/Probability/SLE/Loewner/InitialSlitFuture.lean index f82c495920c2b5ed376dc5b2746a2db5eb130415..d541318b1d6fb6d6698995c940225d526dfada87 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialSlitFuture.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialSlitFuture.lean @@ -1,55 +1,55 @@ -import OAI.Probability.SLE.Loewner.InitialForwardChart -import Mathlib.Topology.Order.ProjIcc - -namespace OAI - -/-! The physical future of an initial simple slit, with an exact real -offset parameter and no choice of conformal chart. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -def slitFutureParameter (s : unitInterval) (r : ℝ) : unitInterval := - projIcc 0 1 zero_le_one ((s:ℝ)+r) - -lemma slitFutureParameter_continuous (s : unitInterval) : - Continuous (slitFutureParameter s) := - continuous_projIcc.comp (continuous_const.add continuous_id) - -@[simp] lemma slitFutureParameter_zero (s : unitInterval) : - slitFutureParameter s 0=s := by - simp only [slitFutureParameter,add_zero,projIcc_val] - -lemma slitFutureParameter_coe (s : unitInterval) {r : ℝ} - (hr : r∈Ioo 0 (1-(s:ℝ))) : - (slitFutureParameter s r:ℝ)=(s:ℝ)+r := by - exact congrArg Subtype.val (projIcc_of_mem zero_le_one - (show (s:ℝ)+r∈Icc (0:ℝ) 1 from ⟨by linarith [s.property.1,hr.1],by linarith [hr.2]⟩)) - -lemma slitFuturePoint_mem_domain (p : C(unitInterval,ℂ)) (hi : Injective p) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) {r : ℝ} (hr : r∈Ioo 0 (1-(s:ℝ))) : - p (slitFutureParameter s r)∈upperHalfPlane \ range (initialSubarc p s) := by - have he := slitFutureParameter_coe s hr - have hlt : s range q) (initialSubarc_mul p b u))).1 - -lemma initialInverseChart_subarc (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (b : unitInterval) (hb : b≠0) - (hib : Injective (initialSubarc p b)) (h0b : initialSubarc p b 0=0) - (hHb : ∀ t : unitInterval, t ≠ 0 → initialSubarc p b t∈upperHalfPlane) - (u : unitInterval) : - EqOn (initialInverseChart (initialSubarc p b) hib h0b hHb u) - (initialInverseChart p hi h0 hH (b*u)) upperHalfPlane := by - by_cases hu : u=0 - · simp only [hu,mul_zero,initialInverseChart_zero] - exact fun _ _ => rfl - have hbu : b*u≠0 := mul_ne_zero hb hu - rw [initialInverseChart,dite_eq_right hu,initialInverseChart,dite_eq_right hbu] - exact ((initialSlitChart (initialSubarc p b) hib h0b hHb u hu).eq_of_range_eq - (initialSlitChart p hi h0 hH (b*u) hbu) - (congrArg (fun q : C(unitInterval,ℂ) => range q) (initialSubarc_mul p b u))).2.1 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartRangeUniqueness +import OAI.Probability.SLE.Loewner.InitialForwardChart + +namespace OAI + +/-! Exact capacity and inverse-chart identities under initial-subarc +reparametrization. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma initialSubarc_mul (p : C(unitInterval,ℂ)) (b u : unitInterval) : + initialSubarc (initialSubarc p b) u=initialSubarc p (b*u) := by + ext v + simp only [initialSubarc_apply,mul_assoc] + +lemma initialCapacity_subarc (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (b : unitInterval) (hb : b≠0) + (hib : Injective (initialSubarc p b)) (h0b : initialSubarc p b 0=0) + (hHb : ∀ t : unitInterval, t ≠ 0 → initialSubarc p b t∈upperHalfPlane) + (u : unitInterval) : + initialCapacity (initialSubarc p b) hib h0b hHb u=initialCapacity p hi h0 hH (b*u) := by + by_cases hu : u=0 + · simp only [hu,mul_zero,initialCapacity_zero] + have hbu : b*u≠0 := mul_ne_zero hb hu + rw [initialCapacity,dite_eq_right hu,initialCapacity,dite_eq_right hbu] + exact ((initialSlitChart (initialSubarc p b) hib h0b hHb u hu).eq_of_range_eq + (initialSlitChart p hi h0 hH (b*u) hbu) + (congrArg (fun q : C(unitInterval,ℂ) => range q) (initialSubarc_mul p b u))).1 + +lemma initialInverseChart_subarc (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (b : unitInterval) (hb : b≠0) + (hib : Injective (initialSubarc p b)) (h0b : initialSubarc p b 0=0) + (hHb : ∀ t : unitInterval, t ≠ 0 → initialSubarc p b t∈upperHalfPlane) + (u : unitInterval) : + EqOn (initialInverseChart (initialSubarc p b) hib h0b hHb u) + (initialInverseChart p hi h0 hH (b*u)) upperHalfPlane := by + by_cases hu : u=0 + · simp only [hu,mul_zero,initialInverseChart_zero] + exact fun _ _ => rfl + have hbu : b*u≠0 := mul_ne_zero hb hu + rw [initialInverseChart,dite_eq_right hu,initialInverseChart,dite_eq_right hbu] + exact ((initialSlitChart (initialSubarc p b) hib h0b hHb u hu).eq_of_range_eq + (initialSlitChart p hi h0 hH (b*u) hbu) + (congrArg (fun q : C(unitInterval,ℂ) => range q) (initialSubarc_mul p b u))).2.1 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InitialTipHeightBound.lean b/lean/OAI/Probability/SLE/Loewner/InitialTipHeightBound.lean index b5357eb249ba46d1ae19a9393115499d8cc74c03..d00bb8db5e39574148583e5d458f69d078809e58 100644 --- a/lean/OAI/Probability/SLE/Loewner/InitialTipHeightBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/InitialTipHeightBound.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Loewner.InitialHeightCapacityIncrement -import OAI.Probability.SLE.Loewner.MovingForwardBoundaryHeight -import OAI.Probability.SLE.Geometry.ApproachIntervalEndpoint - -namespace OAI - -/-! A future point has initial-chart height controlled by the capacity -remaining before that point is swallowed. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -theorem initialForward_tip_height_bound (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s (initialForwardChart p hi h0 hH (r n) (p t)).im) - atTop (𝓝 0) := - initialForward_boundary_height_zero p hi h0 hH hrt tendsto_const_nhds hzd - (slit_tip_not_mem_domain p t) - (fun n => initialForward_uniform_near p hi h0 hH hR hp (r n) (hzd n) - (by linarith [hp t])) - have hcap := ((initialCapacity_continuous p hi h0 hH).tendsto t).comp hrt - have hlim := (hheight.pow 2).add - ((hcap.sub_const (initialCapacity p hi h0 hH s)).const_mul 2) - have hineq (n : ℕ) : - (initialForwardChart p hi h0 hH s (p t)).im^2≤ - (initialForwardChart p hi h0 hH (r n) (p t)).im^2+ - 2*(initialCapacity p hi h0 hH (r n)-initialCapacity p hi h0 hH s) := by - obtain ⟨T,hrT,hTt⟩ := exists_between (hr n).2 - exact initialForward_height_capacity_increment p hi h0 hH (hr n).1 hrT.le - (hTt.trans ht) (point_after_prefix_mem_domain p hi hH hTt) - have hh := ge_of_tendsto hlim (Eventually.of_forall hineq) - simpa only [zero_pow two_ne_zero,zero_add] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialHeightCapacityIncrement +import OAI.Probability.SLE.Loewner.MovingForwardBoundaryHeight +import OAI.Probability.SLE.Geometry.ApproachIntervalEndpoint + +namespace OAI + +/-! A future point has initial-chart height controlled by the capacity +remaining before that point is swallowed. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +theorem initialForward_tip_height_bound (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s (initialForwardChart p hi h0 hH (r n) (p t)).im) + atTop (𝓝 0) := + initialForward_boundary_height_zero p hi h0 hH hrt tendsto_const_nhds hzd + (slit_tip_not_mem_domain p t) + (fun n => initialForward_uniform_near p hi h0 hH hR hp (r n) (hzd n) + (by linarith [hp t])) + have hcap := ((initialCapacity_continuous p hi h0 hH).tendsto t).comp hrt + have hlim := (hheight.pow 2).add + ((hcap.sub_const (initialCapacity p hi h0 hH s)).const_mul 2) + have hineq (n : ℕ) : + (initialForwardChart p hi h0 hH s (p t)).im^2≤ + (initialForwardChart p hi h0 hH (r n) (p t)).im^2+ + 2*(initialCapacity p hi h0 hH (r n)-initialCapacity p hi h0 hH s) := by + obtain ⟨T,hrT,hTt⟩ := exists_between (hr n).2 + exact initialForward_height_capacity_increment p hi h0 hH (hr n).1 hrT.le + (hTt.trans ht) (point_after_prefix_mem_domain p hi hH hTt) + have hh := ge_of_tendsto hlim (Eventually.of_forall hineq) + simpa only [zero_pow two_ne_zero,zero_add] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryArc.lean b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryArc.lean index 7af493c89a94040835a32c6cda20276cd583e7cd..402f61fc12b1c07c5c8d6a4261c2a3a873d51a10 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryArc.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryArc.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Loewner.InverseBoundaryCluster -import OAI.Probability.SLE.Geometry.FiniteEnergyArcExtension - -namespace OAI - -/-! Endpoint values of continuously extended inverse-chart arcs lie on -the physical frontier, independently of a global boundary extension. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma inverse_limit_mem_frontier_filter {A : Type*} {l : Filter A} [l.NeBot] - {D : Set ℂ} (hD : IsOpen D) {g f : ℂ → ℂ} (hg : ContinuousOn g D) - (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) - (hinv : ∀ w∈upperHalfPlane, g (f w)=w) - {w : A → ℂ} (hw : ∀ᶠ a in l, w a∈upperHalfPlane) {x : ℂ} - (hx : x∉upperHalfPlane) (hlim : Tendsto w l (𝓝 x)) {z : ℂ} - (hz : Tendsto (fun a => f (w a)) l (𝓝 z)) : z∈frontier D := by - have hcl : z∈closure D := mem_closure_of_tendsto hz (hw.mono fun _ ha => hfD ha) - have hnot : z∉D := by - intro hzin - have hg' := ((hg z hzin).continuousAt (hD.mem_nhds hzin)).tendsto.comp hz - have he : Tendsto w l (𝓝 (g z)) := hg'.congr' (hw.mono fun a ha => hinv (w a) ha) - have heq : g z=x := tendsto_nhds_unique he hlim - exact hx (heq ▸ hgH hzin) - exact ⟨hcl,by simpa only [hD.interior_eq] using hnot⟩ - -lemma inverse_arc_endpoint_mem_frontier {D : Set ℂ} (hD : IsOpen D) - {g f : ℂ → ℂ} (hg : ContinuousOn g D) - (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) - (hinv : ∀ w∈upperHalfPlane, g (f w)=w) - {w c : ℝ → ℂ} (hw : Continuous w) (hc : Continuous c) - {a b s : ℝ} (hab : a hwin h) hws - ((hw.tendsto s).mono_left nhdsWithin_le_nhds) - exact ((hc.tendsto s).mono_left nhdsWithin_le_nhds).congr' - (hmem.mono fun _ hθ => heq hθ) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseBoundaryCluster +import OAI.Probability.SLE.Geometry.FiniteEnergyArcExtension + +namespace OAI + +/-! Endpoint values of continuously extended inverse-chart arcs lie on +the physical frontier, independently of a global boundary extension. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma inverse_limit_mem_frontier_filter {A : Type*} {l : Filter A} [l.NeBot] + {D : Set ℂ} (hD : IsOpen D) {g f : ℂ → ℂ} (hg : ContinuousOn g D) + (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) + (hinv : ∀ w∈upperHalfPlane, g (f w)=w) + {w : A → ℂ} (hw : ∀ᶠ a in l, w a∈upperHalfPlane) {x : ℂ} + (hx : x∉upperHalfPlane) (hlim : Tendsto w l (𝓝 x)) {z : ℂ} + (hz : Tendsto (fun a => f (w a)) l (𝓝 z)) : z∈frontier D := by + have hcl : z∈closure D := mem_closure_of_tendsto hz (hw.mono fun _ ha => hfD ha) + have hnot : z∉D := by + intro hzin + have hg' := ((hg z hzin).continuousAt (hD.mem_nhds hzin)).tendsto.comp hz + have he : Tendsto w l (𝓝 (g z)) := hg'.congr' (hw.mono fun a ha => hinv (w a) ha) + have heq : g z=x := tendsto_nhds_unique he hlim + exact hx (heq ▸ hgH hzin) + exact ⟨hcl,by simpa only [hD.interior_eq] using hnot⟩ + +lemma inverse_arc_endpoint_mem_frontier {D : Set ℂ} (hD : IsOpen D) + {g f : ℂ → ℂ} (hg : ContinuousOn g D) + (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) + (hinv : ∀ w∈upperHalfPlane, g (f w)=w) + {w c : ℝ → ℂ} (hw : Continuous w) (hc : Continuous c) + {a b s : ℝ} (hab : a hwin h) hws + ((hw.tendsto s).mono_left nhdsWithin_le_nhds) + exact ((hc.tendsto s).mono_left nhdsWithin_le_nhds).congr' + (hmem.mono fun _ hθ => heq hθ) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryBound.lean b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryBound.lean index f38523653df134a51c82d7d9f4349de3eaf889b1..0864badb23e07a5b4f6b0a4a9646120dc056416f 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryBound.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.LoewnerCharts - -namespace OAI - -/-! Hydrodynamic normalization prevents an inverse chart from escaping to -infinity above a bounded set. This is independent of boundary extension. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma inverse_bound_of_hydrodynamic {D : Set ℂ} {g f : ℂ → ℂ} {c : ℂ} - (hmap : MapsTo f upperHalfPlane D) - (hinv : ∀ w∈upperHalfPlane, g (f w)=w) - (hlim : Tendsto (fun z : ℂ => z*(g z-z)) (cocompact ℂ ⊓ 𝓟 D) (𝓝 c)) - (R : ℝ) : ∃ M : ℝ, ∀ w∈upperHalfPlane, ‖w‖≤R → ‖f w‖≤M := by - have he := Metric.tendsto_nhds.mp hlim 1 zero_lt_one - rw [eventually_inf_principal,← Metric.cobounded_eq_cocompact] at he - obtain ⟨N,_,hN⟩ := Filter.hasBasis_cobounded_norm.eventually_iff.mp he - refine ⟨max (max N 1) (R+‖c‖+1),fun w hw hR => ?_⟩ - by_cases hz : ‖f w‖ ≤ max N 1 - · exact hz.trans (le_max_left _ _) - have hz' := lt_of_not_ge hz - have hzN : N≤‖f w‖ := (le_max_left _ _).trans hz'.le - have hz1 : 1≤‖f w‖ := (le_max_right _ _).trans hz'.le - have hsmall := hN hzN (hmap hw) - have hp : ‖f w‖*‖g (f w)-f w‖≤‖c‖+1 := by - have htri := norm_le_norm_sub_add (f w*(g (f w)-f w)) c - rw [norm_mul] at htri - have hd : ‖f w*(g (f w)-f w)-c‖<1 := by simpa only [dist_eq_norm] using hsmall - linarith - have hd : ‖g (f w)-f w‖≤‖c‖+1 := - (le_mul_of_one_le_left (norm_nonneg _) hz1).trans hp - have htri := norm_le_norm_sub_add (f w) (g (f w)) - rw [hinv w hw] at htri hd - rw [norm_sub_rev w (f w)] at hd - exact (show ‖f w‖≤R+‖c‖+1 by linarith).trans (le_max_right _ _) - -lemma IsCapacityTwoWitness.inverse_bounded_over_ball - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (R : ℝ) : - ∃ M : ℝ, ∀ w∈upperHalfPlane, ‖w‖≤R → ‖F t w‖≤M := - inverse_bound_of_hydrodynamic (hw.2.2.2.2.2.2.1 t) - (hw.2.2.2.2.2.2.2.2.1 t) (hw.2.2.2.2.2.2.2.2.2.2.2.1 t) R - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.LoewnerCharts + +namespace OAI + +/-! Hydrodynamic normalization prevents an inverse chart from escaping to +infinity above a bounded set. This is independent of boundary extension. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma inverse_bound_of_hydrodynamic {D : Set ℂ} {g f : ℂ → ℂ} {c : ℂ} + (hmap : MapsTo f upperHalfPlane D) + (hinv : ∀ w∈upperHalfPlane, g (f w)=w) + (hlim : Tendsto (fun z : ℂ => z*(g z-z)) (cocompact ℂ ⊓ 𝓟 D) (𝓝 c)) + (R : ℝ) : ∃ M : ℝ, ∀ w∈upperHalfPlane, ‖w‖≤R → ‖f w‖≤M := by + have he := Metric.tendsto_nhds.mp hlim 1 zero_lt_one + rw [eventually_inf_principal,← Metric.cobounded_eq_cocompact] at he + obtain ⟨N,_,hN⟩ := Filter.hasBasis_cobounded_norm.eventually_iff.mp he + refine ⟨max (max N 1) (R+‖c‖+1),fun w hw hR => ?_⟩ + by_cases hz : ‖f w‖ ≤ max N 1 + · exact hz.trans (le_max_left _ _) + have hz' := lt_of_not_ge hz + have hzN : N≤‖f w‖ := (le_max_left _ _).trans hz'.le + have hz1 : 1≤‖f w‖ := (le_max_right _ _).trans hz'.le + have hsmall := hN hzN (hmap hw) + have hp : ‖f w‖*‖g (f w)-f w‖≤‖c‖+1 := by + have htri := norm_le_norm_sub_add (f w*(g (f w)-f w)) c + rw [norm_mul] at htri + have hd : ‖f w*(g (f w)-f w)-c‖<1 := by simpa only [dist_eq_norm] using hsmall + linarith + have hd : ‖g (f w)-f w‖≤‖c‖+1 := + (le_mul_of_one_le_left (norm_nonneg _) hz1).trans hp + have htri := norm_le_norm_sub_add (f w) (g (f w)) + rw [hinv w hw] at htri hd + rw [norm_sub_rev w (f w)] at hd + exact (show ‖f w‖≤R+‖c‖+1 by linarith).trans (le_max_right _ _) + +lemma IsCapacityTwoWitness.inverse_bounded_over_ball + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (R : ℝ) : + ∃ M : ℝ, ∀ w∈upperHalfPlane, ‖w‖≤R → ‖F t w‖≤M := + inverse_bound_of_hydrodynamic (hw.2.2.2.2.2.2.1 t) + (hw.2.2.2.2.2.2.2.2.1 t) (hw.2.2.2.2.2.2.2.2.2.2.2.1 t) R + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryClosingPath.lean b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryClosingPath.lean index b00dd2b6fc1f9e312cd016aec89e0835f3ca1c9a..72d2394c57ba984e95f2b067826d08cf15d36719 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryClosingPath.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryClosingPath.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Loewner.InverseSmallSemicircles -import OAI.Probability.SLE.Loewner.TraceBoundaryBarrier - -namespace OAI - -/-! A small image semicircle can be closed by a small path outside the -surviving domain. The closing path is supplied by the actual finite trace -and real boundary, not by a further topological input. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.exists_small_inverse_closing_path - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x Q ε : ℝ) (hε : 0<ε) : - ∃ (v : ℝ) (c : ℝ → ℂ) (p : Path (c Real.pi) (c 0)), v (disjoint_left.mp (traceBoundaryBarrier_disjoint_domain γ t R)) (hpB s) hD⟩ - have htri := dist_triangle (p s) (c Real.pi) (c 0) - have hps := hp s - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseSmallSemicircles +import OAI.Probability.SLE.Loewner.TraceBoundaryBarrier + +namespace OAI + +/-! A small image semicircle can be closed by a small path outside the +surviving domain. The closing path is supplied by the actual finite trace +and real boundary, not by a further topological input. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma IsCapacityTwoWitness.exists_small_inverse_closing_path + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x Q ε : ℝ) (hε : 0<ε) : + ∃ (v : ℝ) (c : ℝ → ℂ) (p : Path (c Real.pi) (c 0)), v (disjoint_left.mp (traceBoundaryBarrier_disjoint_domain γ t R)) (hpB s) hD⟩ + have htri := dist_triangle (p s) (c Real.pi) (c 0) + have hps := hp s + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryCluster.lean b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryCluster.lean index d30e68fbeb803af5e9f508dd7ac07181adcddf68..3843652886954416713db88cda88cbe484271f4b 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryCluster.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryCluster.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Loewner.InverseBoundaryBound - -namespace OAI - -/-! Finite inverse-chart cluster points above the real axis lie on the - physical domain boundary. No boundary extension is assumed. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma inverse_limit_mem_frontier {D : Set ℂ} (hD : IsOpen D) - {g f : ℂ → ℂ} (hg : ContinuousOn g D) - (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) - (hinv : ∀ w∈upperHalfPlane, g (f w)=w) - {w : ℕ → ℂ} (hw : ∀ n, w n∈upperHalfPlane) {x : ℝ} - (hlim : Tendsto w atTop (𝓝 (x:ℂ))) {z : ℂ} - (hz : Tendsto (fun n => f (w n)) atTop (𝓝 z)) : z∈frontier D := by - have hcl : z∈closure D := mem_closure_of_tendsto hz (Eventually.of_forall fun n => hfD (hw n)) - have hnot : z∉D := by - intro hzin - have hg' := ((hg z hzin).continuousAt (hD.mem_nhds hzin)).tendsto.comp hz - have he : Tendsto w atTop (𝓝 (g z)) := by - convert hg' using 1 - funext n - exact (hinv (w n) (hw n)).symm - have heq : g z=(x:ℂ) := tendsto_nhds_unique he hlim - have him := hgH hzin - rw [heq] at him - change 0 < (x:ℂ).im at him - norm_num at him - exact ⟨hcl,by simpa only [hD.interior_eq] using hnot⟩ - -lemma IsCapacityTwoWitness.inverse_boundary_subsequence - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) - (w : ℕ → ℂ) (hw : ∀ n, w n∈upperHalfPlane) (x : ℝ) - (hlim : Tendsto w atTop (𝓝 (x:ℂ))) : - ∃ (φ : ℕ → ℕ) (z : ℂ), StrictMono φ ∧ z∈frontier (survivingDomain γ t) ∧ - Tendsto (fun n => F t (w (φ n))) atTop (𝓝 z) := by - obtain ⟨R,hR⟩ := (Metric.isBounded_range_of_tendsto w hlim).exists_norm_le - obtain ⟨M,hM⟩ := h.inverse_bounded_over_ball t R - obtain ⟨z,_,φ,hφ,hconv⟩ := (isCompact_closedBall (0:ℂ) M).tendsto_subseq (fun n => by - rw [mem_closedBall,dist_zero_right] - exact hM (w n) (hw n) (hR _ (mem_range_self n))) - refine ⟨φ,z,hφ,?_,hconv⟩ - exact inverse_limit_mem_frontier (isOpen_survivingDomain h.1 t) - ((h.2.2.2.1 t).continuousOn) (h.2.2.2.2.2.1 t) (h.2.2.2.2.2.2.1 t) - (h.2.2.2.2.2.2.2.2.1 t) (fun n => hw (φ n)) (hlim.comp hφ.tendsto_atTop) hconv - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseBoundaryBound + +namespace OAI + +/-! Finite inverse-chart cluster points above the real axis lie on the + physical domain boundary. No boundary extension is assumed. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma inverse_limit_mem_frontier {D : Set ℂ} (hD : IsOpen D) + {g f : ℂ → ℂ} (hg : ContinuousOn g D) + (hgH : MapsTo g D upperHalfPlane) (hfD : MapsTo f upperHalfPlane D) + (hinv : ∀ w∈upperHalfPlane, g (f w)=w) + {w : ℕ → ℂ} (hw : ∀ n, w n∈upperHalfPlane) {x : ℝ} + (hlim : Tendsto w atTop (𝓝 (x:ℂ))) {z : ℂ} + (hz : Tendsto (fun n => f (w n)) atTop (𝓝 z)) : z∈frontier D := by + have hcl : z∈closure D := mem_closure_of_tendsto hz (Eventually.of_forall fun n => hfD (hw n)) + have hnot : z∉D := by + intro hzin + have hg' := ((hg z hzin).continuousAt (hD.mem_nhds hzin)).tendsto.comp hz + have he : Tendsto w atTop (𝓝 (g z)) := by + convert hg' using 1 + funext n + exact (hinv (w n) (hw n)).symm + have heq : g z=(x:ℂ) := tendsto_nhds_unique he hlim + have him := hgH hzin + rw [heq] at him + change 0 < (x:ℂ).im at him + norm_num at him + exact ⟨hcl,by simpa only [hD.interior_eq] using hnot⟩ + +lemma IsCapacityTwoWitness.inverse_boundary_subsequence + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) + (w : ℕ → ℂ) (hw : ∀ n, w n∈upperHalfPlane) (x : ℝ) + (hlim : Tendsto w atTop (𝓝 (x:ℂ))) : + ∃ (φ : ℕ → ℕ) (z : ℂ), StrictMono φ ∧ z∈frontier (survivingDomain γ t) ∧ + Tendsto (fun n => F t (w (φ n))) atTop (𝓝 z) := by + obtain ⟨R,hR⟩ := (Metric.isBounded_range_of_tendsto w hlim).exists_norm_le + obtain ⟨M,hM⟩ := h.inverse_bounded_over_ball t R + obtain ⟨z,_,φ,hφ,hconv⟩ := (isCompact_closedBall (0:ℂ) M).tendsto_subseq (fun n => by + rw [mem_closedBall,dist_zero_right] + exact hM (w n) (hw n) (hR _ (mem_range_self n))) + refine ⟨φ,z,hφ,?_,hconv⟩ + exact inverse_limit_mem_frontier (isOpen_survivingDomain h.1 t) + ((h.2.2.2.1 t).continuousOn) (h.2.2.2.2.2.1 t) (h.2.2.2.2.2.2.1 t) + (h.2.2.2.2.2.2.2.2.1 t) (fun n => hw (φ n)) (hlim.comp hφ.tendsto_atTop) hconv + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryShrink.lean b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryShrink.lean index 1a8866ad47dec2d78af84c01e9109066dd6f76b7..d2b0939d6eb4d531a840525ec922304b4287285a 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseBoundaryShrink.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseBoundaryShrink.lean @@ -1,106 +1,106 @@ -import OAI.Probability.SLE.Geometry.PolarBoundarySeparation -import OAI.Probability.SLE.Loewner.InverseBoundaryClosingPath - -namespace OAI - -/-! Small closed image semicircles force the images of their entire -half-disks to be small. The Loewner height inequality supplies the exterior -endpoint of each radial crossing. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.inverse_im_ge - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) {w : ℂ} (hwH : w∈upperHalfPlane) : - w.im≤(F t w).im := by - have hz := hw.2.2.2.2.2.2.1 t hwH - have h := hw.imag_ge_terminal t (F t w) hz 0 ⟨le_rfl,t.coe_nonneg⟩ - simpa only [Real.toNNReal_zero,hw.2.2.2.2.2.2.2.2.1 t w hwH,hw.initial_value (F t w) hz.1.1] using h - -lemma IsCapacityTwoWitness.inverse_arc_traps_radial_point - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) - (x r ρ θ : ℝ) (hρ : 0<ρ) (hρr : ρ him.ne' (Complex.arg_eq_zero_iff.mp h.symm).2) - have haπ : z.arg (hqb u).2 - · exact hε - · exact hcb - · exact fun u => (hqb u).1 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.PolarBoundarySeparation +import OAI.Probability.SLE.Loewner.InverseBoundaryClosingPath + +namespace OAI + +/-! Small closed image semicircles force the images of their entire +half-disks to be small. The Loewner height inequality supplies the exterior +endpoint of each radial crossing. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma IsCapacityTwoWitness.inverse_im_ge + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) {w : ℂ} (hwH : w∈upperHalfPlane) : + w.im≤(F t w).im := by + have hz := hw.2.2.2.2.2.2.1 t hwH + have h := hw.imag_ge_terminal t (F t w) hz 0 ⟨le_rfl,t.coe_nonneg⟩ + simpa only [Real.toNNReal_zero,hw.2.2.2.2.2.2.2.2.1 t w hwH,hw.initial_value (F t w) hz.1.1] using h + +lemma IsCapacityTwoWitness.inverse_arc_traps_radial_point + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) + (x r ρ θ : ℝ) (hρ : 0<ρ) (hρr : ρ him.ne' (Complex.arg_eq_zero_iff.mp h.symm).2) + have haπ : z.arg (hqb u).2 + · exact hε + · exact hcb + · exact fun u => (hqb u).1 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseCircleArea.lean b/lean/OAI/Probability/SLE/Loewner/InverseCircleArea.lean index 900a0d52b23ba06a9185081c42f3af818f30b83a..21b3aae807028f8d2d9d765350db67f3682c7e0c 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseCircleArea.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseCircleArea.lean @@ -1,50 +1,50 @@ -import OAI.Probability.SLE.Geometry.HolomorphicRadialFlux - -namespace OAI - -/-! Inversion of circle averages and the elementary signed-area identity -for a map with leading term z and a holomorphic inverse-radius remainder. -/ -noncomputable section -open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real -open scoped Topology Laplacian ComplexConjugate - -namespace SLEExactGauge - -lemma circleAverage_inv_radius {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] - (f : ℂ → E) (r : ℝ) : - circleAverage (fun z => f z⁻¹) 0 r=circleAverage f 0 r⁻¹ := by - calc - _ = circleAverage (fun z : ℂ => f (((r:ℂ)*z+0)⁻¹)) 0 1 := - circleAverage_eq_circleAverage_zero_one - _ = circleAverage (fun z : ℂ => f (((r⁻¹:ℝ):ℂ)*z⁻¹)) 0 1 := by - congr 1 - ext z - simp only [add_zero,mul_inv_rev,Complex.ofReal_inv,mul_comm] - _ = circleAverage (fun z : ℂ => f (((r⁻¹:ℝ):ℂ)*z)) 0 1 := by - exact circleAverage_zero_one_congr_inv (f := fun z : ℂ => f (((r⁻¹:ℝ):ℂ)*z)) - _ = _ := by - simpa only [add_zero] using - (circleAverage_eq_circleAverage_zero_one (f := f) (c := (0:ℂ)) (R := r⁻¹)).symm - -lemma inverse_area_density_identity (z c cp : ℂ) (hz : z≠0) : - 2*((z-z⁻¹*cp)*conj (z+c)).re= - 2*‖z‖^2+2*‖z‖^2*(z⁻¹*c-z⁻¹^2*cp).re- - 2*(z⁻¹*cp*conj c).re := by - have hh : (z-z⁻¹*cp)*conj (z+c)= - ((‖z‖^2:ℝ):ℂ)+((‖z‖^2:ℝ):ℂ)*(conj (z⁻¹*c)-z⁻¹^2*cp)-z⁻¹*cp*conj c := by - have hnorm : ((‖z‖^2:ℝ):ℂ)=z*conj z := by - simpa only [Complex.ofReal_pow] using (Complex.mul_conj' z).symm - rw [hnorm] - simp only [map_add,map_mul,map_inv₀] - field_simp [hz] - ring - rw [hh] - simp only [Complex.add_re,Complex.sub_re,Complex.mul_re, - Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,Complex.conj_re] - ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HolomorphicRadialFlux + +namespace OAI + +/-! Inversion of circle averages and the elementary signed-area identity +for a map with leading term z and a holomorphic inverse-radius remainder. -/ +noncomputable section +open Set Filter Metric Function Complex MeasureTheory InnerProductSpace Real +open scoped Topology Laplacian ComplexConjugate + +namespace SLEExactGauge + +lemma circleAverage_inv_radius {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] + (f : ℂ → E) (r : ℝ) : + circleAverage (fun z => f z⁻¹) 0 r=circleAverage f 0 r⁻¹ := by + calc + _ = circleAverage (fun z : ℂ => f (((r:ℂ)*z+0)⁻¹)) 0 1 := + circleAverage_eq_circleAverage_zero_one + _ = circleAverage (fun z : ℂ => f (((r⁻¹:ℝ):ℂ)*z⁻¹)) 0 1 := by + congr 1 + ext z + simp only [add_zero,mul_inv_rev,Complex.ofReal_inv,mul_comm] + _ = circleAverage (fun z : ℂ => f (((r⁻¹:ℝ):ℂ)*z)) 0 1 := by + exact circleAverage_zero_one_congr_inv (f := fun z : ℂ => f (((r⁻¹:ℝ):ℂ)*z)) + _ = _ := by + simpa only [add_zero] using + (circleAverage_eq_circleAverage_zero_one (f := f) (c := (0:ℂ)) (R := r⁻¹)).symm + +lemma inverse_area_density_identity (z c cp : ℂ) (hz : z≠0) : + 2*((z-z⁻¹*cp)*conj (z+c)).re= + 2*‖z‖^2+2*‖z‖^2*(z⁻¹*c-z⁻¹^2*cp).re- + 2*(z⁻¹*cp*conj c).re := by + have hh : (z-z⁻¹*cp)*conj (z+c)= + ((‖z‖^2:ℝ):ℂ)+((‖z‖^2:ℝ):ℂ)*(conj (z⁻¹*c)-z⁻¹^2*cp)-z⁻¹*cp*conj c := by + have hnorm : ((‖z‖^2:ℝ):ℂ)=z*conj z := by + simpa only [Complex.ofReal_pow] using (Complex.mul_conj' z).symm + rw [hnorm] + simp only [map_add,map_mul,map_inv₀] + field_simp [hz] + ring + rw [hh] + simp only [Complex.add_re,Complex.sub_re,Complex.mul_re, + Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,Complex.conj_re] + ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseDirichletEnergy.lean b/lean/OAI/Probability/SLE/Loewner/InverseDirichletEnergy.lean index 02633e66550c05152caf6654257a20708d7d6f61..0f02c0b48d3f32323d871093b52d41ca13e5b4c5 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseDirichletEnergy.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseDirichletEnergy.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Loewner.InverseBoundaryBound -import OAI.Probability.SLE.Gauge.ConformalSausageTransfer - -namespace OAI - -/-! Finite local Dirichlet energy for the inverse Loewner chart follows -from injectivity, the area formula and hydrodynamic boundedness. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.inverse_local_dirichlet_finite - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x : ℂ) (R : ℝ) : - (∫⁻ z in upperHalfPlane ∩ ball x R, ENNReal.ofReal (‖deriv (F t) z‖^2))<∞ := by - let S := upperHalfPlane ∩ ball x R - have hS : MeasurableSet S := isOpen_upperHalfPlane.measurableSet.inter measurableSet_ball - have hder (z : ℂ) (hz : z∈S) : HasDerivAt (F t) (deriv (F t) z) z := - ((hw.2.2.2.2.1 t).differentiableAt (isOpen_upperHalfPlane.mem_nhds hz.1)).hasDerivAt - have hinj : InjOn (F t) S := by - intro z hz w hw' he - have h1 := hw.2.2.2.2.2.2.2.2.1 t z hz.1 - have h2 := hw.2.2.2.2.2.2.2.2.1 t w hw'.1 - rw [he,h2] at h1 - exact h1.symm - have harea : (∫⁻ z in S, ENNReal.ofReal (‖deriv (F t) z‖^2)) = volume (F t '' S) := by - have he := lintegral_abs_det_fderiv_eq_addHaar_image volume hS - (fun z hz => ((hder z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hinj - simpa only [complex_derivative_real_det,abs_pow,abs_norm] using he - obtain ⟨M,hM⟩ := hw.inverse_bounded_over_ball t (R+‖x‖) - have himage : F t '' S ⊆ closedBall (0:ℂ) M := by - rintro _ ⟨z,hz,rfl⟩ - rw [mem_closedBall,dist_zero_right] - apply hM z hz.1 - have hnear : ‖z-x‖ ((hder z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hinj + simpa only [complex_derivative_real_det,abs_pow,abs_norm] using he + obtain ⟨M,hM⟩ := hw.inverse_bounded_over_ball t (R+‖x‖) + have himage : F t '' S ⊆ closedBall (0:ℂ) M := by + rintro _ ⟨z,hz,rfl⟩ + rw [mem_closedBall,dist_zero_right] + apply hM z hz.1 + have hnear : ‖z-x‖ ∫ y in Ioc 0 h, - ‖deriv (initialInverseChart p hi h0 hH s) ((x:ℂ)+(y:ℂ)*Complex.I)‖^2) - (Ioo a b) ∧ - ∀ᵐ x : ℝ ∂volume.restrict (Ioo a b), - MemLp (fun y : ℝ => deriv (initialInverseChart p hi h0 hH s) - ((x:ℂ)+(y:ℂ)*Complex.I)) 2 (volume.restrict (Ioc 0 h)) := by - let F := initialInverseChart p hi h0 hH s - let J := Ioo a b - let K := Ioc 0 h - let S := upperHalfPlane ∩ ball (0:ℂ) (L+2) - let T := {z : ℂ | z.re∈J ∧ z.im∈K} - let e := Complex.measurableEquivRealProd.symm - have he : MeasurePreserving e volume volume := Complex.volume_preserving_equiv_real_prod.symm _ - have hpre : e ⁻¹' T=J ×ˢ K := by ext z; simp [e,T,J,K] - have hTS : T⊆S := by - intro z hz - refine ⟨hz.2.1,mem_ball_zero_iff.mpr ?_⟩ - have hb := norm_le_abs_re_add_abs_im z - rw [abs_of_pos hz.2.1] at hb - have hx' := hx z.re hz.1 - have hy' : z.im≤1 := hz.2.2.trans hh1 - linarith - have hiS : IntegrableOn (fun z => ‖deriv F z‖^2) S := - initialInverse_energy_integrable p hi h0 hH hR hp s (L+2) - have hiT := hiS.mono_set hTS - have hip : Integrable (fun z : ℝ × ℝ => ‖deriv F (e z)‖^2) - ((volume.restrict J).prod (volume.restrict K)) := by - rw [Measure.prod_restrict] - exact hpre ▸ (he.integrableOn_comp_preimage e.measurableEmbedding).mpr hiT - have heq : (∫ x in J, ∫ y in K, ‖deriv F (e (x,y))‖^2)= - ∫ z in T, ‖deriv F z‖^2 := by - rw [← integral_prod _ hip,Measure.prod_restrict,←hpre] - simpa only [Measure.volume_eq_prod] using - he.setIntegral_preimage_emb e.measurableEmbedding (fun z => ‖deriv F z‖^2) T - have hbound : (∫ x in J, ∫ y in K, ‖deriv F (e (x,y))‖^2)≤ - (volume (closedBall (0:ℂ) (L+2+3*R))).toReal := by - rw [heq] - exact (setIntegral_mono_set hiS (ae_of_all _ (fun z => sq_nonneg _)) hTS.eventuallySubset).trans - (initialInverse_energy_integral_le p hi h0 hH hR hp s (L+2)) - have hsections : ∀ᵐ x : ℝ ∂volume.restrict J, - MemLp (fun y : ℝ => deriv F (e (x,y))) 2 (volume.restrict K) := by - filter_upwards [hip.prod_right_ae] with x hxi - have hm : AEStronglyMeasurable (fun y : ℝ => deriv F (e (x,y))) (volume.restrict K) := - ((measurable_deriv F).comp (e.measurable.comp measurable_prodMk_left)).aestronglyMeasurable - exact (memLp_two_iff_integrable_sq_norm hm).mpr hxi - have heval (x y : ℝ) : e (x,y)=(x:ℂ)+(y:ℂ)*Complex.I := by - apply Complex.ext <;> simp [e] - constructor - · simpa only [heval] using hbound - · constructor - · simpa only [heval,F,J,K,IntegrableOn] using hip.integral_prod_left - · simpa only [heval] using hsections - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseVerticalDifferentiation + +namespace OAI + +/-! Fubini transfers the common inverse-chart Dirichlet bound to +vertical sections over a bounded real interval. -/ +noncomputable section +open Set Filter Function Complex Metric MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +theorem initialInverse_rectangle_energy (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 ∫ y in Ioc 0 h, + ‖deriv (initialInverseChart p hi h0 hH s) ((x:ℂ)+(y:ℂ)*Complex.I)‖^2) + (Ioo a b) ∧ + ∀ᵐ x : ℝ ∂volume.restrict (Ioo a b), + MemLp (fun y : ℝ => deriv (initialInverseChart p hi h0 hH s) + ((x:ℂ)+(y:ℂ)*Complex.I)) 2 (volume.restrict (Ioc 0 h)) := by + let F := initialInverseChart p hi h0 hH s + let J := Ioo a b + let K := Ioc 0 h + let S := upperHalfPlane ∩ ball (0:ℂ) (L+2) + let T := {z : ℂ | z.re∈J ∧ z.im∈K} + let e := Complex.measurableEquivRealProd.symm + have he : MeasurePreserving e volume volume := Complex.volume_preserving_equiv_real_prod.symm _ + have hpre : e ⁻¹' T=J ×ˢ K := by ext z; simp [e,T,J,K] + have hTS : T⊆S := by + intro z hz + refine ⟨hz.2.1,mem_ball_zero_iff.mpr ?_⟩ + have hb := norm_le_abs_re_add_abs_im z + rw [abs_of_pos hz.2.1] at hb + have hx' := hx z.re hz.1 + have hy' : z.im≤1 := hz.2.2.trans hh1 + linarith + have hiS : IntegrableOn (fun z => ‖deriv F z‖^2) S := + initialInverse_energy_integrable p hi h0 hH hR hp s (L+2) + have hiT := hiS.mono_set hTS + have hip : Integrable (fun z : ℝ × ℝ => ‖deriv F (e z)‖^2) + ((volume.restrict J).prod (volume.restrict K)) := by + rw [Measure.prod_restrict] + exact hpre ▸ (he.integrableOn_comp_preimage e.measurableEmbedding).mpr hiT + have heq : (∫ x in J, ∫ y in K, ‖deriv F (e (x,y))‖^2)= + ∫ z in T, ‖deriv F z‖^2 := by + rw [← integral_prod _ hip,Measure.prod_restrict,←hpre] + simpa only [Measure.volume_eq_prod] using + he.setIntegral_preimage_emb e.measurableEmbedding (fun z => ‖deriv F z‖^2) T + have hbound : (∫ x in J, ∫ y in K, ‖deriv F (e (x,y))‖^2)≤ + (volume (closedBall (0:ℂ) (L+2+3*R))).toReal := by + rw [heq] + exact (setIntegral_mono_set hiS (ae_of_all _ (fun z => sq_nonneg _)) hTS.eventuallySubset).trans + (initialInverse_energy_integral_le p hi h0 hH hR hp s (L+2)) + have hsections : ∀ᵐ x : ℝ ∂volume.restrict J, + MemLp (fun y : ℝ => deriv F (e (x,y))) 2 (volume.restrict K) := by + filter_upwards [hip.prod_right_ae] with x hxi + have hm : AEStronglyMeasurable (fun y : ℝ => deriv F (e (x,y))) (volume.restrict K) := + ((measurable_deriv F).comp (e.measurable.comp measurable_prodMk_left)).aestronglyMeasurable + exact (memLp_two_iff_integrable_sq_norm hm).mpr hxi + have heval (x y : ℝ) : e (x,y)=(x:ℂ)+(y:ℂ)*Complex.I := by + apply Complex.ext <;> simp [e] + constructor + · simpa only [heval] using hbound + · constructor + · simpa only [heval,F,J,K,IntegrableOn] using hip.integral_prod_left + · simpa only [heval] using hsections + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseSmallSemicircles.lean b/lean/OAI/Probability/SLE/Loewner/InverseSmallSemicircles.lean index 70d0ae110823d444975168dce4889b47647a967c..48735af7d36e65a1b7fd5f5b3f9f7222193cb25f 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseSmallSemicircles.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseSmallSemicircles.lean @@ -1,90 +1,90 @@ -import OAI.Probability.SLE.Loewner.InverseStripEnergy -import OAI.Probability.SLE.Geometry.ShrinkingStripEnergy -import OAI.Probability.SLE.Loewner.InverseBoundaryArc - -namespace OAI - -/-! Arbitrarily small closed image semicircles at each real boundary point. -The endpoint values are obtained from finite energy, not from an assumed -continuous extension of the inverse map. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.inverse_log_holomorphic - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x : ℝ) : - DifferentiableOn ℂ (fun z => F t ((x:ℂ)+Complex.exp z)) - {z | 0 F t ((x:ℂ)+Complex.exp z) - have hf := hw.inverse_log_holomorphic t x - obtain ⟨v,hv,hd,hsmall⟩ := exists_small_holomorphic_section hf - (hw.inverse_strip_dirichlet_finite t x 0) Q ε hε - let w : ℝ → ℂ := fun θ => (x:ℂ)+Complex.exp ((v:ℂ)+(θ:ℂ)*Complex.I) - have hS : IsOpen {z : ℂ | 0 F t (w θ)) θ := by - change DifferentiableAt ℝ (f ∘ fun θ : ℝ => (v:ℂ)+(θ:ℂ)*Complex.I) θ - have hz : (v:ℂ)+(θ:ℂ)*Complex.I∈{z : ℂ | 0 - simp [w,upperHalfPlane,Complex.exp_add,Complex.exp_ofReal_mul_I,← Complex.ofReal_exp] - have hend (s : ℝ) (hs : s∈Icc 0 Real.pi) (hs' : s=0 ∨ s=Real.pi) : - c s∈frontier (survivingDomain γ t) := - inverse_arc_endpoint_mem_frontier (isOpen_survivingDomain hw.1 t) - (hw.2.2.2.1 t).continuousOn (hw.2.2.2.2.2.1 t) (hw.2.2.2.2.2.2.1 t) - (hw.2.2.2.2.2.2.2.2.1 t) hwc hc Real.pi_pos hs hwin (hwend s hs') he - refine ⟨v,c,hv,hc,he,?_,hend 0 ⟨le_rfl,Real.pi_pos.le⟩ (Or.inl rfl), - hend Real.pi ⟨Real.pi_pos.le,le_rfl⟩ (Or.inr rfl),?_,?_⟩ - · apply continuous_angular_oscillation_on_endpoints hc ε - intro θ hθ φ hφ - rw [he hθ,he hφ] - exact (hsmall θ hθ φ hφ).le - · intro θ hθ - rw [he hθ] - exact hw.2.2.2.2.2.2.1 t (hwin hθ) - · intro θ hθ φ hφ heq - rw [he hθ,he hφ] at heq - change F t (w θ)=F t (w φ) at heq - have hwθ := hw.2.2.2.2.2.2.2.2.1 t (w θ) (hwin hθ) - have hwφ := hw.2.2.2.2.2.2.2.2.1 t (w φ) (hwin hφ) - have hww : w θ=w φ := by rw [heq,hwφ] at hwθ; exact hwθ.symm - have harg := exp_injective_strip - (x₁ := (v:ℂ)+(θ:ℂ)*Complex.I) (x₂ := (v:ℂ)+(φ:ℂ)*Complex.I) - (by simpa using hθ) (by simpa using hφ) (add_left_cancel hww) - have him := congrArg Complex.im harg - simpa using him - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseStripEnergy +import OAI.Probability.SLE.Geometry.ShrinkingStripEnergy +import OAI.Probability.SLE.Loewner.InverseBoundaryArc + +namespace OAI + +/-! Arbitrarily small closed image semicircles at each real boundary point. +The endpoint values are obtained from finite energy, not from an assumed +continuous extension of the inverse map. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma IsCapacityTwoWitness.inverse_log_holomorphic + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x : ℝ) : + DifferentiableOn ℂ (fun z => F t ((x:ℂ)+Complex.exp z)) + {z | 0 F t ((x:ℂ)+Complex.exp z) + have hf := hw.inverse_log_holomorphic t x + obtain ⟨v,hv,hd,hsmall⟩ := exists_small_holomorphic_section hf + (hw.inverse_strip_dirichlet_finite t x 0) Q ε hε + let w : ℝ → ℂ := fun θ => (x:ℂ)+Complex.exp ((v:ℂ)+(θ:ℂ)*Complex.I) + have hS : IsOpen {z : ℂ | 0 F t (w θ)) θ := by + change DifferentiableAt ℝ (f ∘ fun θ : ℝ => (v:ℂ)+(θ:ℂ)*Complex.I) θ + have hz : (v:ℂ)+(θ:ℂ)*Complex.I∈{z : ℂ | 0 + simp [w,upperHalfPlane,Complex.exp_add,Complex.exp_ofReal_mul_I,← Complex.ofReal_exp] + have hend (s : ℝ) (hs : s∈Icc 0 Real.pi) (hs' : s=0 ∨ s=Real.pi) : + c s∈frontier (survivingDomain γ t) := + inverse_arc_endpoint_mem_frontier (isOpen_survivingDomain hw.1 t) + (hw.2.2.2.1 t).continuousOn (hw.2.2.2.2.2.1 t) (hw.2.2.2.2.2.2.1 t) + (hw.2.2.2.2.2.2.2.2.1 t) hwc hc Real.pi_pos hs hwin (hwend s hs') he + refine ⟨v,c,hv,hc,he,?_,hend 0 ⟨le_rfl,Real.pi_pos.le⟩ (Or.inl rfl), + hend Real.pi ⟨Real.pi_pos.le,le_rfl⟩ (Or.inr rfl),?_,?_⟩ + · apply continuous_angular_oscillation_on_endpoints hc ε + intro θ hθ φ hφ + rw [he hθ,he hφ] + exact (hsmall θ hθ φ hφ).le + · intro θ hθ + rw [he hθ] + exact hw.2.2.2.2.2.2.1 t (hwin hθ) + · intro θ hθ φ hφ heq + rw [he hθ,he hφ] at heq + change F t (w θ)=F t (w φ) at heq + have hwθ := hw.2.2.2.2.2.2.2.2.1 t (w θ) (hwin hθ) + have hwφ := hw.2.2.2.2.2.2.2.2.1 t (w φ) (hwin hφ) + have hww : w θ=w φ := by rw [heq,hwφ] at hwθ; exact hwθ.symm + have harg := exp_injective_strip + (x₁ := (v:ℂ)+(θ:ℂ)*Complex.I) (x₂ := (v:ℂ)+(φ:ℂ)*Complex.I) + (by simpa using hθ) (by simpa using hφ) (add_left_cancel hww) + have him := congrArg Complex.im harg + simpa using him + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseStripEnergy.lean b/lean/OAI/Probability/SLE/Loewner/InverseStripEnergy.lean index 9258a6539df7e4f54a85e4ffb99e9000d70b64bc..2ace6f846e1f2213297b3629927e38578addf97e 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseStripEnergy.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseStripEnergy.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Loewner.InverseDirichletEnergy -import OAI.Probability.SLE.DyadicStrips - -namespace OAI - -/-! Finite energy in logarithmic boundary coordinates. The exponential -chart is injective on the half-plane strip, so no new growth input is used. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.inverse_strip_dirichlet_finite - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x q : ℝ) : - (∫⁻ z in {z : ℂ | z.re F t ((x:ℂ)+Complex.exp w)) z‖^2))<∞ := by - let S : Set ℂ := {z | z.re (x:ℂ)+Complex.exp z - let f : ℂ → ℂ := fun z => F t (A z) - have hS : IsOpen S := (isOpen_lt Complex.continuous_re continuous_const).inter - ((isOpen_lt continuous_const Complex.continuous_im).inter - (isOpen_lt Complex.continuous_im continuous_const)) - have hAH : MapsTo A S upperHalfPlane := by - intro z hz - change 0<((x:ℂ)+Complex.exp z).im - have he : 0<(Complex.exp z).im := exp_strip_halfplane hz.2 - simpa only [Complex.add_im,Complex.ofReal_im,zero_add] using he - have hf : DifferentiableOn ℂ f S := (hw.2.2.2.2.1 t).comp - ((differentiable_const (x:ℂ)).add Complex.differentiable_exp).differentiableOn hAH - have hi : InjOn f S := by - intro z hz w hw' he - have hAz := hw.2.2.2.2.2.2.2.2.1 t (A z) (hAH hz) - have hAw := hw.2.2.2.2.2.2.2.2.1 t (A w) (hAH hw') - have heA : A z=A w := by - rw [show F t (A z)=F t (A w) from he,hAw] at hAz - exact hAz.symm - exact exp_injective_strip hz.2 hw'.2 (add_left_cancel heA) - have harea : (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))=volume (f '' S) := by - have hd (z : ℂ) (hz : z∈S) := - (hf.differentiableAt (hS.mem_nhds hz)).hasDerivAt - have he := lintegral_abs_det_fderiv_eq_addHaar_image volume hS.measurableSet - (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi - simpa only [complex_derivative_real_det,abs_pow,abs_norm] using he - obtain ⟨M,hM⟩ := hw.inverse_bounded_over_ball t (‖(x:ℂ)‖+Real.exp q) - have himage : f '' S ⊆ closedBall (0:ℂ) M := by - rintro _ ⟨z,hz,rfl⟩ - rw [mem_closedBall,dist_zero_right] - apply hM (A z) (hAH hz) - calc - ‖A z‖ ≤ ‖(x:ℂ)‖+‖Complex.exp z‖ := norm_add_le _ _ - _ ≤ ‖(x:ℂ)‖+Real.exp q := by - rw [Complex.norm_exp] - exact add_le_add le_rfl (Real.exp_le_exp.mpr hz.1.le) - change (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))<∞ - rw [harea] - exact (measure_mono himage).trans_lt (isCompact_closedBall (0:ℂ) M).measure_lt_top - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseDirichletEnergy +import OAI.Probability.SLE.DyadicStrips + +namespace OAI + +/-! Finite energy in logarithmic boundary coordinates. The exponential +chart is injective on the half-plane strip, so no new growth input is used. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma IsCapacityTwoWitness.inverse_strip_dirichlet_finite + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (x q : ℝ) : + (∫⁻ z in {z : ℂ | z.re F t ((x:ℂ)+Complex.exp w)) z‖^2))<∞ := by + let S : Set ℂ := {z | z.re (x:ℂ)+Complex.exp z + let f : ℂ → ℂ := fun z => F t (A z) + have hS : IsOpen S := (isOpen_lt Complex.continuous_re continuous_const).inter + ((isOpen_lt continuous_const Complex.continuous_im).inter + (isOpen_lt Complex.continuous_im continuous_const)) + have hAH : MapsTo A S upperHalfPlane := by + intro z hz + change 0<((x:ℂ)+Complex.exp z).im + have he : 0<(Complex.exp z).im := exp_strip_halfplane hz.2 + simpa only [Complex.add_im,Complex.ofReal_im,zero_add] using he + have hf : DifferentiableOn ℂ f S := (hw.2.2.2.2.1 t).comp + ((differentiable_const (x:ℂ)).add Complex.differentiable_exp).differentiableOn hAH + have hi : InjOn f S := by + intro z hz w hw' he + have hAz := hw.2.2.2.2.2.2.2.2.1 t (A z) (hAH hz) + have hAw := hw.2.2.2.2.2.2.2.2.1 t (A w) (hAH hw') + have heA : A z=A w := by + rw [show F t (A z)=F t (A w) from he,hAw] at hAz + exact hAz.symm + exact exp_injective_strip hz.2 hw'.2 (add_left_cancel heA) + have harea : (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))=volume (f '' S) := by + have hd (z : ℂ) (hz : z∈S) := + (hf.differentiableAt (hS.mem_nhds hz)).hasDerivAt + have he := lintegral_abs_det_fderiv_eq_addHaar_image volume hS.measurableSet + (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi + simpa only [complex_derivative_real_det,abs_pow,abs_norm] using he + obtain ⟨M,hM⟩ := hw.inverse_bounded_over_ball t (‖(x:ℂ)‖+Real.exp q) + have himage : f '' S ⊆ closedBall (0:ℂ) M := by + rintro _ ⟨z,hz,rfl⟩ + rw [mem_closedBall,dist_zero_right] + apply hM (A z) (hAH hz) + calc + ‖A z‖ ≤ ‖(x:ℂ)‖+‖Complex.exp z‖ := norm_add_le _ _ + _ ≤ ‖(x:ℂ)‖+Real.exp q := by + rw [Complex.norm_exp] + exact add_le_add le_rfl (Real.exp_le_exp.mpr hz.1.le) + change (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))<∞ + rw [harea] + exact (measure_mono himage).trans_lt (isCompact_closedBall (0:ℂ) M).measure_lt_top + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseThinCollapse.lean b/lean/OAI/Probability/SLE/Loewner/InverseThinCollapse.lean index eb8b165b2f1f8d3f067417997126d8a95a2bf82b..59938e90b64602fe376d744c8276beb09b93e450 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseThinCollapse.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseThinCollapse.lean @@ -1,80 +1,80 @@ -import OAI.Probability.SLE.Geometry.BoundarySeparationRectangle -import OAI.Probability.SLE.Loewner.InitialInverseTimeContinuity - -namespace OAI - -/-! Thin crossings whose physical images shrink to one point force -collapse of a whole real boundary interval in the limiting inverse. -/ -noncomputable section -open Set Filter Function Complex Metric MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -theorem initialInverse_thin_crossing_collapse (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 (u:ℂ)+(h:ℂ)*I) '' Icc a' b' - have hK : IsCompact K := isCompact_Icc.image (by fun_prop) - have hKH : K⊆upperHalfPlane := by - rintro _ ⟨u,hu,rfl⟩ - simpa [upperHalfPlane] using hh - have hu := (tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane).mp - (initialInverseChart_locallyUniform_continuous p hi h0 hH s) K hKH hK - have hu' := ht.eventually ((Metric.tendstoUniformlyOn_iff.mp hu) δ hδ) - have he' := hε.eventually (gt_mem_nhds hδ) - have hη' := hη.eventually (gt_mem_nhds hh) - obtain ⟨n,hn,he,hηn,hlown⟩ := (hu'.and (he'.and (hη'.and hlow))).exists - have hhigh : ∀ u∈Ioo a' b', δ+ε n≤ - ‖initialInverseChart p hi h0 hH (t n) ((u:ℂ)+(h:ℂ)*I)-c‖ := by - intro u huI - have hdist := hn _ (mem_image_of_mem _ ⟨huI.1.le,huI.2.le⟩) - have hs := hsep u ⟨huI.1.le,huI.2.le⟩ h ⟨hh,hhH⟩ - have htri := norm_sub_le_norm_sub_add_norm_sub - (initialInverseChart p hi h0 hH s ((u:ℂ)+(h:ℂ)*I)) - (initialInverseChart p hi h0 hH (t n) ((u:ℂ)+(h:ℂ)*I)) c - rw [dist_eq_norm] at hdist - linarith - have hlow' : ∀ u∈Ioo a' b', ∃ y∈Ioc 0 h, - ‖initialInverseChart p hi h0 hH (t n) ((u:ℂ)+(y:ℂ)*I)-c‖≤ε n := by - intro u huI - obtain ⟨y,hy,hyc⟩ := hlown u ⟨haa.trans huI.1,huI.2.trans hbb⟩ - exact ⟨y,⟨hy.1,hy.2.trans hηn.le⟩,hyc⟩ - have hbound := initialInverse_thin_crossing_bound p hi h0 hH hR hp (t n) - hab hh hh1 hδ.le c hxL hhigh hlow' - exact (not_le_of_gt hsmall) hbound - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.BoundarySeparationRectangle +import OAI.Probability.SLE.Loewner.InitialInverseTimeContinuity + +namespace OAI + +/-! Thin crossings whose physical images shrink to one point force +collapse of a whole real boundary interval in the limiting inverse. -/ +noncomputable section +open Set Filter Function Complex Metric MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +theorem initialInverse_thin_crossing_collapse (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 (u:ℂ)+(h:ℂ)*I) '' Icc a' b' + have hK : IsCompact K := isCompact_Icc.image (by fun_prop) + have hKH : K⊆upperHalfPlane := by + rintro _ ⟨u,hu,rfl⟩ + simpa [upperHalfPlane] using hh + have hu := (tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane).mp + (initialInverseChart_locallyUniform_continuous p hi h0 hH s) K hKH hK + have hu' := ht.eventually ((Metric.tendstoUniformlyOn_iff.mp hu) δ hδ) + have he' := hε.eventually (gt_mem_nhds hδ) + have hη' := hη.eventually (gt_mem_nhds hh) + obtain ⟨n,hn,he,hηn,hlown⟩ := (hu'.and (he'.and (hη'.and hlow))).exists + have hhigh : ∀ u∈Ioo a' b', δ+ε n≤ + ‖initialInverseChart p hi h0 hH (t n) ((u:ℂ)+(h:ℂ)*I)-c‖ := by + intro u huI + have hdist := hn _ (mem_image_of_mem _ ⟨huI.1.le,huI.2.le⟩) + have hs := hsep u ⟨huI.1.le,huI.2.le⟩ h ⟨hh,hhH⟩ + have htri := norm_sub_le_norm_sub_add_norm_sub + (initialInverseChart p hi h0 hH s ((u:ℂ)+(h:ℂ)*I)) + (initialInverseChart p hi h0 hH (t n) ((u:ℂ)+(h:ℂ)*I)) c + rw [dist_eq_norm] at hdist + linarith + have hlow' : ∀ u∈Ioo a' b', ∃ y∈Ioc 0 h, + ‖initialInverseChart p hi h0 hH (t n) ((u:ℂ)+(y:ℂ)*I)-c‖≤ε n := by + intro u huI + obtain ⟨y,hy,hyc⟩ := hlown u ⟨haa.trans huI.1,huI.2.trans hbb⟩ + exact ⟨y,⟨hy.1,hy.2.trans hηn.le⟩,hyc⟩ + have hbound := initialInverse_thin_crossing_bound p hi h0 hH hR hp (t n) + hab hh hh1 hδ.le c hxL hhigh hlow' + exact (not_le_of_gt hsmall) hbound + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseVerticalDifferentiation.lean b/lean/OAI/Probability/SLE/Loewner/InverseVerticalDifferentiation.lean index d394f167544523f01a636362706241abbd3226cf..1104a13afc13e96cbfde59439dc246fd707f79ff 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseVerticalDifferentiation.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseVerticalDifferentiation.lean @@ -1,36 +1,36 @@ -import OAI.Probability.SLE.Loewner.InitialInverseEnergyIntegral -import OAI.Probability.SLE.Loewner.VerticalIntervalEnergy - -namespace OAI - -/-! Exact derivatives on vertical lines in the upper half-plane. -/ -noncomputable section -open Set Filter Function Complex MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -lemma holomorphic_vertical_hasDerivAt {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {y : ℝ} (hy : 0 F ((x:ℂ)+(t:ℂ)*Complex.I)) - (Complex.I*deriv F ((x:ℂ)+(y:ℂ)*Complex.I)) y := by - have hp : (x:ℂ)+(y:ℂ)*Complex.I∈upperHalfPlane := by - change 0<((x:ℂ)+(y:ℂ)*Complex.I).im - simpa only [add_im,ofReal_im,mul_im,ofReal_re,I_im,I_re,mul_one,mul_zero,add_zero,zero_add] using hy - have hc := ((Complex.ofRealCLM.hasDerivAt (x := y)).mul_const Complex.I).const_add (x:ℂ) - have hd := ((hF.differentiableAt (isOpen_upperHalfPlane.mem_nhds hp)).hasDerivAt.hasFDerivAt).restrictScalars ℝ - have hh := hd.comp_hasDerivAt y hc - simpa only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply, - Complex.ofRealCLM_apply,ofReal_one,one_mul,mul_one,smul_eq_mul,Function.comp_def,mul_comm] using hh - -lemma holomorphic_vertical_deriv_norm {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {y : ℝ} (hy : 0 F ((x:ℂ)+(t:ℂ)*Complex.I)) y‖= - ‖deriv F ((x:ℂ)+(y:ℂ)*Complex.I)‖ := by - rw [(holomorphic_vertical_hasDerivAt hF x hy).deriv,norm_mul,norm_I,one_mul] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseEnergyIntegral +import OAI.Probability.SLE.Loewner.VerticalIntervalEnergy + +namespace OAI + +/-! Exact derivatives on vertical lines in the upper half-plane. -/ +noncomputable section +open Set Filter Function Complex MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +lemma holomorphic_vertical_hasDerivAt {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {y : ℝ} (hy : 0 F ((x:ℂ)+(t:ℂ)*Complex.I)) + (Complex.I*deriv F ((x:ℂ)+(y:ℂ)*Complex.I)) y := by + have hp : (x:ℂ)+(y:ℂ)*Complex.I∈upperHalfPlane := by + change 0<((x:ℂ)+(y:ℂ)*Complex.I).im + simpa only [add_im,ofReal_im,mul_im,ofReal_re,I_im,I_re,mul_one,mul_zero,add_zero,zero_add] using hy + have hc := ((Complex.ofRealCLM.hasDerivAt (x := y)).mul_const Complex.I).const_add (x:ℂ) + have hd := ((hF.differentiableAt (isOpen_upperHalfPlane.mem_nhds hp)).hasDerivAt.hasFDerivAt).restrictScalars ℝ + have hh := hd.comp_hasDerivAt y hc + simpa only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply, + Complex.ofRealCLM_apply,ofReal_one,one_mul,mul_one,smul_eq_mul,Function.comp_def,mul_comm] using hh + +lemma holomorphic_vertical_deriv_norm {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {y : ℝ} (hy : 0 F ((x:ℂ)+(t:ℂ)*Complex.I)) y‖= + ‖deriv F ((x:ℂ)+(y:ℂ)*Complex.I)‖ := by + rw [(holomorphic_vertical_hasDerivAt hF x hy).deriv,norm_mul,norm_I,one_mul] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/InverseVerticalEnergy.lean b/lean/OAI/Probability/SLE/Loewner/InverseVerticalEnergy.lean index c5c2884eda8d1507efe61944e9aa67b27189d209..885dfe00fa04555326c9f1b6e1c625ffe798e883 100644 --- a/lean/OAI/Probability/SLE/Loewner/InverseVerticalEnergy.lean +++ b/lean/OAI/Probability/SLE/Loewner/InverseVerticalEnergy.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Loewner.InverseVerticalDifferentiation - -namespace OAI - -/-! Vertical energy controls separation from a small physical image. -/ -noncomputable section -open Set Filter Function Complex Metric MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -lemma holomorphic_vertical_energy {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {h y : ℝ} - (hh : 0 deriv F ((x:ℂ)+(t:ℂ)*I)) 2 - (volume.restrict (Ioc 0 h))) : - ‖F ((x:ℂ)+(h:ℂ)*I)-F ((x:ℂ)+(y:ℂ)*I)‖^2 ≤ - h*(∫ t in Ioc 0 h, ‖deriv F ((x:ℂ)+(t:ℂ)*I)‖^2) := by - let f := fun t : ℝ => F ((x:ℂ)+(t:ℂ)*I) - have hn : ∀ᵐ t : ℝ ∂volume.restrict (Ioc 0 h), - ‖deriv f t‖=‖deriv F ((x:ℂ)+(t:ℂ)*I)‖ := by - filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht - exact holomorphic_vertical_deriv_norm hF x ht.1 - have hd : MemLp (deriv f) 2 (volume.restrict (Ioc 0 h)) := - (memLp_congr_norm (measurable_deriv f).aestronglyMeasurable - hE.aestronglyMeasurable hn).mpr hE - have hb := complex_interval_energy_oscillation hh hy ⟨hh,le_rfl⟩ - (fun t ht => (holomorphic_vertical_hasDerivAt hF x - (ordConnected_Ioc.uIcc_subset hy ⟨hh,le_rfl⟩ ht).1).differentiableAt) hd - have he := integral_congr_ae (hn.mono fun t ht => congrArg (fun r : ℝ => r^2) ht) - rw [he] at hb - exact hb - -lemma holomorphic_vertical_separation {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {h δ ε : ℝ} - (hh : 0 deriv F ((x:ℂ)+(t:ℂ)*I)) 2 - (volume.restrict (Ioc 0 h))) : - δ^2 ≤ h*(∫ t in Ioc 0 h, ‖deriv F ((x:ℂ)+(t:ℂ)*I)‖^2) := by - obtain ⟨y,hy,hyc⟩ := hlow - have htriangle := norm_sub_le_norm_sub_add_norm_sub (F ((x:ℂ)+(h:ℂ)*I)) - (F ((x:ℂ)+(y:ℂ)*I)) c - have hsep : δ≤‖F ((x:ℂ)+(h:ℂ)*I)-F ((x:ℂ)+(y:ℂ)*I)‖ := by linarith - exact (pow_le_pow_left₀ hδ hsep 2).trans (holomorphic_vertical_energy hF x hh hy hE) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InverseVerticalDifferentiation + +namespace OAI + +/-! Vertical energy controls separation from a small physical image. -/ +noncomputable section +open Set Filter Function Complex Metric MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma holomorphic_vertical_energy {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {h y : ℝ} + (hh : 0 deriv F ((x:ℂ)+(t:ℂ)*I)) 2 + (volume.restrict (Ioc 0 h))) : + ‖F ((x:ℂ)+(h:ℂ)*I)-F ((x:ℂ)+(y:ℂ)*I)‖^2 ≤ + h*(∫ t in Ioc 0 h, ‖deriv F ((x:ℂ)+(t:ℂ)*I)‖^2) := by + let f := fun t : ℝ => F ((x:ℂ)+(t:ℂ)*I) + have hn : ∀ᵐ t : ℝ ∂volume.restrict (Ioc 0 h), + ‖deriv f t‖=‖deriv F ((x:ℂ)+(t:ℂ)*I)‖ := by + filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht + exact holomorphic_vertical_deriv_norm hF x ht.1 + have hd : MemLp (deriv f) 2 (volume.restrict (Ioc 0 h)) := + (memLp_congr_norm (measurable_deriv f).aestronglyMeasurable + hE.aestronglyMeasurable hn).mpr hE + have hb := complex_interval_energy_oscillation hh hy ⟨hh,le_rfl⟩ + (fun t ht => (holomorphic_vertical_hasDerivAt hF x + (ordConnected_Ioc.uIcc_subset hy ⟨hh,le_rfl⟩ ht).1).differentiableAt) hd + have he := integral_congr_ae (hn.mono fun t ht => congrArg (fun r : ℝ => r^2) ht) + rw [he] at hb + exact hb + +lemma holomorphic_vertical_separation {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (x : ℝ) {h δ ε : ℝ} + (hh : 0 deriv F ((x:ℂ)+(t:ℂ)*I)) 2 + (volume.restrict (Ioc 0 h))) : + δ^2 ≤ h*(∫ t in Ioc 0 h, ‖deriv F ((x:ℂ)+(t:ℂ)*I)‖^2) := by + obtain ⟨y,hy,hyc⟩ := hlow + have htriangle := norm_sub_le_norm_sub_add_norm_sub (F ((x:ℂ)+(h:ℂ)*I)) + (F ((x:ℂ)+(y:ℂ)*I)) c + have hsep : δ≤‖F ((x:ℂ)+(h:ℂ)*I)-F ((x:ℂ)+(y:ℂ)*I)‖ := by linarith + exact (pow_le_pow_left₀ hδ hsep 2).trans (holomorphic_vertical_energy hF x hh hy hE) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LinearReverseTrajectory.lean b/lean/OAI/Probability/SLE/Loewner/LinearReverseTrajectory.lean index 71e750d68448c341f743378766eccca173ceb368..5c07aaf6b98b400b98800df7461cb9173642fd6d 100644 --- a/lean/OAI/Probability/SLE/Loewner/LinearReverseTrajectory.lean +++ b/lean/OAI/Probability/SLE/Loewner/LinearReverseTrajectory.lean @@ -1,105 +1,105 @@ -import OAI.Probability.SLE.Loewner.SmoothRawTraceContinuity -import OAI.Probability.SLE.Loewner.PositiveReverseUniqueness - -namespace OAI - -/-! Linear drivers and their autonomous centered reverse trajectories. -These give concrete steering curves for the geometric support construction. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -def linearDriver (c : ℝ) : Driver := - ⟨⟨fun t => c*(t:ℝ), continuous_const.mul continuous_subtype_val⟩, by simp⟩ - -@[simp] lemma linearDriver_apply (c : ℝ) (t : ℝ≥0) : - linearDriver c t=c*(t:ℝ) := rfl - -lemma linearDriver_value (c t : ℝ) (ht : 0 ≤ t) : - driverValue (linearDriver c) t=c*t := by - simp only [driverValue,linearDriver_apply,Real.coe_toNNReal _ ht] - -lemma linearDriver_derivative (c t : ℝ) (ht : 0 ≤ t) : - HasDerivWithinAt (driverValue (linearDriver c)) c (Ici 0) t := by - have h := ((hasDerivAt_id t).const_mul c).hasDerivWithinAt (s := Ici 0) - simpa only [mul_one] using h.congr - (fun s hs => (linearDriver_value c s hs)) (linearDriver_value c t ht) - -private lemma approximation_height_pos (n : ℕ) : 0 < 1/((n:ℝ)+1) := by positivity - -def linearReverseBase (c : ℝ) (n : ℕ) : ℝ → ℂ := - (exists_reverse_trajectory (fun s => -c*s) (by fun_prop) - (((1/((n:ℝ)+1):ℝ):ℂ)*I) (by simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero] using approximation_height_pos n)).choose - -lemma linearReverseBase_spec (c : ℝ) (n : ℕ) : - linearReverseBase c n 0=(((1/((n:ℝ)+1):ℝ):ℂ)*I) ∧ - (∀ t, 0 ≤ t → 1/((n:ℝ)+1) ≤ (linearReverseBase c n t).im) ∧ - ∀ t, 0 ≤ t → HasDerivWithinAt (linearReverseBase c n) - (-2/(linearReverseBase c n t-((-c*t:ℝ):ℂ))) (Ici 0) t := by - simpa only [linearReverseBase,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im, - Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero] using - (exists_reverse_trajectory (fun s => -c*s) (by fun_prop) - (((1/((n:ℝ)+1):ℝ):ℂ)*I) (by simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero] using approximation_height_pos n)).choose_spec - -def linearReverseApprox (c : ℝ) (n : ℕ) (t : ℝ) : ℂ := - linearReverseBase c n t+((c*t:ℝ):ℂ) - -lemma linearReverseApprox_derivative (c : ℝ) (n : ℕ) (t : ℝ) (ht : 0 ≤ t) : - HasDerivWithinAt (linearReverseApprox c n) - ((c:ℂ)-2/linearReverseApprox c n t) (Ici 0) t := by - have hd := (linearReverseBase_spec c n).2.2 t ht - have hc := (((hasDerivAt_id t).const_mul c).ofReal_comp).hasDerivWithinAt (s := Ici 0) - have h := hd.add hc - change HasDerivWithinAt (linearReverseApprox c n) - (-2/(linearReverseBase c n t-((-c*t:ℝ):ℂ))+((c*1:ℝ):ℂ)) (Ici 0) t at h - exact h.congr_deriv (by - simp only [linearReverseApprox,Complex.ofReal_neg,Complex.ofReal_mul,mul_one, - neg_div] - ring) - -lemma linearReverseApprox_height (c : ℝ) (n : ℕ) (t : ℝ) (ht : 0 ≤ t) : - 0 < (linearReverseApprox c n t).im := by - simpa only [linearReverseApprox,Complex.add_im,Complex.ofReal_im,add_zero] using - (approximation_height_pos n).trans_le ((linearReverseBase_spec c n).2.1 t ht) - -lemma linearReverseApprox_eq_inverse (c : ℝ) (n : ℕ) (T : ℝ≥0) : - linearReverseApprox c n T = inverseFlow 1 (linearDriver c) T - ((linearDriver c T:ℂ)+(((1/((n:ℝ)+1):ℝ):ℂ)*I)) := by - let q := linearReverseBase c n - let r (s : ℝ) := q s+((c*(T:ℝ):ℝ):ℂ) - have hq := linearReverseBase_spec c n - have hr (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt r - (-2/(r s-(driverValue (linearDriver c) ((T:ℝ)-s):ℂ))) (Icc 0 (T:ℝ)) s := by - have h := ((hq.2.2 s hs.1).add_const ((c*(T:ℝ):ℝ):ℂ)).mono (show Icc 0 (T:ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) - have he : r s-(driverValue (linearDriver c) ((T:ℝ)-s):ℂ)=q s-((-c*s:ℝ):ℂ) := by - rw [linearDriver_value _ _ (sub_nonneg.mpr hs.2)] - dsimp [r] - push_cast - ring - simpa only [r,q,he] using h - have hrp (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : 0 < (r s).im := by - simpa only [r,Complex.add_im,Complex.ofReal_im,add_zero] using - (approximation_height_pos n).trans_le (hq.2.1 s hs.1) - obtain ⟨v,hv0,hvp,hvd,hvend⟩ := exists_raw_reverse_trajectory (linearDriver c) T - ((linearDriver c T:ℂ)+(((1/((n:ℝ)+1):ℝ):ℂ)*I)) - (by simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_add] using approximation_height_pos n) - have hvpos (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : 0 < (v s).im := by - apply (approximation_height_pos n).trans_le - simpa only [Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im, - Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_add] using hvp s hs.1 - have he := positive_reverse_unique T.coe_nonneg r v - (fun s => driverValue (linearDriver c) ((T:ℝ)-s)) - (fun s hs => (hr s hs).continuousWithinAt) - (fun s hs => (hvd s hs.1).continuousWithinAt.mono Icc_subset_Ici_self) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hr s (Ico_subset_Icc_self hs))) - (fun s hs => (hvd s hs.1).mono (Ici_subset_Ici.mpr hs.1)) hrp hvpos - (by simp only [r,q,hq.1,hv0,linearDriver_apply]; ring) - exact (he T ⟨T.coe_nonneg,le_rfl⟩).trans hvend.symm - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawTraceContinuity +import OAI.Probability.SLE.Loewner.PositiveReverseUniqueness + +namespace OAI + +/-! Linear drivers and their autonomous centered reverse trajectories. +These give concrete steering curves for the geometric support construction. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +def linearDriver (c : ℝ) : Driver := + ⟨⟨fun t => c*(t:ℝ), continuous_const.mul continuous_subtype_val⟩, by simp⟩ + +@[simp] lemma linearDriver_apply (c : ℝ) (t : ℝ≥0) : + linearDriver c t=c*(t:ℝ) := rfl + +lemma linearDriver_value (c t : ℝ) (ht : 0 ≤ t) : + driverValue (linearDriver c) t=c*t := by + simp only [driverValue,linearDriver_apply,Real.coe_toNNReal _ ht] + +lemma linearDriver_derivative (c t : ℝ) (ht : 0 ≤ t) : + HasDerivWithinAt (driverValue (linearDriver c)) c (Ici 0) t := by + have h := ((hasDerivAt_id t).const_mul c).hasDerivWithinAt (s := Ici 0) + simpa only [mul_one] using h.congr + (fun s hs => (linearDriver_value c s hs)) (linearDriver_value c t ht) + +private lemma approximation_height_pos (n : ℕ) : 0 < 1/((n:ℝ)+1) := by positivity + +def linearReverseBase (c : ℝ) (n : ℕ) : ℝ → ℂ := + (exists_reverse_trajectory (fun s => -c*s) (by fun_prop) + (((1/((n:ℝ)+1):ℝ):ℂ)*I) (by simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero] using approximation_height_pos n)).choose + +lemma linearReverseBase_spec (c : ℝ) (n : ℕ) : + linearReverseBase c n 0=(((1/((n:ℝ)+1):ℝ):ℂ)*I) ∧ + (∀ t, 0 ≤ t → 1/((n:ℝ)+1) ≤ (linearReverseBase c n t).im) ∧ + ∀ t, 0 ≤ t → HasDerivWithinAt (linearReverseBase c n) + (-2/(linearReverseBase c n t-((-c*t:ℝ):ℂ))) (Ici 0) t := by + simpa only [linearReverseBase,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im, + Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero] using + (exists_reverse_trajectory (fun s => -c*s) (by fun_prop) + (((1/((n:ℝ)+1):ℝ):ℂ)*I) (by simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero] using approximation_height_pos n)).choose_spec + +def linearReverseApprox (c : ℝ) (n : ℕ) (t : ℝ) : ℂ := + linearReverseBase c n t+((c*t:ℝ):ℂ) + +lemma linearReverseApprox_derivative (c : ℝ) (n : ℕ) (t : ℝ) (ht : 0 ≤ t) : + HasDerivWithinAt (linearReverseApprox c n) + ((c:ℂ)-2/linearReverseApprox c n t) (Ici 0) t := by + have hd := (linearReverseBase_spec c n).2.2 t ht + have hc := (((hasDerivAt_id t).const_mul c).ofReal_comp).hasDerivWithinAt (s := Ici 0) + have h := hd.add hc + change HasDerivWithinAt (linearReverseApprox c n) + (-2/(linearReverseBase c n t-((-c*t:ℝ):ℂ))+((c*1:ℝ):ℂ)) (Ici 0) t at h + exact h.congr_deriv (by + simp only [linearReverseApprox,Complex.ofReal_neg,Complex.ofReal_mul,mul_one, + neg_div] + ring) + +lemma linearReverseApprox_height (c : ℝ) (n : ℕ) (t : ℝ) (ht : 0 ≤ t) : + 0 < (linearReverseApprox c n t).im := by + simpa only [linearReverseApprox,Complex.add_im,Complex.ofReal_im,add_zero] using + (approximation_height_pos n).trans_le ((linearReverseBase_spec c n).2.1 t ht) + +lemma linearReverseApprox_eq_inverse (c : ℝ) (n : ℕ) (T : ℝ≥0) : + linearReverseApprox c n T = inverseFlow 1 (linearDriver c) T + ((linearDriver c T:ℂ)+(((1/((n:ℝ)+1):ℝ):ℂ)*I)) := by + let q := linearReverseBase c n + let r (s : ℝ) := q s+((c*(T:ℝ):ℝ):ℂ) + have hq := linearReverseBase_spec c n + have hr (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : HasDerivWithinAt r + (-2/(r s-(driverValue (linearDriver c) ((T:ℝ)-s):ℂ))) (Icc 0 (T:ℝ)) s := by + have h := ((hq.2.2 s hs.1).add_const ((c*(T:ℝ):ℝ):ℂ)).mono (show Icc 0 (T:ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) + have he : r s-(driverValue (linearDriver c) ((T:ℝ)-s):ℂ)=q s-((-c*s:ℝ):ℂ) := by + rw [linearDriver_value _ _ (sub_nonneg.mpr hs.2)] + dsimp [r] + push_cast + ring + simpa only [r,q,he] using h + have hrp (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : 0 < (r s).im := by + simpa only [r,Complex.add_im,Complex.ofReal_im,add_zero] using + (approximation_height_pos n).trans_le (hq.2.1 s hs.1) + obtain ⟨v,hv0,hvp,hvd,hvend⟩ := exists_raw_reverse_trajectory (linearDriver c) T + ((linearDriver c T:ℂ)+(((1/((n:ℝ)+1):ℝ):ℂ)*I)) + (by simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_add] using approximation_height_pos n) + have hvpos (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : 0 < (v s).im := by + apply (approximation_height_pos n).trans_le + simpa only [Complex.add_im,Complex.ofReal_im,Complex.ofReal_re,Complex.mul_im, + Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_add] using hvp s hs.1 + have he := positive_reverse_unique T.coe_nonneg r v + (fun s => driverValue (linearDriver c) ((T:ℝ)-s)) + (fun s hs => (hr s hs).continuousWithinAt) + (fun s hs => (hvd s hs.1).continuousWithinAt.mono Icc_subset_Ici_self) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hr s (Ico_subset_Icc_self hs))) + (fun s hs => (hvd s hs.1).mono (Ici_subset_Ici.mpr hs.1)) hrp hvpos + (by simp only [r,q,hq.1,hv0,linearDriver_apply]; ring) + exact (he T ⟨T.coe_nonneg,le_rfl⟩).trans hvend.symm + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LocalReverseCone.lean b/lean/OAI/Probability/SLE/Loewner/LocalReverseCone.lean index 3abd105923b43628a422267ce0a2863197c00268..28dea6c7003a02e448ef2cbc11818fccaad969b9 100644 --- a/lean/OAI/Probability/SLE/Loewner/LocalReverseCone.lean +++ b/lean/OAI/Probability/SLE/Loewner/LocalReverseCone.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Loewner.ReverseBoundaryCone - -namespace OAI - -/-! A continuous positive reverse solution starting at the driver is -non-tangential on a sufficiently short initial interval. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology - -namespace SLEExactGauge - -theorem exists_initial_reverse_cone {T L : ℝ} (hT : 0 < T) (hL : 0 ≤ L) - (q : ℝ → ℂ) (u u' : ℝ → ℝ) - (hc : ContinuousOn q (Icc 0 T)) (huc : ContinuousOn u (Icc 0 T)) - (hq0 : q 0=(u 0 : ℂ)) - (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Ici t) t) - (hu : ∀ t ∈ Ioc 0 T, HasDerivWithinAt u (u' t) (Ici t) t) - (hp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) - (hb : ∀ t ∈ Ioc 0 T, |u' t| ≤ L) : - ∃ a : ℝ, 0 < a ∧ a ≤ T ∧ - ∀ t ∈ Icc 0 a, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2 := by - let M : ℝ := 1/(2*(L+1)) - have hMpos : 0 < M := by dsimp [M]; positivity - have hsmall : 2*L*M ≤ 1 := by - have hd : 0 < 2*(L+1) := by positivity - calc - 2*L*M = (2*L)/(2*(L+1)) := by dsimp [M]; ring - _ ≤ 1 := (div_le_one hd).mpr (by linarith) - have hc0 := ((hc.sub (Complex.continuous_ofReal.comp_continuousOn huc)) 0 ⟨le_rfl,hT.le⟩).norm - have hv : ‖q 0-(u 0 : ℂ)‖ < M := by simpa only [hq0,sub_self,norm_zero] using hMpos - have he : ∀ᶠ t in 𝓝[Icc 0 T] 0, ‖q t-(u t : ℂ)‖ < M := - hc0.eventually (Iio_mem_nhds hv) - obtain ⟨δ,hδ,hδb⟩ := Metric.mem_nhdsWithin_iff.mp he - let a : ℝ := min T (δ/2) - have ha : 0 < a := lt_min hT (half_pos hδ) - have haT : a ≤ T := min_le_left _ _ - have hsub : Icc 0 a ⊆ Icc 0 T := Icc_subset_Icc_right haT - have hsub' : Ioc 0 a ⊆ Ioc 0 T := Ioc_subset_Ioc_right haT - refine ⟨a,ha,haT,reverse_boundary_cone_of_small_center ha.le hL hsmall q u u' - (hc.mono hsub) (huc.mono hsub) hq0 - (fun t ht => hq t (hsub' ht)) (fun t ht => hu t (hsub' ht)) - (fun t ht => hp t (hsub' ht)) (fun t ht => hb t (hsub' ht)) ?_⟩ - intro t ht - apply le_of_lt - apply hδb - refine ⟨?_,hsub (Ioc_subset_Icc_self ht)⟩ - rw [mem_ball,Real.dist_eq,sub_zero,abs_of_pos ht.1] - exact (ht.2.trans (min_le_right _ _)).trans_lt (half_lt_self hδ) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReverseBoundaryCone + +namespace OAI + +/-! A continuous positive reverse solution starting at the driver is +non-tangential on a sufficiently short initial interval. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology + +namespace SLEExactGauge + +theorem exists_initial_reverse_cone {T L : ℝ} (hT : 0 < T) (hL : 0 ≤ L) + (q : ℝ → ℂ) (u u' : ℝ → ℝ) + (hc : ContinuousOn q (Icc 0 T)) (huc : ContinuousOn u (Icc 0 T)) + (hq0 : q 0=(u 0 : ℂ)) + (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Ici t) t) + (hu : ∀ t ∈ Ioc 0 T, HasDerivWithinAt u (u' t) (Ici t) t) + (hp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) + (hb : ∀ t ∈ Ioc 0 T, |u' t| ≤ L) : + ∃ a : ℝ, 0 < a ∧ a ≤ T ∧ + ∀ t ∈ Icc 0 a, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2 := by + let M : ℝ := 1/(2*(L+1)) + have hMpos : 0 < M := by dsimp [M]; positivity + have hsmall : 2*L*M ≤ 1 := by + have hd : 0 < 2*(L+1) := by positivity + calc + 2*L*M = (2*L)/(2*(L+1)) := by dsimp [M]; ring + _ ≤ 1 := (div_le_one hd).mpr (by linarith) + have hc0 := ((hc.sub (Complex.continuous_ofReal.comp_continuousOn huc)) 0 ⟨le_rfl,hT.le⟩).norm + have hv : ‖q 0-(u 0 : ℂ)‖ < M := by simpa only [hq0,sub_self,norm_zero] using hMpos + have he : ∀ᶠ t in 𝓝[Icc 0 T] 0, ‖q t-(u t : ℂ)‖ < M := + hc0.eventually (Iio_mem_nhds hv) + obtain ⟨δ,hδ,hδb⟩ := Metric.mem_nhdsWithin_iff.mp he + let a : ℝ := min T (δ/2) + have ha : 0 < a := lt_min hT (half_pos hδ) + have haT : a ≤ T := min_le_left _ _ + have hsub : Icc 0 a ⊆ Icc 0 T := Icc_subset_Icc_right haT + have hsub' : Ioc 0 a ⊆ Ioc 0 T := Ioc_subset_Ioc_right haT + refine ⟨a,ha,haT,reverse_boundary_cone_of_small_center ha.le hL hsmall q u u' + (hc.mono hsub) (huc.mono hsub) hq0 + (fun t ht => hq t (hsub' ht)) (fun t ht => hu t (hsub' ht)) + (fun t ht => hp t (hsub' ht)) (fun t ht => hb t (hsub' ht)) ?_⟩ + intro t ht + apply le_of_lt + apply hδb + refine ⟨?_,hsub (Ioc_subset_Icc_self ht)⟩ + rw [mem_ball,Real.dist_eq,sub_zero,abs_of_pos ht.1] + exact (ht.2.trans (min_le_right _ _)).trans_lt (half_lt_self hδ) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerCompactDriverNeighborhood.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerCompactDriverNeighborhood.lean index 89fd9044bb9bc79e50a99493178119cd27f87054..56c227137f30999ab57b203678f5aa02e1341cba 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerCompactDriverNeighborhood.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerCompactDriverNeighborhood.lean @@ -1,40 +1,40 @@ -import OAI.Probability.SLE.Loewner.LoewnerCompactSurvival -import OAI.Probability.SLE.Loewner.LoewnerInverseDriverContinuity - -namespace OAI - -/-! Compact survival holds throughout a uniform neighborhood of a fixed -driver, with no regularity assumption on the perturbed trace beyond its -actual capacity witness. -/ -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem driver_neighborhood_preserves_compact - {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) - {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : - ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → - (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → K ⊆ survivingDomain γ T := by - classical - by_contra hn - push Not at hn - have hc (n : ℕ) := hn ((n+1:ℝ)⁻¹) (by positivity) - choose U γ G F hw hclose hfail using hc - have hδ : Tendsto (fun n : ℕ => (n+1:ℝ)⁻¹) atTop (𝓝 0) := by - exact tendsto_inv_atTop_zero.comp - (tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop) - have hl := inverse_maps_locally_uniform_of_drivers U γ G F hw hv T - (fun n : ℕ => (n+1:ℝ)⁻¹) hδ hclose - have he := compact_survival_of_inverse_convergence U γ G F hw hv T hl hK hKD - obtain ⟨n,hn⟩ := he.exists - exact hfail n hn - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerCompactSurvival +import OAI.Probability.SLE.Loewner.LoewnerInverseDriverContinuity + +namespace OAI + +/-! Compact survival holds throughout a uniform neighborhood of a fixed +driver, with no regularity assumption on the perturbed trace beyond its +actual capacity witness. -/ +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem driver_neighborhood_preserves_compact + {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) + {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : + ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → + (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → K ⊆ survivingDomain γ T := by + classical + by_contra hn + push Not at hn + have hc (n : ℕ) := hn ((n+1:ℝ)⁻¹) (by positivity) + choose U γ G F hw hclose hfail using hc + have hδ : Tendsto (fun n : ℕ => (n+1:ℝ)⁻¹) atTop (𝓝 0) := by + exact tendsto_inv_atTop_zero.comp + (tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop) + have hl := inverse_maps_locally_uniform_of_drivers U γ G F hw hv T + (fun n : ℕ => (n+1:ℝ)⁻¹) hδ hclose + have he := compact_survival_of_inverse_convergence U γ G F hw hv T hl hK hKD + obtain ⟨n,hn⟩ := he.exists + exact hfail n hn + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerCompactSurvival.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerCompactSurvival.lean index 43b4023e70d00a68336628184b99bbc6fa0d39db..caeff28b090dad66caa1adb93ec22ec939dce149 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerCompactSurvival.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerCompactSurvival.lean @@ -1,42 +1,42 @@ -import OAI.Probability.SLE.Loewner.LoewnerKernelContinuity - -namespace OAI - -/-! Compact subsets of the limiting surviving domain stay in the -surviving domains of all sufficiently close drivers. -/ -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem compact_survival_of_inverse_convergence - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) - (hl : TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane) - {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : - ∀ᶠ n in atTop, K ⊆ survivingDomain (γ n) T := by - classical - have hp (z : K) : ∃ r : ℝ, 0 < r ∧ - ∀ᶠ n in atTop, Metric.ball (z:ℂ) r ⊆ survivingDomain (γ n) T := by - obtain ⟨r,hr,he⟩ := inverse_image_neighborhood_persists U γ G F h hv T hl - (hv.2.2.2.2.2.1 T (hKD z.property)) - rw [hv.2.2.2.2.2.2.2.1 T z (hKD z.property)] at he - exact ⟨r,hr,he⟩ - choose r hr hp using hp - obtain ⟨s,hs⟩ := hK.elim_finite_subcover (fun z : K => Metric.ball (z:ℂ) (r z)) - (fun _ => Metric.isOpen_ball) (fun z hz => - mem_iUnion.mpr ⟨⟨z,hz⟩,Metric.mem_ball_self (hr ⟨z,hz⟩)⟩) - have he : ∀ᶠ n in atTop, ∀ z ∈ s, Metric.ball (z:ℂ) (r z) ⊆ survivingDomain (γ n) T := by - exact s.eventually_all.mpr (fun z _ => hp z) - filter_upwards [he] with n hn z hz - obtain ⟨w,hw,hwz⟩ := mem_iUnion₂.mp (hs hz) - exact hn w hw hwz - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerKernelContinuity + +namespace OAI + +/-! Compact subsets of the limiting surviving domain stay in the +surviving domains of all sufficiently close drivers. -/ +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem compact_survival_of_inverse_convergence + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) + (hl : TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane) + {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : + ∀ᶠ n in atTop, K ⊆ survivingDomain (γ n) T := by + classical + have hp (z : K) : ∃ r : ℝ, 0 < r ∧ + ∀ᶠ n in atTop, Metric.ball (z:ℂ) r ⊆ survivingDomain (γ n) T := by + obtain ⟨r,hr,he⟩ := inverse_image_neighborhood_persists U γ G F h hv T hl + (hv.2.2.2.2.2.1 T (hKD z.property)) + rw [hv.2.2.2.2.2.2.2.1 T z (hKD z.property)] at he + exact ⟨r,hr,he⟩ + choose r hr hp using hp + obtain ⟨s,hs⟩ := hK.elim_finite_subcover (fun z : K => Metric.ball (z:ℂ) (r z)) + (fun _ => Metric.isOpen_ball) (fun z hz => + mem_iUnion.mpr ⟨⟨z,hz⟩,Metric.mem_ball_self (hr ⟨z,hz⟩)⟩) + have he : ∀ᶠ n in atTop, ∀ z ∈ s, Metric.ball (z:ℂ) (r z) ⊆ survivingDomain (γ n) T := by + exact s.eventually_all.mpr (fun z _ => hp z) + filter_upwards [he] with n hn z hz + obtain ⟨w,hw,hwz⟩ := mem_iUnion₂.mp (hs hz) + exact hn w hw hwz + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerDriverBounds.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerDriverBounds.lean index bc2a7d1ed7acfe98c970a9b03ea1fb8b5951c01b..5d770d8a1e121eb9f2da3ce4a986127699f4a6e1 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerDriverBounds.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerDriverBounds.lean @@ -1,98 +1,98 @@ -import OAI.Probability.SLE.LoewnerReflection - -namespace OAI - -/-! Deterministic confinement from the driving function, obtained from the -reverse Loewner equation before taking the actual boundary trace limit. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - -lemma IsCapacityTwoWitness.reverse_path_unclipped - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} - (hw : w ∈ upperHalfPlane) {s : ℝ} (hs : s ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun t : ℝ => G ((T : ℝ) - t).toNNReal (F T w)) - (-2 / (G ((T : ℝ) - s).toNNReal (F T w) - (U ((T : ℝ) - s).toNNReal : ℂ))) - (Icc 0 (T : ℝ)) s := by - have hz := h.2.2.2.2.2.2.1 T hw - have hi := h.imag_ge_terminal T (F T w) hz ((T : ℝ) - s) - ⟨by linarith [hs.2], by linarith [hs.1]⟩ - rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hi - simpa only [reverseField, reverseCenter_eq hi, div_eq_mul_inv] using - h.reverse_path_derivative T w hw le_rfl s hs - -lemma IsCapacityTwoWitness.inverse_re_le - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} - (hw : w ∈ upperHalfPlane) {M : ℝ} - (hwM : w.re ≤ M) (hU : ∀ t ∈ Icc 0 T, U t ≤ M) : (F T w).re ≤ M := by - let q (s : ℝ) := G ((T : ℝ) - s).toNNReal (F T w) - have hq (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) := h.reverse_path_unclipped T hw hs - have hq0 : q 0 = w := by - simp only [q, sub_zero, Real.toNNReal_coe] - exact h.2.2.2.2.2.2.2.2.1 T w hw - have hqT : q T = F T w := by - simp only [q, sub_self, Real.toNNReal_zero] - exact h.initial_value _ (h.2.2.2.2.2.2.1 T hw).1.1 - apply le_of_forall_pos_le_add - intro ε hε - have hr (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun t => (q t).re) - ((-2 * ((q s).re - U ((T : ℝ) - s).toNNReal)) / - normSq (q s - (U ((T : ℝ) - s).toNNReal : ℂ))) (Icc 0 (T : ℝ)) s := by - have hd := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt s (hq s hs) - change HasDerivWithinAt (fun t => (q t).re) - ((-2 / (q s - (U ((T : ℝ) - s).toNNReal : ℂ))).re) _ _ at hd - simpa only [Function.comp_def, Complex.div_re, Complex.neg_re, Complex.re_ofNat, - Complex.neg_im, Complex.im_ofNat, neg_zero, zero_mul, zero_div, add_zero, - Complex.sub_re, Complex.ofReal_re] using hd - have hc : ContinuousOn (fun t => (q t).re) (Icc 0 (T : ℝ)) := - fun s hs => (hr s hs).continuousWithinAt - have hbound := image_le_of_deriv_right_lt_deriv_boundary hc - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hr s ⟨hs.1,hs.2.le⟩)) - (B := fun _ => M + ε) (B' := fun _ => 0) - (by rw [hq0]; linarith) - (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) - (show (T : ℝ) ∈ Icc 0 (T : ℝ) from ⟨T.coe_nonneg,le_rfl⟩) - · simpa only [hqT] using hbound - · have ht : ((T : ℝ)-s).toNNReal ∈ Icc 0 T := by - refine ⟨zero_le,?_⟩ - exact_mod_cast (show (((T : ℝ)-s).toNNReal : ℝ) ≤ T by - rw [Real.coe_toNNReal _ (by linarith [hs.2])]; linarith [hs.1]) - have him : 0 < (q s).im := h.imag_positive _ _ - (survivingDomain_antitone γ ht.2 (h.2.2.2.2.2.2.1 T hw)) - have hne : q s - (U ((T : ℝ)-s).toNNReal : ℂ) ≠ 0 := by - intro hn - have hh := congrArg Complex.im hn - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hh - exact him.ne' hh - apply div_neg_of_neg_of_pos _ (Complex.normSq_pos.mpr hne) - rw [he] - nlinarith [hU _ ht] - -lemma IsCapacityTwoWitness.trace_re_le - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {M : ℝ} - (hU : ∀ t ∈ Icc 0 T, U t ≤ M) : (γ T).re ≤ M := by - have hl := Complex.continuous_re.continuousAt.tendsto.comp - (h.2.2.2.2.2.2.2.2.2.2.2.2 T) - apply le_of_tendsto hl - filter_upwards [self_mem_nhdsWithin] with y hy - apply h.inverse_re_le T (show (U T : ℂ)+(y:ℂ)*I ∈ upperHalfPlane by - simpa [upperHalfPlane] using hy) (by simpa using hU T ⟨zero_le,le_rfl⟩) hU - -lemma IsCapacityTwoWitness.trace_abs_re_le - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {M : ℝ} - (hU : ∀ t ∈ Icc 0 T, |U t| ≤ M) : |(γ T).re| ≤ M := by - apply abs_le.mpr - refine ⟨?_,h.trace_re_le T (fun t ht => (le_abs_self _).trans (hU t ht))⟩ - have hh := h.mirror.trace_re_le T (fun t ht => (neg_le_abs _).trans (hU t ht)) - simpa only [Function.comp_apply,loewnerMirror_re,neg_le] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.LoewnerReflection + +namespace OAI + +/-! Deterministic confinement from the driving function, obtained from the +reverse Loewner equation before taking the actual boundary trace limit. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + +lemma IsCapacityTwoWitness.reverse_path_unclipped + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} + (hw : w ∈ upperHalfPlane) {s : ℝ} (hs : s ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun t : ℝ => G ((T : ℝ) - t).toNNReal (F T w)) + (-2 / (G ((T : ℝ) - s).toNNReal (F T w) - (U ((T : ℝ) - s).toNNReal : ℂ))) + (Icc 0 (T : ℝ)) s := by + have hz := h.2.2.2.2.2.2.1 T hw + have hi := h.imag_ge_terminal T (F T w) hz ((T : ℝ) - s) + ⟨by linarith [hs.2], by linarith [hs.1]⟩ + rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hi + simpa only [reverseField, reverseCenter_eq hi, div_eq_mul_inv] using + h.reverse_path_derivative T w hw le_rfl s hs + +lemma IsCapacityTwoWitness.inverse_re_le + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} + (hw : w ∈ upperHalfPlane) {M : ℝ} + (hwM : w.re ≤ M) (hU : ∀ t ∈ Icc 0 T, U t ≤ M) : (F T w).re ≤ M := by + let q (s : ℝ) := G ((T : ℝ) - s).toNNReal (F T w) + have hq (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) := h.reverse_path_unclipped T hw hs + have hq0 : q 0 = w := by + simp only [q, sub_zero, Real.toNNReal_coe] + exact h.2.2.2.2.2.2.2.2.1 T w hw + have hqT : q T = F T w := by + simp only [q, sub_self, Real.toNNReal_zero] + exact h.initial_value _ (h.2.2.2.2.2.2.1 T hw).1.1 + apply le_of_forall_pos_le_add + intro ε hε + have hr (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun t => (q t).re) + ((-2 * ((q s).re - U ((T : ℝ) - s).toNNReal)) / + normSq (q s - (U ((T : ℝ) - s).toNNReal : ℂ))) (Icc 0 (T : ℝ)) s := by + have hd := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt s (hq s hs) + change HasDerivWithinAt (fun t => (q t).re) + ((-2 / (q s - (U ((T : ℝ) - s).toNNReal : ℂ))).re) _ _ at hd + simpa only [Function.comp_def, Complex.div_re, Complex.neg_re, Complex.re_ofNat, + Complex.neg_im, Complex.im_ofNat, neg_zero, zero_mul, zero_div, add_zero, + Complex.sub_re, Complex.ofReal_re] using hd + have hc : ContinuousOn (fun t => (q t).re) (Icc 0 (T : ℝ)) := + fun s hs => (hr s hs).continuousWithinAt + have hbound := image_le_of_deriv_right_lt_deriv_boundary hc + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hr s ⟨hs.1,hs.2.le⟩)) + (B := fun _ => M + ε) (B' := fun _ => 0) + (by rw [hq0]; linarith) + (fun _ => hasDerivAt_const _ _) (fun s hs he => ?_) + (show (T : ℝ) ∈ Icc 0 (T : ℝ) from ⟨T.coe_nonneg,le_rfl⟩) + · simpa only [hqT] using hbound + · have ht : ((T : ℝ)-s).toNNReal ∈ Icc 0 T := by + refine ⟨zero_le,?_⟩ + exact_mod_cast (show (((T : ℝ)-s).toNNReal : ℝ) ≤ T by + rw [Real.coe_toNNReal _ (by linarith [hs.2])]; linarith [hs.1]) + have him : 0 < (q s).im := h.imag_positive _ _ + (survivingDomain_antitone γ ht.2 (h.2.2.2.2.2.2.1 T hw)) + have hne : q s - (U ((T : ℝ)-s).toNNReal : ℂ) ≠ 0 := by + intro hn + have hh := congrArg Complex.im hn + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hh + exact him.ne' hh + apply div_neg_of_neg_of_pos _ (Complex.normSq_pos.mpr hne) + rw [he] + nlinarith [hU _ ht] + +lemma IsCapacityTwoWitness.trace_re_le + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {M : ℝ} + (hU : ∀ t ∈ Icc 0 T, U t ≤ M) : (γ T).re ≤ M := by + have hl := Complex.continuous_re.continuousAt.tendsto.comp + (h.2.2.2.2.2.2.2.2.2.2.2.2 T) + apply le_of_tendsto hl + filter_upwards [self_mem_nhdsWithin] with y hy + apply h.inverse_re_le T (show (U T : ℂ)+(y:ℂ)*I ∈ upperHalfPlane by + simpa [upperHalfPlane] using hy) (by simpa using hU T ⟨zero_le,le_rfl⟩) hU + +lemma IsCapacityTwoWitness.trace_abs_re_le + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {M : ℝ} + (hU : ∀ t ∈ Icc 0 T, |U t| ≤ M) : |(γ T).re| ≤ M := by + apply abs_le.mpr + refine ⟨?_,h.trace_re_le T (fun t ht => (le_abs_self _).trans (hU t ht))⟩ + have hh := h.mirror.trace_re_le T (fun t ht => (neg_le_abs _).trans (hU t ht)) + simpa only [Function.comp_apply,loewnerMirror_re,neg_le] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerHeightClearance.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerHeightClearance.lean index 4ebc56ecfca4fb1dec142f63ebfbf48086076625..5d5a954cf01155bebfda374ea8a1dbd43213be1d 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerHeightClearance.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerHeightClearance.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.Gauge.RibbonSurvival -import OAI.Probability.SLE.Geometry.NormalizedMarkedState -import OAI.Probability.SLE.Geometry.HyperbolicGuideCompactness - -namespace OAI - -/-! A hypothetical upper bound on the trace height gives a physical -clearance ball and hence a lower bound on the inverse derivative. -/ -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma survives_above_trace_height - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (H : ℝ) - (hγ : ∀ t ∈ Icc 0 T, (γ t).im ≤ H) {z : ℂ} - (hz : 0 < z.im) (hH : H < z.im) : z ∈ survivingDomain γ T := by - have hf : range (upwardRay z) ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by - rintro w ⟨s,rfl⟩ - have hi : (upwardRay z s).im = z.im+s := by simp [upwardRay] - refine ⟨?_,?_⟩ - · change 0 < (upwardRay z s).im - rw [hi] - exact add_pos_of_pos_of_nonneg hz s.coe_nonneg - · rintro ⟨t,ht,he⟩ - have hh := hγ t ht - rw [he,hi] at hh - linarith [s.coe_nonneg] - have hh := survives_of_unbounded_free_curve γ T (upwardRay z) - (upwardRay_continuous z) hf (upwardRay_unbounded z) (mem_range_self 0) - simpa [upwardRay] using hh - -lemma IsCapacityTwoWitness.inverse_clearance_bound - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} - (hw : w ∈ upperHalfPlane) {r : ℝ} (hr : 0 < r) - (hball : ball (F T w) r ⊆ survivingDomain γ T) : - r ≤ (128*Real.pi)*w.im*‖deriv (F T) w‖ := by - have hD := h.2.2.2.2.2.2.1 T hw - have hid := h.inverse_derivative_identity T (F T w) hD - rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hid - have hnorm := congrArg norm hid - rw [norm_mul,norm_one] at hnorm - have hi : InjOn (G T) (survivingDomain γ T) := by - intro z hz v hv he - rw [← h.2.2.2.2.2.2.2.1 T z hz,he,h.2.2.2.2.2.2.2.1 T v hv] - have hd := HyperbolicGuideCompactness.inverse_chart_derivative_bound hr - ((h.2.2.2.1 T).mono hball) (hi.mono hball) - (fun z hz => h.2.2.2.2.2.1 T (hball hz)) - rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hd - have hm := mul_le_mul_of_nonneg_right hd (norm_nonneg (deriv (F T) w)) - have hone : 1 ≤ (128*Real.pi/r)*w.im*‖deriv (F T) w‖ := by - simpa only [mul_comm ‖deriv (G T) (F T w)‖,hnorm,mul_assoc] using hm - calc - r = 1*r := by ring - _ ≤ ((128*Real.pi/r)*w.im*‖deriv (F T) w‖)*r := - mul_le_mul_of_nonneg_right hone hr.le - _ = (128*Real.pi)*w.im*‖deriv (F T) w‖ := by field_simp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.RibbonSurvival +import OAI.Probability.SLE.Geometry.NormalizedMarkedState +import OAI.Probability.SLE.Geometry.HyperbolicGuideCompactness + +namespace OAI + +/-! A hypothetical upper bound on the trace height gives a physical +clearance ball and hence a lower bound on the inverse derivative. -/ +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma survives_above_trace_height + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (H : ℝ) + (hγ : ∀ t ∈ Icc 0 T, (γ t).im ≤ H) {z : ℂ} + (hz : 0 < z.im) (hH : H < z.im) : z ∈ survivingDomain γ T := by + have hf : range (upwardRay z) ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by + rintro w ⟨s,rfl⟩ + have hi : (upwardRay z s).im = z.im+s := by simp [upwardRay] + refine ⟨?_,?_⟩ + · change 0 < (upwardRay z s).im + rw [hi] + exact add_pos_of_pos_of_nonneg hz s.coe_nonneg + · rintro ⟨t,ht,he⟩ + have hh := hγ t ht + rw [he,hi] at hh + linarith [s.coe_nonneg] + have hh := survives_of_unbounded_free_curve γ T (upwardRay z) + (upwardRay_continuous z) hf (upwardRay_unbounded z) (mem_range_self 0) + simpa [upwardRay] using hh + +lemma IsCapacityTwoWitness.inverse_clearance_bound + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} + (hw : w ∈ upperHalfPlane) {r : ℝ} (hr : 0 < r) + (hball : ball (F T w) r ⊆ survivingDomain γ T) : + r ≤ (128*Real.pi)*w.im*‖deriv (F T) w‖ := by + have hD := h.2.2.2.2.2.2.1 T hw + have hid := h.inverse_derivative_identity T (F T w) hD + rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hid + have hnorm := congrArg norm hid + rw [norm_mul,norm_one] at hnorm + have hi : InjOn (G T) (survivingDomain γ T) := by + intro z hz v hv he + rw [← h.2.2.2.2.2.2.2.1 T z hz,he,h.2.2.2.2.2.2.2.1 T v hv] + have hd := HyperbolicGuideCompactness.inverse_chart_derivative_bound hr + ((h.2.2.2.1 T).mono hball) (hi.mono hball) + (fun z hz => h.2.2.2.2.2.1 T (hball hz)) + rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hd + have hm := mul_le_mul_of_nonneg_right hd (norm_nonneg (deriv (F T) w)) + have hone : 1 ≤ (128*Real.pi/r)*w.im*‖deriv (F T) w‖ := by + simpa only [mul_comm ‖deriv (G T) (F T w)‖,hnorm,mul_assoc] using hm + calc + r = 1*r := by ring + _ ≤ ((128*Real.pi/r)*w.im*‖deriv (F T) w‖)*r := + mul_le_mul_of_nonneg_right hone hr.le + _ = (128*Real.pi)*w.im*‖deriv (F T) w‖ := by field_simp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerInverseCompactness.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerInverseCompactness.lean index 42fe5ce309df71f4d6c1758d2502dbfd6f18fd3a..a3b0c56caafecca91d98cb88409a2cbd09f473d8 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerInverseCompactness.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerInverseCompactness.lean @@ -1,62 +1,62 @@ -import OAI.Probability.SLE.Loewner.LoewnerInverseStability -import OAI.Probability.SLE.Geometry.HolomorphicFamilyCompactness - -namespace OAI - -/-! Compactness of inverse maps at a fixed capacity time follows directly -from their reverse-flow displacement estimate. -/ -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -local instance : LocallyCompactSpace upperHalfPlane := - isOpen_upperHalfPlane.locallyCompactSpace - -lemma inverse_family_compact_bounds - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) (T : ℝ≥0) - (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) : - ∃ C : ℝ, ∀ n z, z ∈ K → ‖F n T z‖ ≤ C := by - by_cases hne : K.Nonempty - · obtain ⟨w,hw,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn - have ha : 0 < w.im := hKH hw - obtain ⟨R,_,hR⟩ := hK.isBounded.exists_pos_norm_lt - refine ⟨R+(2*w.im⁻¹)*T, fun n z hz => ?_⟩ - have hb := (h n).inverse_displacement_bound T ha z (hmin hz) - have ht := norm_le_norm_sub_add (F n T z) z - rw [← dist_eq_norm] at ht - exact le_trans ht (add_le_add hb (hR z hz).le) |>.trans_eq (add_comm _ _) - · exact ⟨0,fun _ z hz => (hne ⟨z,hz⟩).elim⟩ - -theorem inverse_family_subsequence - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) (T : ℝ≥0) : - ∃ (σ : ℕ → ℕ) (f : ℂ → ℂ), StrictMono σ ∧ - DifferentiableOn ℂ f upperHalfPlane ∧ - TendstoLocallyUniformlyOn (fun n => F (σ n) T) f atTop upperHalfPlane := by - classical - obtain ⟨g,σ,hσ,hconv⟩ := holomorphic_halfPlane_family_subsequence - (fun n => F n T) (fun n => (h n).2.2.2.2.1 T) - (inverse_family_compact_bounds U γ G F h T) - let f : ℂ → ℂ := fun w => if hw : w ∈ upperHalfPlane then g ⟨w,hw⟩ else 0 - have hf : TendstoLocallyUniformlyOn (fun n => F (σ n) T) f atTop upperHalfPlane := by - rw [tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe] - apply tendstoLocallyUniformly_iff_forall_isCompact.mpr - intro K hK - have hh := ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn.mp hconv K hK - change TendstoUniformlyOn (fun n (w : upperHalfPlane) => F (σ n) T w) - (fun w : upperHalfPlane => f w) atTop K - have he : (fun w : upperHalfPlane => f w) = (g : upperHalfPlane → ℂ) := by - funext w - exact dite_eq_left w.property - rwa [he] - exact ⟨σ,f,hσ,hf.differentiableOn - (Eventually.of_forall fun n => (h (σ n)).2.2.2.2.1 T) isOpen_upperHalfPlane,hf⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerInverseStability +import OAI.Probability.SLE.Geometry.HolomorphicFamilyCompactness + +namespace OAI + +/-! Compactness of inverse maps at a fixed capacity time follows directly +from their reverse-flow displacement estimate. -/ +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +local instance : LocallyCompactSpace upperHalfPlane := + isOpen_upperHalfPlane.locallyCompactSpace + +lemma inverse_family_compact_bounds + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) (T : ℝ≥0) + (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) : + ∃ C : ℝ, ∀ n z, z ∈ K → ‖F n T z‖ ≤ C := by + by_cases hne : K.Nonempty + · obtain ⟨w,hw,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn + have ha : 0 < w.im := hKH hw + obtain ⟨R,_,hR⟩ := hK.isBounded.exists_pos_norm_lt + refine ⟨R+(2*w.im⁻¹)*T, fun n z hz => ?_⟩ + have hb := (h n).inverse_displacement_bound T ha z (hmin hz) + have ht := norm_le_norm_sub_add (F n T z) z + rw [← dist_eq_norm] at ht + exact le_trans ht (add_le_add hb (hR z hz).le) |>.trans_eq (add_comm _ _) + · exact ⟨0,fun _ z hz => (hne ⟨z,hz⟩).elim⟩ + +theorem inverse_family_subsequence + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) (T : ℝ≥0) : + ∃ (σ : ℕ → ℕ) (f : ℂ → ℂ), StrictMono σ ∧ + DifferentiableOn ℂ f upperHalfPlane ∧ + TendstoLocallyUniformlyOn (fun n => F (σ n) T) f atTop upperHalfPlane := by + classical + obtain ⟨g,σ,hσ,hconv⟩ := holomorphic_halfPlane_family_subsequence + (fun n => F n T) (fun n => (h n).2.2.2.2.1 T) + (inverse_family_compact_bounds U γ G F h T) + let f : ℂ → ℂ := fun w => if hw : w ∈ upperHalfPlane then g ⟨w,hw⟩ else 0 + have hf : TendstoLocallyUniformlyOn (fun n => F (σ n) T) f atTop upperHalfPlane := by + rw [tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe] + apply tendstoLocallyUniformly_iff_forall_isCompact.mpr + intro K hK + have hh := ContinuousMap.tendsto_iff_forall_isCompact_tendstoUniformlyOn.mp hconv K hK + change TendstoUniformlyOn (fun n (w : upperHalfPlane) => F (σ n) T w) + (fun w : upperHalfPlane => f w) atTop K + have he : (fun w : upperHalfPlane => f w) = (g : upperHalfPlane → ℂ) := by + funext w + exact dite_eq_left w.property + rwa [he] + exact ⟨σ,f,hσ,hf.differentiableOn + (Eventually.of_forall fun n => (h (σ n)).2.2.2.2.1 T) isOpen_upperHalfPlane,hf⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerInverseDriverContinuity.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerInverseDriverContinuity.lean index 9b3c752eb66aaea8cf3949479c4163671c6e3c4e..5584914977f6de50f35820591c189b77713074b0 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerInverseDriverContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerInverseDriverContinuity.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Loewner.LoewnerInverseCompactness - -namespace OAI - -/-! Uniformly close drivers give locally uniformly close inverse maps at -every fixed capacity time. This is an interior statement. -/ -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma inverse_maps_driver_distance - {U V : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F H J : ℝ≥0 → ℂ → ℂ} - (hu : IsCapacityTwoWitness U γ G F) (hv : IsCapacityTwoWitness V β H J) - (T : ℝ≥0) {a : ℝ} (ha : 0 < a) (w : ℂ) (haw : a ≤ w.im) - (δ : ℝ) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) : - dist (F T w) (J T w) ≤ gronwallBound 0 (4*a⁻¹^2) (2*a⁻¹^2*δ) T := by - have hh := hu.inverse_approximation_bound T ha w haw V δ hUV - (fun s => H ((T:ℝ)-s).toNNReal (J T w)) - (by simpa only [sub_zero,Real.toNNReal_coe] using hv.2.2.2.2.2.2.2.2.1 T w (ha.trans_le haw)) - (fun s hs => hv.reverse_path_derivative T w ha haw s hs) - simpa only [sub_self,Real.toNNReal_zero, - hv.initial_value _ (hv.2.2.2.2.2.2.1 T (ha.trans_le haw)).1.1] using hh - -theorem inverse_maps_locally_uniform_of_drivers - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) - (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) - (hUV : ∀ n t, t ∈ Icc 0 T → |U n t-V t| ≤ δ n) : - TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane := by - rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane] - intro K hKH hK - by_cases hne : K.Nonempty - · obtain ⟨w,hw,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn - have ha : 0 < w.im := hKH hw - have hz : Tendsto (fun n => gronwallBound 0 (4*w.im⁻¹^2) - (2*w.im⁻¹^2*δ n) T) atTop (𝓝 0) := by - have hh := (gronwallBound_continuous_ε 0 (4*w.im⁻¹^2) T).continuousAt.tendsto.comp - (hδ.const_mul (2*w.im⁻¹^2)) - simpa only [Function.comp_def,mul_zero,gronwallBound_ε0_δ0] using hh - rw [Metric.tendstoUniformlyOn_iff] - intro ε hε - filter_upwards [hz.eventually (Iio_mem_nhds hε)] with n hn z hzk - simpa only [dist_comm] using - (inverse_maps_driver_distance (h n) hv T ha z (hmin hzk) (δ n) (hUV n)).trans_lt hn - · have hKe : K=∅ := Set.not_nonempty_iff_eq_empty.mp hne - simp only [hKe,tendstoUniformlyOn_empty] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerInverseCompactness + +namespace OAI + +/-! Uniformly close drivers give locally uniformly close inverse maps at +every fixed capacity time. This is an interior statement. -/ +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma inverse_maps_driver_distance + {U V : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F H J : ℝ≥0 → ℂ → ℂ} + (hu : IsCapacityTwoWitness U γ G F) (hv : IsCapacityTwoWitness V β H J) + (T : ℝ≥0) {a : ℝ} (ha : 0 < a) (w : ℂ) (haw : a ≤ w.im) + (δ : ℝ) (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) : + dist (F T w) (J T w) ≤ gronwallBound 0 (4*a⁻¹^2) (2*a⁻¹^2*δ) T := by + have hh := hu.inverse_approximation_bound T ha w haw V δ hUV + (fun s => H ((T:ℝ)-s).toNNReal (J T w)) + (by simpa only [sub_zero,Real.toNNReal_coe] using hv.2.2.2.2.2.2.2.2.1 T w (ha.trans_le haw)) + (fun s hs => hv.reverse_path_derivative T w ha haw s hs) + simpa only [sub_self,Real.toNNReal_zero, + hv.initial_value _ (hv.2.2.2.2.2.2.1 T (ha.trans_le haw)).1.1] using hh + +theorem inverse_maps_locally_uniform_of_drivers + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) + (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) + (hUV : ∀ n t, t ∈ Icc 0 T → |U n t-V t| ≤ δ n) : + TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane := by + rw [tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane] + intro K hKH hK + by_cases hne : K.Nonempty + · obtain ⟨w,hw,hmin⟩ := hK.exists_isMinOn hne Complex.continuous_im.continuousOn + have ha : 0 < w.im := hKH hw + have hz : Tendsto (fun n => gronwallBound 0 (4*w.im⁻¹^2) + (2*w.im⁻¹^2*δ n) T) atTop (𝓝 0) := by + have hh := (gronwallBound_continuous_ε 0 (4*w.im⁻¹^2) T).continuousAt.tendsto.comp + (hδ.const_mul (2*w.im⁻¹^2)) + simpa only [Function.comp_def,mul_zero,gronwallBound_ε0_δ0] using hh + rw [Metric.tendstoUniformlyOn_iff] + intro ε hε + filter_upwards [hz.eventually (Iio_mem_nhds hε)] with n hn z hzk + simpa only [dist_comm] using + (inverse_maps_driver_distance (h n) hv T ha z (hmin hzk) (δ n) (hUV n)).trans_lt hn + · have hKe : K=∅ := Set.not_nonempty_iff_eq_empty.mp hne + simp only [hKe,tendstoUniformlyOn_empty] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerInverseStability.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerInverseStability.lean index 4a1098c06cc8750122453b8e959e4c7554495ca2..241914d494f8b1ba07c2e38f472b4f43c7fe8515 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerInverseStability.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerInverseStability.lean @@ -1,95 +1,95 @@ -import OAI.Probability.SLE.Loewner.LoewnerDriverBounds - -namespace OAI - -/-! Uniform perturbation estimates for inverse maps above a positive height. -They require no regularity of a boundary trace beyond the capacity witness. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma reverseField_driver_dist {a : ℝ} (ha : 0 < a) (u v : ℝ) (z : ℂ) : - dist (reverseField a u z) (reverseField a v z) ≤ 2 * a⁻¹ ^ 2 * |u-v| := by - have hc : reverseCenter a v z - reverseCenter a u z = ((u-v : ℝ) : ℂ) := by - apply Complex.ext <;> simp [reverseCenter] - have hu : ‖(reverseCenter a u z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv] - exact inv_anti₀ ha (reverseCenter_norm a u z) - have hv : ‖(reverseCenter a v z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv] - exact inv_anti₀ ha (reverseCenter_norm a v z) - rw [dist_eq_norm,reverseField,reverseField,← mul_sub,norm_mul,norm_neg,norm_ofNat, - inv_sub_inv' (reverseCenter_ne_zero ha _ _) (reverseCenter_ne_zero ha _ _), - norm_mul,norm_mul,hc,Complex.norm_real,Real.norm_eq_abs] - calc - 2 * (‖(reverseCenter a u z)⁻¹‖ * |u-v| * ‖(reverseCenter a v z)⁻¹‖) ≤ - 2 * (a⁻¹ * |u-v| * a⁻¹) := by gcongr - _ = _ := by ring - -lemma IsCapacityTwoWitness.inverse_approximation_bound - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) - (w : ℂ) (haw : a ≤ w.im) (V : ℝ≥0 → ℝ) (δ : ℝ) - (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) - (Q : ℝ → ℂ) (hQ0 : Q 0=w) - (hQ : ∀ s ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q - (reverseField a (V ((T:ℝ)-s).toNNReal) (Q s)) (Icc 0 (T:ℝ)) s) : - dist (F T w) (Q T) ≤ gronwallBound 0 (4*a⁻¹^2) (2*a⁻¹^2*δ) T := by - let q (s : ℝ) := G ((T:ℝ)-s).toNNReal (F T w) - have hq (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) := - h.reverse_path_derivative T w ha haw s hs - have hq0 : q 0=w := by - simp only [q,sub_zero,Real.toNNReal_coe] - exact h.2.2.2.2.2.2.2.2.1 T w (ha.trans_le haw) - have hqT : q T=F T w := by - simp only [q,sub_self,Real.toNNReal_zero] - exact h.initial_value _ (h.2.2.2.2.2.2.1 T (ha.trans_le haw)).1.1 - have he := dist_le_of_approx_trajectories_ODE - (K := (⟨4*a⁻¹^2, by positivity⟩ : ℝ≥0)) - (v := fun s => reverseField a (U ((T:ℝ)-s).toNNReal)) - (fun _ => reverseField_lipschitz ha _) - (f := q) (fun s hs => (hq s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hq s ⟨hs.1,hs.2.le⟩)) - (εf := 0) (fun _ _ => by simp [q]) - (g := Q) (fun s hs => (hQ s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQ s ⟨hs.1,hs.2.le⟩)) - (εg := 2*a⁻¹^2*δ) (fun s hs => ?_) - (δ := 0) (by rw [hq0,hQ0,dist_self]) - T ⟨T.coe_nonneg,le_rfl⟩ - · change dist (q T) (Q T) ≤ - gronwallBound 0 (4*a⁻¹^2) (0+2*a⁻¹^2*δ) ((T:ℝ)-0) at he - simpa only [hqT,zero_add,sub_zero] using he - · have ht : ((T:ℝ)-s).toNNReal ∈ Icc 0 T := by - refine ⟨zero_le,?_⟩ - exact_mod_cast (show (((T:ℝ)-s).toNNReal : ℝ) ≤ T by - rw [Real.coe_toNNReal _ (by linarith [hs.2])]; linarith [hs.1]) - rw [dist_comm] - exact (reverseField_driver_dist ha _ _ _).trans - (mul_le_mul_of_nonneg_left (hUV _ ht) (by positivity)) - -lemma IsCapacityTwoWitness.inverse_displacement_bound - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) - (w : ℂ) (haw : a ≤ w.im) : dist (F T w) w ≤ (2*a⁻¹)*T := by - let q (s : ℝ) := G ((T:ℝ)-s).toNNReal (F T w) - have hq (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) := - h.reverse_path_derivative T w ha haw s hs - have hb := norm_image_sub_le_of_norm_deriv_le_segment' - (fun s hs => hq s hs) (fun s _ => reverseField_bound ha _ _) T - ⟨T.coe_nonneg,le_rfl⟩ - have hq0 : q 0=w := by - simp only [q,sub_zero,Real.toNNReal_coe] - exact h.2.2.2.2.2.2.2.2.1 T w (ha.trans_le haw) - have hqT : q T=F T w := by - simp only [q,sub_self,Real.toNNReal_zero] - exact h.initial_value _ (h.2.2.2.2.2.2.1 T (ha.trans_le haw)).1.1 - change ‖q T-q 0‖ ≤ _ at hb - simpa only [hqT,hq0,sub_zero,dist_eq_norm] using hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerDriverBounds + +namespace OAI + +/-! Uniform perturbation estimates for inverse maps above a positive height. +They require no regularity of a boundary trace beyond the capacity witness. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma reverseField_driver_dist {a : ℝ} (ha : 0 < a) (u v : ℝ) (z : ℂ) : + dist (reverseField a u z) (reverseField a v z) ≤ 2 * a⁻¹ ^ 2 * |u-v| := by + have hc : reverseCenter a v z - reverseCenter a u z = ((u-v : ℝ) : ℂ) := by + apply Complex.ext <;> simp [reverseCenter] + have hu : ‖(reverseCenter a u z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv] + exact inv_anti₀ ha (reverseCenter_norm a u z) + have hv : ‖(reverseCenter a v z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv] + exact inv_anti₀ ha (reverseCenter_norm a v z) + rw [dist_eq_norm,reverseField,reverseField,← mul_sub,norm_mul,norm_neg,norm_ofNat, + inv_sub_inv' (reverseCenter_ne_zero ha _ _) (reverseCenter_ne_zero ha _ _), + norm_mul,norm_mul,hc,Complex.norm_real,Real.norm_eq_abs] + calc + 2 * (‖(reverseCenter a u z)⁻¹‖ * |u-v| * ‖(reverseCenter a v z)⁻¹‖) ≤ + 2 * (a⁻¹ * |u-v| * a⁻¹) := by gcongr + _ = _ := by ring + +lemma IsCapacityTwoWitness.inverse_approximation_bound + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) + (w : ℂ) (haw : a ≤ w.im) (V : ℝ≥0 → ℝ) (δ : ℝ) + (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) + (Q : ℝ → ℂ) (hQ0 : Q 0=w) + (hQ : ∀ s ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q + (reverseField a (V ((T:ℝ)-s).toNNReal) (Q s)) (Icc 0 (T:ℝ)) s) : + dist (F T w) (Q T) ≤ gronwallBound 0 (4*a⁻¹^2) (2*a⁻¹^2*δ) T := by + let q (s : ℝ) := G ((T:ℝ)-s).toNNReal (F T w) + have hq (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) := + h.reverse_path_derivative T w ha haw s hs + have hq0 : q 0=w := by + simp only [q,sub_zero,Real.toNNReal_coe] + exact h.2.2.2.2.2.2.2.2.1 T w (ha.trans_le haw) + have hqT : q T=F T w := by + simp only [q,sub_self,Real.toNNReal_zero] + exact h.initial_value _ (h.2.2.2.2.2.2.1 T (ha.trans_le haw)).1.1 + have he := dist_le_of_approx_trajectories_ODE + (K := (⟨4*a⁻¹^2, by positivity⟩ : ℝ≥0)) + (v := fun s => reverseField a (U ((T:ℝ)-s).toNNReal)) + (fun _ => reverseField_lipschitz ha _) + (f := q) (fun s hs => (hq s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hq s ⟨hs.1,hs.2.le⟩)) + (εf := 0) (fun _ _ => by simp [q]) + (g := Q) (fun s hs => (hQ s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQ s ⟨hs.1,hs.2.le⟩)) + (εg := 2*a⁻¹^2*δ) (fun s hs => ?_) + (δ := 0) (by rw [hq0,hQ0,dist_self]) + T ⟨T.coe_nonneg,le_rfl⟩ + · change dist (q T) (Q T) ≤ + gronwallBound 0 (4*a⁻¹^2) (0+2*a⁻¹^2*δ) ((T:ℝ)-0) at he + simpa only [hqT,zero_add,sub_zero] using he + · have ht : ((T:ℝ)-s).toNNReal ∈ Icc 0 T := by + refine ⟨zero_le,?_⟩ + exact_mod_cast (show (((T:ℝ)-s).toNNReal : ℝ) ≤ T by + rw [Real.coe_toNNReal _ (by linarith [hs.2])]; linarith [hs.1]) + rw [dist_comm] + exact (reverseField_driver_dist ha _ _ _).trans + (mul_le_mul_of_nonneg_left (hUV _ ht) (by positivity)) + +lemma IsCapacityTwoWitness.inverse_displacement_bound + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {a : ℝ} (ha : 0 < a) + (w : ℂ) (haw : a ≤ w.im) : dist (F T w) w ≤ (2*a⁻¹)*T := by + let q (s : ℝ) := G ((T:ℝ)-s).toNNReal (F T w) + have hq (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) := + h.reverse_path_derivative T w ha haw s hs + have hb := norm_image_sub_le_of_norm_deriv_le_segment' + (fun s hs => hq s hs) (fun s _ => reverseField_bound ha _ _) T + ⟨T.coe_nonneg,le_rfl⟩ + have hq0 : q 0=w := by + simp only [q,sub_zero,Real.toNNReal_coe] + exact h.2.2.2.2.2.2.2.2.1 T w (ha.trans_le haw) + have hqT : q T=F T w := by + simp only [q,sub_self,Real.toNNReal_zero] + exact h.initial_value _ (h.2.2.2.2.2.2.1 T (ha.trans_le haw)).1.1 + change ‖q T-q 0‖ ≤ _ at hb + simpa only [hqT,hq0,sub_zero,dist_eq_norm] using hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerKernelContinuity.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerKernelContinuity.lean index e14e03701cf04b5f12153915932e5855fe9bd05c..7996a977d6abd5abcbb17caf38d341454ccbeb4b 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerKernelContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerKernelContinuity.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Loewner.LoewnerInverseDriverContinuity -import OAI.Probability.SLE.Loewner.UnivalentLocallyUniformLimit - -namespace OAI - -/-! Compact interior regions of a limiting Loewner domain persist under -uniform perturbations of the driver. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem inverse_image_neighborhood_persists - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) - (hl : TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane) - {w : ℂ} (hw : w ∈ upperHalfPlane) : - ∃ r : ℝ, 0 < r ∧ ∀ᶠ n in atTop, ball (J T w) r ⊆ survivingDomain (γ n) T := by - let R := w.im/2 - have hwpos : 0 < w.im := hw - have hR : 0 < R := half_pos hw - have hKH : closedBall w R ⊆ upperHalfPlane := by - intro z hz - have hz' : ‖z-w‖ ≤ R := by simpa only [mem_closedBall,dist_eq_norm] using hz - have hd := (Complex.abs_im_le_norm (z-w)).trans hz' - simp only [Complex.sub_im] at hd - change 0 < z.im - dsimp [R] at hd - linarith [(abs_le.mp hd).1] - have hJi : InjOn (J T) upperHalfPlane := by - intro z hz v hv' he - rw [← hv.2.2.2.2.2.2.2.2.1 T z hz,he,hv.2.2.2.2.2.2.2.2.1 T v hv'] - have hsep (z : ℂ) (hz : z ∈ sphere w R) : J T z ≠ J T w := by - intro he - have heq := hJi (hKH (sphere_subset_closedBall hz)) hw he - exact (ne_of_mem_sphere hz hR.ne') heq - have hc : ContinuousOn (fun z => ‖J T z-J T w‖) (sphere w R) := - (((hv.2.2.2.2.1 T).continuousOn.mono (sphere_subset_closedBall.trans hKH)).sub - continuousOn_const).norm - obtain ⟨v,hvS,hmin⟩ := (isCompact_sphere w R).exists_isMinOn - (NormedSpace.sphere_nonempty.mpr hR.le) hc - let d : ℝ := ‖J T v-J T w‖ - have hd : 0 < d := norm_pos_iff.mpr (sub_ne_zero.mpr (hsep v hvS)) - have hsmall := (Metric.tendstoUniformlyOn_iff.mp - ((tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane).mp hl - (closedBall w R) hKH (isCompact_closedBall _ _))) (d/8) (by positivity) - refine ⟨d/8,by positivity,?_⟩ - filter_upwards [hsmall] with n hn - have hn' (z : ℂ) (hz : z ∈ closedBall w R) : ‖F n T z-J T z‖ < d/8 := by - simpa only [dist_eq_norm,norm_sub_rev] using hn z hz - have hcenter := hn' w (mem_closedBall_self hR.le) - have hboundary (z : ℂ) (hz : z ∈ sphere w R) : d/2 ≤ ‖F n T z-F n T w‖ := by - have herror := hn' z (sphere_subset_closedBall hz) - have htri : ‖J T z-J T w‖ ≤ ‖J T z-F n T z‖+ - ‖F n T z-F n T w‖+‖F n T w-J T w‖ := by - calc - _ = ‖(J T z-F n T z)+(F n T z-F n T w)+(F n T w-J T w)‖ := by congr 1; ring - _ ≤ _ := (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) - rw [norm_sub_rev (J T z) (F n T z)] at htri - have hdmin : d ≤ ‖J T z-J T w‖ := hmin hz - linarith - have hFi : InjOn (F n T) upperHalfPlane := by - intro z hz v hv' he - rw [← (h n).2.2.2.2.2.2.2.2.1 T z hz,he,(h n).2.2.2.2.2.2.2.2.1 T v hv'] - have hdc : DiffContOnCl ℂ (F n T) (ball w R) := - ⟨((h n).2.2.2.2.1 T).mono (ball_subset_closedBall.trans hKH),by - rw [closure_ball w hR.ne'] - exact ((h n).2.2.2.2.1 T).continuousOn.mono hKH⟩ - have hnot : ∃ᶠ z in 𝓝 w, F n T z ≠ F n T w := - not_eventually.mp (univalent_not_eventually_constant isOpen_upperHalfPlane hFi hw) - intro z hz - have hz' : z ∈ ball (F n T w) ((d/2)/2) := by - have hdist := dist_triangle z (J T w) (F n T w) - rw [mem_ball,dist_eq_norm] at hz ⊢ - simp only [dist_eq_norm,norm_sub_rev (J T w) (F n T w)] at hdist - linarith - obtain ⟨x,hx,he⟩ := hdc.ball_subset_image_closedBall hR hboundary hnot hz' - rw [← he] - exact (h n).2.2.2.2.2.2.1 T (hKH hx) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerInverseDriverContinuity +import OAI.Probability.SLE.Loewner.UnivalentLocallyUniformLimit + +namespace OAI + +/-! Compact interior regions of a limiting Loewner domain persist under +uniform perturbations of the driver. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem inverse_image_neighborhood_persists + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) + (hl : TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane) + {w : ℂ} (hw : w ∈ upperHalfPlane) : + ∃ r : ℝ, 0 < r ∧ ∀ᶠ n in atTop, ball (J T w) r ⊆ survivingDomain (γ n) T := by + let R := w.im/2 + have hwpos : 0 < w.im := hw + have hR : 0 < R := half_pos hw + have hKH : closedBall w R ⊆ upperHalfPlane := by + intro z hz + have hz' : ‖z-w‖ ≤ R := by simpa only [mem_closedBall,dist_eq_norm] using hz + have hd := (Complex.abs_im_le_norm (z-w)).trans hz' + simp only [Complex.sub_im] at hd + change 0 < z.im + dsimp [R] at hd + linarith [(abs_le.mp hd).1] + have hJi : InjOn (J T) upperHalfPlane := by + intro z hz v hv' he + rw [← hv.2.2.2.2.2.2.2.2.1 T z hz,he,hv.2.2.2.2.2.2.2.2.1 T v hv'] + have hsep (z : ℂ) (hz : z ∈ sphere w R) : J T z ≠ J T w := by + intro he + have heq := hJi (hKH (sphere_subset_closedBall hz)) hw he + exact (ne_of_mem_sphere hz hR.ne') heq + have hc : ContinuousOn (fun z => ‖J T z-J T w‖) (sphere w R) := + (((hv.2.2.2.2.1 T).continuousOn.mono (sphere_subset_closedBall.trans hKH)).sub + continuousOn_const).norm + obtain ⟨v,hvS,hmin⟩ := (isCompact_sphere w R).exists_isMinOn + (NormedSpace.sphere_nonempty.mpr hR.le) hc + let d : ℝ := ‖J T v-J T w‖ + have hd : 0 < d := norm_pos_iff.mpr (sub_ne_zero.mpr (hsep v hvS)) + have hsmall := (Metric.tendstoUniformlyOn_iff.mp + ((tendstoLocallyUniformlyOn_iff_forall_isCompact isOpen_upperHalfPlane).mp hl + (closedBall w R) hKH (isCompact_closedBall _ _))) (d/8) (by positivity) + refine ⟨d/8,by positivity,?_⟩ + filter_upwards [hsmall] with n hn + have hn' (z : ℂ) (hz : z ∈ closedBall w R) : ‖F n T z-J T z‖ < d/8 := by + simpa only [dist_eq_norm,norm_sub_rev] using hn z hz + have hcenter := hn' w (mem_closedBall_self hR.le) + have hboundary (z : ℂ) (hz : z ∈ sphere w R) : d/2 ≤ ‖F n T z-F n T w‖ := by + have herror := hn' z (sphere_subset_closedBall hz) + have htri : ‖J T z-J T w‖ ≤ ‖J T z-F n T z‖+ + ‖F n T z-F n T w‖+‖F n T w-J T w‖ := by + calc + _ = ‖(J T z-F n T z)+(F n T z-F n T w)+(F n T w-J T w)‖ := by congr 1; ring + _ ≤ _ := (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) + rw [norm_sub_rev (J T z) (F n T z)] at htri + have hdmin : d ≤ ‖J T z-J T w‖ := hmin hz + linarith + have hFi : InjOn (F n T) upperHalfPlane := by + intro z hz v hv' he + rw [← (h n).2.2.2.2.2.2.2.2.1 T z hz,he,(h n).2.2.2.2.2.2.2.2.1 T v hv'] + have hdc : DiffContOnCl ℂ (F n T) (ball w R) := + ⟨((h n).2.2.2.2.1 T).mono (ball_subset_closedBall.trans hKH),by + rw [closure_ball w hR.ne'] + exact ((h n).2.2.2.2.1 T).continuousOn.mono hKH⟩ + have hnot : ∃ᶠ z in 𝓝 w, F n T z ≠ F n T w := + not_eventually.mp (univalent_not_eventually_constant isOpen_upperHalfPlane hFi hw) + intro z hz + have hz' : z ∈ ball (F n T w) ((d/2)/2) := by + have hdist := dist_triangle z (J T w) (F n T w) + rw [mem_ball,dist_eq_norm] at hz ⊢ + simp only [dist_eq_norm,norm_sub_rev (J T w) (F n T w)] at hdist + linarith + obtain ⟨x,hx,he⟩ := hdc.ball_subset_image_closedBall hR hboundary hnot hz' + rw [← he] + exact (h n).2.2.2.2.2.2.1 T (hKH hx) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerSmallDriverHeight.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerSmallDriverHeight.lean index 83e1c11535fba887c86f82ad28bb8932f5d6c48b..b863cf9782c38180c1164a51e9e368355ea3a513 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerSmallDriverHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerSmallDriverHeight.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Loewner.LoewnerInverseCompactness -import OAI.Probability.SLE.Loewner.VerticalMapDerivative -import OAI.Probability.SLE.Loewner.LoewnerHeightClearance - -namespace OAI - -/-! Drivers converging uniformly to zero cannot all generate traces below -a height strictly smaller than the corresponding vertical slit. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem small_drivers_not_all_height_bounded - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - (T : ℝ≥0) (H : ℝ) (hH : 0 ≤ H) (hHT : H < 2*Real.sqrt T) - (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) - (hU : ∀ n t, t ∈ Icc 0 T → |U n t| ≤ δ n) : - ¬ ∀ n t, t ∈ Icc 0 T → (γ n t).im ≤ H := by - intro hlow - obtain ⟨σ,f,hσ,hf,hlim⟩ := inverse_family_subsequence U γ G F h T - have haxis (y : ℝ) (hy : 0 < y) : f ((y:ℂ)*I)=verticalReversePath y T := by - exact tendsto_nhds_unique - (hlim.tendsto_at (by simpa [upperHalfPlane] using hy)) - ((inverse_vertical_tendsto U γ G F h T δ hδ hU hy).comp hσ.tendsto_atTop) - let r : ℝ := (2*Real.sqrt T-H)/4 - have hr : 0 < r := by dsimp [r]; linarith - have hT : 0 < (T:ℝ) := by - by_contra hn - have he : (T:ℝ)=0 := le_antisymm (le_of_not_gt hn) T.coe_nonneg - simp only [he,Real.sqrt_zero,mul_zero] at hHT - linarith - have hsqrt (y : ℝ) : 2*Real.sqrt T ≤ Real.sqrt (y^2+4*T) := by - have h1 := Real.sq_sqrt T.coe_nonneg - have h2 := Real.sq_sqrt (show 0 ≤ y^2+4*(T:ℝ) by positivity) - nlinarith [Real.sqrt_nonneg (T:ℝ),Real.sqrt_nonneg (y^2+4*(T:ℝ))] - have hbound (y : ℝ) (hy : 0 < y) : - r ≤ (128*Real.pi)*y*(y/Real.sqrt (y^2+4*T)) := by - have hiy : (y:ℂ)*I ∈ upperHalfPlane := by simpa [upperHalfPlane] using hy - have hi := Complex.continuous_im.continuousAt.tendsto.comp (hlim.tendsto_at hiy) - rw [haxis y hy] at hi - have hheight : H+2*r < (verticalReversePath y T).im := by - simp only [verticalReversePath,Complex.mul_im,Complex.ofReal_re,Complex.I_im, - Complex.ofReal_im,Complex.I_re,mul_one,mul_zero,add_zero] - dsimp [r] - linarith [hsqrt y] - have hev : ∀ᶠ n in atTop, H+2*r < (F (σ n) T ((y:ℂ)*I)).im := - hi.eventually (Ioi_mem_nhds hheight) - have hb : ∀ᶠ n in atTop, - r ≤ (128*Real.pi)*y*‖deriv (F (σ n) T) ((y:ℂ)*I)‖ := by - filter_upwards [hev] with n hn - have hball : ball (F (σ n) T ((y:ℂ)*I)) r ⊆ survivingDomain (γ (σ n)) T := by - intro z hz - have hzdist : ‖z-F (σ n) T ((y:ℂ)*I)‖ < r := by - simpa only [mem_ball,dist_eq_norm] using hz - have hzheight := (abs_lt.mp ((Complex.abs_im_le_norm - (z-F (σ n) T ((y:ℂ)*I))).trans_lt hzdist)).1 - simp only [Complex.sub_im] at hzheight - apply survives_above_trace_height (γ (σ n)) T H (hlow (σ n)) - · linarith - · linarith - simpa only [Complex.mul_im,Complex.ofReal_re,Complex.I_im,mul_one, - Complex.ofReal_im,Complex.I_re,mul_zero,add_zero] using - (h (σ n)).inverse_clearance_bound T hiy hr hball - have hd := ((hlim.deriv (Eventually.of_forall fun n => (h (σ n)).2.2.2.2.1 T) - isOpen_upperHalfPlane).tendsto_at hiy).norm - have hh := ge_of_tendsto (hd.const_mul ((128*Real.pi)*y)) hb - rw [vertical_axis_derivative f T hf haxis hy,Complex.norm_real,Real.norm_eq_abs, - abs_of_nonneg (div_nonneg hy.le (Real.sqrt_nonneg _))] at hh - exact hh - let y : ℕ → ℝ := fun n => 1/((n:ℝ)+1) - have hy : Tendsto y atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat - have hden := ((hy.pow 2).add_const (4*(T:ℝ))).sqrt - have hden0 : Real.sqrt (0^2+4*(T:ℝ)) ≠ 0 := (Real.sqrt_pos.mpr (by positivity)).ne' - have hl := ((hy.const_mul (128*Real.pi)).mul (hy.div hden hden0)) - have hzero : r ≤ 0 := by - apply ge_of_tendsto (by simpa using hl) - exact Eventually.of_forall (fun n => hbound (y n) (by dsimp [y]; positivity)) - exact (not_le_of_gt hr) hzero - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerInverseCompactness +import OAI.Probability.SLE.Loewner.VerticalMapDerivative +import OAI.Probability.SLE.Loewner.LoewnerHeightClearance + +namespace OAI + +/-! Drivers converging uniformly to zero cannot all generate traces below +a height strictly smaller than the corresponding vertical slit. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem small_drivers_not_all_height_bounded + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + (T : ℝ≥0) (H : ℝ) (hH : 0 ≤ H) (hHT : H < 2*Real.sqrt T) + (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) + (hU : ∀ n t, t ∈ Icc 0 T → |U n t| ≤ δ n) : + ¬ ∀ n t, t ∈ Icc 0 T → (γ n t).im ≤ H := by + intro hlow + obtain ⟨σ,f,hσ,hf,hlim⟩ := inverse_family_subsequence U γ G F h T + have haxis (y : ℝ) (hy : 0 < y) : f ((y:ℂ)*I)=verticalReversePath y T := by + exact tendsto_nhds_unique + (hlim.tendsto_at (by simpa [upperHalfPlane] using hy)) + ((inverse_vertical_tendsto U γ G F h T δ hδ hU hy).comp hσ.tendsto_atTop) + let r : ℝ := (2*Real.sqrt T-H)/4 + have hr : 0 < r := by dsimp [r]; linarith + have hT : 0 < (T:ℝ) := by + by_contra hn + have he : (T:ℝ)=0 := le_antisymm (le_of_not_gt hn) T.coe_nonneg + simp only [he,Real.sqrt_zero,mul_zero] at hHT + linarith + have hsqrt (y : ℝ) : 2*Real.sqrt T ≤ Real.sqrt (y^2+4*T) := by + have h1 := Real.sq_sqrt T.coe_nonneg + have h2 := Real.sq_sqrt (show 0 ≤ y^2+4*(T:ℝ) by positivity) + nlinarith [Real.sqrt_nonneg (T:ℝ),Real.sqrt_nonneg (y^2+4*(T:ℝ))] + have hbound (y : ℝ) (hy : 0 < y) : + r ≤ (128*Real.pi)*y*(y/Real.sqrt (y^2+4*T)) := by + have hiy : (y:ℂ)*I ∈ upperHalfPlane := by simpa [upperHalfPlane] using hy + have hi := Complex.continuous_im.continuousAt.tendsto.comp (hlim.tendsto_at hiy) + rw [haxis y hy] at hi + have hheight : H+2*r < (verticalReversePath y T).im := by + simp only [verticalReversePath,Complex.mul_im,Complex.ofReal_re,Complex.I_im, + Complex.ofReal_im,Complex.I_re,mul_one,mul_zero,add_zero] + dsimp [r] + linarith [hsqrt y] + have hev : ∀ᶠ n in atTop, H+2*r < (F (σ n) T ((y:ℂ)*I)).im := + hi.eventually (Ioi_mem_nhds hheight) + have hb : ∀ᶠ n in atTop, + r ≤ (128*Real.pi)*y*‖deriv (F (σ n) T) ((y:ℂ)*I)‖ := by + filter_upwards [hev] with n hn + have hball : ball (F (σ n) T ((y:ℂ)*I)) r ⊆ survivingDomain (γ (σ n)) T := by + intro z hz + have hzdist : ‖z-F (σ n) T ((y:ℂ)*I)‖ < r := by + simpa only [mem_ball,dist_eq_norm] using hz + have hzheight := (abs_lt.mp ((Complex.abs_im_le_norm + (z-F (σ n) T ((y:ℂ)*I))).trans_lt hzdist)).1 + simp only [Complex.sub_im] at hzheight + apply survives_above_trace_height (γ (σ n)) T H (hlow (σ n)) + · linarith + · linarith + simpa only [Complex.mul_im,Complex.ofReal_re,Complex.I_im,mul_one, + Complex.ofReal_im,Complex.I_re,mul_zero,add_zero] using + (h (σ n)).inverse_clearance_bound T hiy hr hball + have hd := ((hlim.deriv (Eventually.of_forall fun n => (h (σ n)).2.2.2.2.1 T) + isOpen_upperHalfPlane).tendsto_at hiy).norm + have hh := ge_of_tendsto (hd.const_mul ((128*Real.pi)*y)) hb + rw [vertical_axis_derivative f T hf haxis hy,Complex.norm_real,Real.norm_eq_abs, + abs_of_nonneg (div_nonneg hy.le (Real.sqrt_nonneg _))] at hh + exact hh + let y : ℕ → ℝ := fun n => 1/((n:ℝ)+1) + have hy : Tendsto y atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat + have hden := ((hy.pow 2).add_const (4*(T:ℝ))).sqrt + have hden0 : Real.sqrt (0^2+4*(T:ℝ)) ≠ 0 := (Real.sqrt_pos.mpr (by positivity)).ne' + have hl := ((hy.const_mul (128*Real.pi)).mul (hy.div hden hden0)) + have hzero : r ≤ 0 := by + apply ge_of_tendsto (by simpa using hl) + exact Eventually.of_forall (fun n => hbound (y n) (by dsimp [y]; positivity)) + exact (not_le_of_gt hr) hzero + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerTipHullApproximation.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerTipHullApproximation.lean index b6f2daf7ce6372f1680cd2d2b957b244a2a5bcaf..2072d7a46c71a2917c60ace298217abd755529f9 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerTipHullApproximation.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerTipHullApproximation.lean @@ -1,83 +1,83 @@ -import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryDerivative -import OAI.Probability.SLE.Loewner.LoewnerHeightClearance -import OAI.Probability.SLE.Loewner.LoewnerInverseDriverContinuity - -namespace OAI - -/-! Kernel convergence cannot leave a fixed surviving ball around the -limiting tip. The boundary derivative estimate gives a quantitative -contradiction to such a ball. -/ -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem inverse_convergence_excludes_common_tip_ball - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) - (hl : TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane) - {r : ℝ} (hr : 0 < r) : - ¬ ∀ n, ball (β T) r ⊆ survivingDomain (γ n) T := by - intro hballs - have hnear : ∀ᶠ y : ℝ in 𝓝[>] 0, - dist (J T ((V T : ℂ)+(y : ℂ)*Complex.I)) (β T) < r/4 := - (hv.2.2.2.2.2.2.2.2.2.2.2.2 T).eventually (ball_mem_nhds _ (by positivity)) - have hbound : ∀ᶠ y : ℝ in 𝓝[>] 0, - r/2 ≤ (128*Real.pi)*(y*‖deriv (J T) ((V T : ℂ)+(y : ℂ)*Complex.I)‖) := by - filter_upwards [hnear,self_mem_nhdsWithin] with y hy hy0 - let w : ℂ := (V T : ℂ)+(y : ℂ)*Complex.I - have hw : w ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy0 - have hvalue := hl.tendsto_at hw - have hd := ((hl.deriv (Eventually.of_forall fun n => (h n).2.2.2.2.1 T) - isOpen_upperHalfPlane).tendsto_at hw).norm - have hb : ∀ᶠ n in atTop, r/2 ≤ (128*Real.pi)*y*‖deriv (F n T) w‖ := by - filter_upwards [hvalue.eventually (ball_mem_nhds _ (by positivity : 0 < r/4))] - with n hn - have hball : ball (F n T w) (r/2) ⊆ survivingDomain (γ n) T := by - intro z hz - apply hballs n - rw [mem_ball] at hz ⊢ - have ht := dist_triangle z (F n T w) (β T) - have ht' := dist_triangle (F n T w) (J T w) (β T) - change dist (F n T w) (J T w) < r/4 at hn - change dist (J T w) (β T) < r/4 at hy - linarith - simpa only [w,Complex.add_im,Complex.ofReal_im,Complex.mul_im, - Complex.ofReal_re,Complex.I_re,Complex.I_im,mul_one,mul_zero, - zero_add,add_zero] using (h n).inverse_clearance_bound T hw - (by positivity : 0 < r/2) hball - have hh := ge_of_tendsto (hd.const_mul ((128*Real.pi)*y)) hb - simpa only [mul_assoc] using hh - have hzero : r/2 ≤ 0 := by - have hh := ge_of_tendsto ((hv.inverse_tip_derivative_vanishes T).const_mul - (128*Real.pi)) hbound - simpa only [mul_zero] using hh - linarith - -theorem driver_neighborhood_hits_tip_hull - {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) {r : ℝ} (hr : 0 < r) : - ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → - (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → - ¬ ball (β T) r ⊆ survivingDomain γ T := by - classical - by_contra hn - push Not at hn - have hc (n : ℕ) := hn ((n+1:ℝ)⁻¹) (by positivity) - choose U γ G F hw hclose hball using hc - have hδ : Tendsto (fun n : ℕ => (n+1:ℝ)⁻¹) atTop (𝓝 0) := - tendsto_inv_atTop_zero.comp - (tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop) - have hl := inverse_maps_locally_uniform_of_drivers U γ G F hw hv T - (fun n : ℕ => (n+1:ℝ)⁻¹) hδ hclose - exact inverse_convergence_excludes_common_tip_ball U γ G F hw hv T hl hr hball - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryDerivative +import OAI.Probability.SLE.Loewner.LoewnerHeightClearance +import OAI.Probability.SLE.Loewner.LoewnerInverseDriverContinuity + +namespace OAI + +/-! Kernel convergence cannot leave a fixed surviving ball around the +limiting tip. The boundary derivative estimate gives a quantitative +contradiction to such a ball. -/ +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem inverse_convergence_excludes_common_tip_ball + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) + (hl : TendstoLocallyUniformlyOn (fun n => F n T) (J T) atTop upperHalfPlane) + {r : ℝ} (hr : 0 < r) : + ¬ ∀ n, ball (β T) r ⊆ survivingDomain (γ n) T := by + intro hballs + have hnear : ∀ᶠ y : ℝ in 𝓝[>] 0, + dist (J T ((V T : ℂ)+(y : ℂ)*Complex.I)) (β T) < r/4 := + (hv.2.2.2.2.2.2.2.2.2.2.2.2 T).eventually (ball_mem_nhds _ (by positivity)) + have hbound : ∀ᶠ y : ℝ in 𝓝[>] 0, + r/2 ≤ (128*Real.pi)*(y*‖deriv (J T) ((V T : ℂ)+(y : ℂ)*Complex.I)‖) := by + filter_upwards [hnear,self_mem_nhdsWithin] with y hy hy0 + let w : ℂ := (V T : ℂ)+(y : ℂ)*Complex.I + have hw : w ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy0 + have hvalue := hl.tendsto_at hw + have hd := ((hl.deriv (Eventually.of_forall fun n => (h n).2.2.2.2.1 T) + isOpen_upperHalfPlane).tendsto_at hw).norm + have hb : ∀ᶠ n in atTop, r/2 ≤ (128*Real.pi)*y*‖deriv (F n T) w‖ := by + filter_upwards [hvalue.eventually (ball_mem_nhds _ (by positivity : 0 < r/4))] + with n hn + have hball : ball (F n T w) (r/2) ⊆ survivingDomain (γ n) T := by + intro z hz + apply hballs n + rw [mem_ball] at hz ⊢ + have ht := dist_triangle z (F n T w) (β T) + have ht' := dist_triangle (F n T w) (J T w) (β T) + change dist (F n T w) (J T w) < r/4 at hn + change dist (J T w) (β T) < r/4 at hy + linarith + simpa only [w,Complex.add_im,Complex.ofReal_im,Complex.mul_im, + Complex.ofReal_re,Complex.I_re,Complex.I_im,mul_one,mul_zero, + zero_add,add_zero] using (h n).inverse_clearance_bound T hw + (by positivity : 0 < r/2) hball + have hh := ge_of_tendsto (hd.const_mul ((128*Real.pi)*y)) hb + simpa only [mul_assoc] using hh + have hzero : r/2 ≤ 0 := by + have hh := ge_of_tendsto ((hv.inverse_tip_derivative_vanishes T).const_mul + (128*Real.pi)) hbound + simpa only [mul_zero] using hh + linarith + +theorem driver_neighborhood_hits_tip_hull + {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) {r : ℝ} (hr : 0 < r) : + ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → + (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → + ¬ ball (β T) r ⊆ survivingDomain γ T := by + classical + by_contra hn + push Not at hn + have hc (n : ℕ) := hn ((n+1:ℝ)⁻¹) (by positivity) + choose U γ G F hw hclose hball using hc + have hδ : Tendsto (fun n : ℕ => (n+1:ℝ)⁻¹) atTop (𝓝 0) := + tendsto_inv_atTop_zero.comp + (tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop) + have hl := inverse_maps_locally_uniform_of_drivers U γ G F hw hv T + (fun n : ℕ => (n+1:ℝ)⁻¹) hδ hclose + exact inverse_convergence_excludes_common_tip_ball U γ G F hw hv T hl hr hball + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerTipVisit.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerTipVisit.lean index 8b046a796c75ab3d40c184abcf81e49730e315a3..049c2534dcd25aa2aeddb1903dff1e333c020655 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerTipVisit.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerTipVisit.lean @@ -1,83 +1,83 @@ -import OAI.Probability.SLE.Loewner.LoewnerTipHullApproximation -import OAI.Probability.SLE.Loewner.LoewnerCompactDriverNeighborhood - -namespace OAI - -/-! A perturbed hull near an interior reference tip must be visited by -the actual trace: a surviving point in the same connected ball rules out -the possibility that the ball is entirely swallowed. -/ -noncomputable section -open Set Filter Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma trace_meets_connected_region - (γ : ℝ≥0 → ℂ) (T : ℝ≥0) {S : Set ℂ} - (hS : IsPreconnected S) (hSH : S ⊆ upperHalfPlane) - {w : ℂ} (hwS : w ∈ S) (hwD : w ∈ survivingDomain γ T) - (hnot : ¬ S ⊆ survivingDomain γ T) : - ∃ t ∈ Icc 0 T, γ t ∈ S := by - by_contra hn - push Not at hn - have hf : S ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by - intro z hz - refine ⟨hSH hz,?_⟩ - rintro ⟨t,ht,rfl⟩ - exact hn t ht hz - apply hnot - intro z hz - have hc := hS.subset_connectedComponentIn hwS hf hz - refine ⟨hf hz,?_⟩ - rw [← connectedComponentIn_eq hc] - exact hwD.2 - -lemma ball_subset_halfPlane {q : ℂ} {r : ℝ} (hr : r ≤ q.im) : - ball q r ⊆ upperHalfPlane := by - intro z hz - have hd : ‖z-q‖ < r := by simpa only [mem_ball,dist_eq_norm] using hz - have hi := (abs_lt.mp ((Complex.abs_im_le_norm (z-q)).trans_lt hd)).1 - simp only [Complex.sub_im] at hi - change 0 < z.im - linarith - -theorem driver_neighborhood_visits_tip - {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} - (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) - {r : ℝ} (hr : 0 < r) (hrH : r ≤ (β T).im) - {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : - ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → - (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → - K ⊆ survivingDomain γ T ∧ ∃ t ∈ Icc 0 T, γ t ∈ ball (β T) r := by - have he : ∀ᶠ y : ℝ in 𝓝[>] 0, 0 < y ∧ - J T ((V T : ℂ)+(y : ℂ)*Complex.I) ∈ ball (β T) r := by - filter_upwards [self_mem_nhdsWithin, - (hv.2.2.2.2.2.2.2.2.2.2.2.2 T).eventually (ball_mem_nhds _ hr)] with y hy hb - exact ⟨hy,hb⟩ - obtain ⟨y,hy,hyw⟩ := he.exists - let w := J T ((V T : ℂ)+(y : ℂ)*Complex.I) - have hw : w ∈ survivingDomain β T := hv.2.2.2.2.2.2.1 T - (by simpa [upperHalfPlane] using hy) - obtain ⟨δ₁,hδ₁,hkeep⟩ := driver_neighborhood_preserves_compact hv T hK hKD - obtain ⟨δ₂,hδ₂,hwkeep⟩ := driver_neighborhood_preserves_compact hv T - (isCompact_singleton : IsCompact ({w} : Set ℂ)) (singleton_subset_iff.mpr hw) - obtain ⟨δ₃,hδ₃,hhit⟩ := driver_neighborhood_hits_tip_hull hv T hr - refine ⟨min δ₁ (min δ₂ δ₃),lt_min hδ₁ (lt_min hδ₂ hδ₃),?_⟩ - intro U γ G F h hclose - have h₁ : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ₁ := - fun t ht => (hclose t ht).trans (min_le_left _ _) - have h₂ : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ₂ := - fun t ht => (hclose t ht).trans ((min_le_right _ _).trans (min_le_left _ _)) - have h₃ : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ₃ := - fun t ht => (hclose t ht).trans ((min_le_right _ _).trans (min_le_right _ _)) - refine ⟨hkeep U γ G F h h₁,?_⟩ - exact trace_meets_connected_region γ T (convex_ball _ _).isPreconnected - (ball_subset_halfPlane hrH) hyw - ((hwkeep U γ G F h h₂) (mem_singleton w)) (hhit U γ G F h h₃) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerTipHullApproximation +import OAI.Probability.SLE.Loewner.LoewnerCompactDriverNeighborhood + +namespace OAI + +/-! A perturbed hull near an interior reference tip must be visited by +the actual trace: a surviving point in the same connected ball rules out +the possibility that the ball is entirely swallowed. -/ +noncomputable section +open Set Filter Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma trace_meets_connected_region + (γ : ℝ≥0 → ℂ) (T : ℝ≥0) {S : Set ℂ} + (hS : IsPreconnected S) (hSH : S ⊆ upperHalfPlane) + {w : ℂ} (hwS : w ∈ S) (hwD : w ∈ survivingDomain γ T) + (hnot : ¬ S ⊆ survivingDomain γ T) : + ∃ t ∈ Icc 0 T, γ t ∈ S := by + by_contra hn + push Not at hn + have hf : S ⊆ upperHalfPlane \ (γ '' Icc 0 T) := by + intro z hz + refine ⟨hSH hz,?_⟩ + rintro ⟨t,ht,rfl⟩ + exact hn t ht hz + apply hnot + intro z hz + have hc := hS.subset_connectedComponentIn hwS hf hz + refine ⟨hf hz,?_⟩ + rw [← connectedComponentIn_eq hc] + exact hwD.2 + +lemma ball_subset_halfPlane {q : ℂ} {r : ℝ} (hr : r ≤ q.im) : + ball q r ⊆ upperHalfPlane := by + intro z hz + have hd : ‖z-q‖ < r := by simpa only [mem_ball,dist_eq_norm] using hz + have hi := (abs_lt.mp ((Complex.abs_im_le_norm (z-q)).trans_lt hd)).1 + simp only [Complex.sub_im] at hi + change 0 < z.im + linarith + +theorem driver_neighborhood_visits_tip + {V : ℝ≥0 → ℝ} {β : ℝ≥0 → ℂ} {H J : ℝ≥0 → ℂ → ℂ} + (hv : IsCapacityTwoWitness V β H J) (T : ℝ≥0) + {r : ℝ} (hr : 0 < r) (hrH : r ≤ (β T).im) + {K : Set ℂ} (hK : IsCompact K) (hKD : K ⊆ survivingDomain β T) : + ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → + (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → + K ⊆ survivingDomain γ T ∧ ∃ t ∈ Icc 0 T, γ t ∈ ball (β T) r := by + have he : ∀ᶠ y : ℝ in 𝓝[>] 0, 0 < y ∧ + J T ((V T : ℂ)+(y : ℂ)*Complex.I) ∈ ball (β T) r := by + filter_upwards [self_mem_nhdsWithin, + (hv.2.2.2.2.2.2.2.2.2.2.2.2 T).eventually (ball_mem_nhds _ hr)] with y hy hb + exact ⟨hy,hb⟩ + obtain ⟨y,hy,hyw⟩ := he.exists + let w := J T ((V T : ℂ)+(y : ℂ)*Complex.I) + have hw : w ∈ survivingDomain β T := hv.2.2.2.2.2.2.1 T + (by simpa [upperHalfPlane] using hy) + obtain ⟨δ₁,hδ₁,hkeep⟩ := driver_neighborhood_preserves_compact hv T hK hKD + obtain ⟨δ₂,hδ₂,hwkeep⟩ := driver_neighborhood_preserves_compact hv T + (isCompact_singleton : IsCompact ({w} : Set ℂ)) (singleton_subset_iff.mpr hw) + obtain ⟨δ₃,hδ₃,hhit⟩ := driver_neighborhood_hits_tip_hull hv T hr + refine ⟨min δ₁ (min δ₂ δ₃),lt_min hδ₁ (lt_min hδ₂ hδ₃),?_⟩ + intro U γ G F h hclose + have h₁ : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ₁ := + fun t ht => (hclose t ht).trans (min_le_left _ _) + have h₂ : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ₂ := + fun t ht => (hclose t ht).trans ((min_le_right _ _).trans (min_le_left _ _)) + have h₃ : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ₃ := + fun t ht => (hclose t ht).trans ((min_le_right _ _).trans (min_le_right _ _)) + refine ⟨hkeep U γ G F h h₁,?_⟩ + exact trace_meets_connected_region γ T (convex_ball _ _).isPreconnected + (ball_subset_halfPlane hrH) hyw + ((hwkeep U γ G F h h₂) (mem_singleton w)) (hhit U γ G F h h₃) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerTraceHeight.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerTraceHeight.lean index b75e36ec5874c9c2d32a6b688fa4271b166aadec..a8c6036817a0c915c0adba7bb2bed6e90271a46d 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerTraceHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerTraceHeight.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Loewner.LoewnerDriverBounds - -namespace OAI - -/-! The deterministic height bound for a capacity-two trace. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - -lemma IsCapacityTwoWitness.inverse_height_sq_le - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} - (hw : w ∈ upperHalfPlane) : (F T w).im^2 ≤ w.im^2+4*T := by - let q (s : ℝ) := G ((T:ℝ)-s).toNNReal (F T w) - have hd (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : - HasDerivWithinAt (fun t => (q t).im^2) - (4*(q s).im^2/normSq (q s-(U ((T:ℝ)-s).toNNReal : ℂ))) - (Icc 0 (T:ℝ)) s := by - have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt s - (h.reverse_path_unclipped T hw hs) - change HasDerivWithinAt (fun t => (q t).im) - ((-2/(q s-(U ((T:ℝ)-s).toNNReal : ℂ))).im) _ _ at hi - have hh := hi.pow 2 - convert! hh using 1 - simp only [Complex.div_im,Complex.neg_re,Complex.re_ofNat,Complex.neg_im, - Complex.im_ofNat,neg_zero,zero_mul,Complex.sub_im,Complex.ofReal_im,sub_zero] - ring - have hq0 : q 0=w := by - simp only [q,sub_zero,Real.toNNReal_coe] - exact h.2.2.2.2.2.2.2.2.1 T w hw - have hqT : q T=F T w := by - simp only [q,sub_self,Real.toNNReal_zero] - exact h.initial_value _ (h.2.2.2.2.2.2.1 T hw).1.1 - have hb := image_le_of_deriv_right_le_deriv_boundary - (fun s hs => (hd s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hd s ⟨hs.1,hs.2.le⟩)) - (B := fun s => w.im^2+4*s) (B' := fun _ => 4) - (by simp [hq0]) (by fun_prop) - (fun s _ => by - convert (((hasDerivAt_id s).const_mul 4).const_add (w.im^2)).hasDerivWithinAt using 1 <;> simp) - (fun s hs => ?_) (show (T:ℝ) ∈ Icc 0 (T:ℝ) from ⟨T.coe_nonneg,le_rfl⟩) - · simpa only [hqT] using hb - · apply (div_le_iff₀ ?_).mpr - · simp only [Complex.normSq_apply,Complex.sub_re,Complex.ofReal_re, - Complex.sub_im,Complex.ofReal_im,sub_zero] - nlinarith [sq_nonneg ((q s).re-U ((T:ℝ)-s).toNNReal)] - · apply Complex.normSq_pos.mpr - have hi := h.imag_ge_terminal T (F T w) (h.2.2.2.2.2.2.1 T hw) - ((T:ℝ)-s) ⟨by linarith [hs.2],by linarith [hs.1]⟩ - rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hi - intro he - have him := congrArg Complex.im he - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at him - change w.im ≤ (q s).im at hi - change 0 < w.im at hw - linarith - -theorem IsCapacityTwoWitness.trace_height_le - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) : (γ T).im ≤ 2*Real.sqrt T := by - let y (n : ℕ) : ℝ := 1/((n:ℝ)+1) - have hy0 : Tendsto y atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat - have hy : Tendsto y atTop (𝓝[>] (0:ℝ)) := tendsto_nhdsWithin_iff.mpr - ⟨hy0,Eventually.of_forall (fun n => by change 0 < y n; dsimp [y]; positivity)⟩ - have hl := (Complex.continuous_im.continuousAt.tendsto.comp - ((h.2.2.2.2.2.2.2.2.2.2.2.2 T).comp hy)).pow 2 - have hr := (hy0.pow 2).add_const (4*(T:ℝ)) - have hb : (γ T).im^2 ≤ 4*(T:ℝ) := by - apply le_of_tendsto_of_tendsto hl (by simpa using hr) - apply Eventually.of_forall - intro n - have hh := h.inverse_height_sq_le T - (w := (U T:ℂ)+(y n:ℂ)*I) (by - change 0 < ((U T:ℂ)+(y n:ℂ)*I).im - simpa using (show 0 < y n by dsimp [y]; positivity)) - simpa using hh - have hs := Real.sq_sqrt T.coe_nonneg - nlinarith [Real.sqrt_nonneg (T:ℝ)] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerDriverBounds + +namespace OAI + +/-! The deterministic height bound for a capacity-two trace. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + +lemma IsCapacityTwoWitness.inverse_height_sq_le + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {w : ℂ} + (hw : w ∈ upperHalfPlane) : (F T w).im^2 ≤ w.im^2+4*T := by + let q (s : ℝ) := G ((T:ℝ)-s).toNNReal (F T w) + have hd (s : ℝ) (hs : s ∈ Icc 0 (T:ℝ)) : + HasDerivWithinAt (fun t => (q t).im^2) + (4*(q s).im^2/normSq (q s-(U ((T:ℝ)-s).toNNReal : ℂ))) + (Icc 0 (T:ℝ)) s := by + have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt s + (h.reverse_path_unclipped T hw hs) + change HasDerivWithinAt (fun t => (q t).im) + ((-2/(q s-(U ((T:ℝ)-s).toNNReal : ℂ))).im) _ _ at hi + have hh := hi.pow 2 + convert! hh using 1 + simp only [Complex.div_im,Complex.neg_re,Complex.re_ofNat,Complex.neg_im, + Complex.im_ofNat,neg_zero,zero_mul,Complex.sub_im,Complex.ofReal_im,sub_zero] + ring + have hq0 : q 0=w := by + simp only [q,sub_zero,Real.toNNReal_coe] + exact h.2.2.2.2.2.2.2.2.1 T w hw + have hqT : q T=F T w := by + simp only [q,sub_self,Real.toNNReal_zero] + exact h.initial_value _ (h.2.2.2.2.2.2.1 T hw).1.1 + have hb := image_le_of_deriv_right_le_deriv_boundary + (fun s hs => (hd s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hd s ⟨hs.1,hs.2.le⟩)) + (B := fun s => w.im^2+4*s) (B' := fun _ => 4) + (by simp [hq0]) (by fun_prop) + (fun s _ => by + convert (((hasDerivAt_id s).const_mul 4).const_add (w.im^2)).hasDerivWithinAt using 1 <;> simp) + (fun s hs => ?_) (show (T:ℝ) ∈ Icc 0 (T:ℝ) from ⟨T.coe_nonneg,le_rfl⟩) + · simpa only [hqT] using hb + · apply (div_le_iff₀ ?_).mpr + · simp only [Complex.normSq_apply,Complex.sub_re,Complex.ofReal_re, + Complex.sub_im,Complex.ofReal_im,sub_zero] + nlinarith [sq_nonneg ((q s).re-U ((T:ℝ)-s).toNNReal)] + · apply Complex.normSq_pos.mpr + have hi := h.imag_ge_terminal T (F T w) (h.2.2.2.2.2.2.1 T hw) + ((T:ℝ)-s) ⟨by linarith [hs.2],by linarith [hs.1]⟩ + rw [h.2.2.2.2.2.2.2.2.1 T w hw] at hi + intro he + have him := congrArg Complex.im he + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at him + change w.im ≤ (q s).im at hi + change 0 < w.im at hw + linarith + +theorem IsCapacityTwoWitness.trace_height_le + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) : (γ T).im ≤ 2*Real.sqrt T := by + let y (n : ℕ) : ℝ := 1/((n:ℝ)+1) + have hy0 : Tendsto y atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat + have hy : Tendsto y atTop (𝓝[>] (0:ℝ)) := tendsto_nhdsWithin_iff.mpr + ⟨hy0,Eventually.of_forall (fun n => by change 0 < y n; dsimp [y]; positivity)⟩ + have hl := (Complex.continuous_im.continuousAt.tendsto.comp + ((h.2.2.2.2.2.2.2.2.2.2.2.2 T).comp hy)).pow 2 + have hr := (hy0.pow 2).add_const (4*(T:ℝ)) + have hb : (γ T).im^2 ≤ 4*(T:ℝ) := by + apply le_of_tendsto_of_tendsto hl (by simpa using hr) + apply Eventually.of_forall + intro n + have hh := h.inverse_height_sq_le T + (w := (U T:ℂ)+(y n:ℂ)*I) (by + change 0 < ((U T:ℂ)+(y n:ℂ)*I).im + simpa using (show 0 < y n by dsimp [y]; positivity)) + simpa using hh + have hs := Real.sq_sqrt T.coe_nonneg + nlinarith [Real.sqrt_nonneg (T:ℝ)] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/LoewnerVerticalLimit.lean b/lean/OAI/Probability/SLE/Loewner/LoewnerVerticalLimit.lean index 6a02e1304aaf31ef9691275e38bb7fc1c7075b13..6d3858d426586cfa2010d44bbb6797d046e27e66 100644 --- a/lean/OAI/Probability/SLE/Loewner/LoewnerVerticalLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/LoewnerVerticalLimit.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Loewner.LoewnerInverseStability - -namespace OAI - -/-! The zero-driver inverse trajectory, and convergence to it on the -imaginary axis under a uniform perturbation of the driving function. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -def verticalReversePath (y s : ℝ) : ℂ := (Real.sqrt (y^2+4*s) : ℂ)*I - -lemma verticalReversePath_zero {y : ℝ} (hy : 0 < y) : - verticalReversePath y 0=(y:ℂ)*I := by - simp [verticalReversePath,Real.sqrt_sq hy.le] - -lemma verticalReversePath_derivative {y s : ℝ} (hy : 0 < y) (hs : 0 ≤ s) : - HasDerivAt (verticalReversePath y) - (reverseField y 0 (verticalReversePath y s)) s := by - have hp : 0 < y^2+4*s := by positivity - have hr : 0 < Real.sqrt (y^2+4*s) := Real.sqrt_pos.mpr hp - have hle : y ≤ Real.sqrt (y^2+4*s) := by - calc - y = Real.sqrt (y^2) := (Real.sqrt_sq hy.le).symm - _ ≤ Real.sqrt (y^2+4*s) := Real.sqrt_le_sqrt (by linarith) - have hd := (((hasDerivAt_const s (y^2)).add - ((hasDerivAt_id s).const_mul 4)).sqrt hp.ne').ofReal_comp.mul_const I - have hd' : HasDerivAt (verticalReversePath y) - ((((4)/(2*Real.sqrt (y^2+4*s)) : ℝ) : ℂ)*I) s := by - convert! hd using 1 - simp only [Pi.add_apply, id_eq, zero_add, mul_one] - have he : reverseField y 0 (verticalReversePath y s) = - (((4)/(2*Real.sqrt (y^2+4*s)) : ℝ) : ℂ)*I := by - have hi : y ≤ (verticalReversePath y s).im := by - simpa [verticalReversePath] using hle - rw [reverseField,reverseCenter_eq hi,Complex.ofReal_zero,sub_zero] - dsimp [verticalReversePath] - have hc : (Real.sqrt (y^2+4*s) : ℂ) ≠ 0 := by exact_mod_cast hr.ne' - push_cast - field_simp [hc] - norm_num [Complex.I_sq] - exact he ▸ hd' - -theorem IsCapacityTwoWitness.inverse_vertical_approximation - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {y : ℝ} (hy : 0 < y) - (δ : ℝ) (hU : ∀ t ∈ Icc 0 T, |U t| ≤ δ) : - dist (F T ((y:ℂ)*I)) (verticalReversePath y T) ≤ - gronwallBound 0 (4*y⁻¹^2) (2*y⁻¹^2*δ) T := by - apply h.inverse_approximation_bound T hy ((y:ℂ)*I) (by simp) - (fun _ => 0) δ (by simpa only [sub_zero] using hU) - (verticalReversePath y) (verticalReversePath_zero hy) - intro s hs - exact (verticalReversePath_derivative hy hs.1).hasDerivWithinAt - -theorem inverse_vertical_tendsto - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) - (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - (T : ℝ≥0) (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) - (hU : ∀ n t, t ∈ Icc 0 T → |U n t| ≤ δ n) - {y : ℝ} (hy : 0 < y) : - Tendsto (fun n => F n T ((y:ℂ)*I)) atTop (𝓝 (verticalReversePath y T)) := by - rw [tendsto_iff_dist_tendsto_zero] - have hb := fun n => (h n).inverse_vertical_approximation T hy (δ n) (hU n) - have ht := (gronwallBound_continuous_ε 0 (4*y⁻¹^2) T).continuousAt.tendsto.comp - (hδ.const_mul (2*y⁻¹^2)) - simp only [mul_zero,gronwallBound_ε0_δ0] at ht - exact squeeze_zero (fun _ => dist_nonneg) hb ht - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerInverseStability + +namespace OAI + +/-! The zero-driver inverse trajectory, and convergence to it on the +imaginary axis under a uniform perturbation of the driving function. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +def verticalReversePath (y s : ℝ) : ℂ := (Real.sqrt (y^2+4*s) : ℂ)*I + +lemma verticalReversePath_zero {y : ℝ} (hy : 0 < y) : + verticalReversePath y 0=(y:ℂ)*I := by + simp [verticalReversePath,Real.sqrt_sq hy.le] + +lemma verticalReversePath_derivative {y s : ℝ} (hy : 0 < y) (hs : 0 ≤ s) : + HasDerivAt (verticalReversePath y) + (reverseField y 0 (verticalReversePath y s)) s := by + have hp : 0 < y^2+4*s := by positivity + have hr : 0 < Real.sqrt (y^2+4*s) := Real.sqrt_pos.mpr hp + have hle : y ≤ Real.sqrt (y^2+4*s) := by + calc + y = Real.sqrt (y^2) := (Real.sqrt_sq hy.le).symm + _ ≤ Real.sqrt (y^2+4*s) := Real.sqrt_le_sqrt (by linarith) + have hd := (((hasDerivAt_const s (y^2)).add + ((hasDerivAt_id s).const_mul 4)).sqrt hp.ne').ofReal_comp.mul_const I + have hd' : HasDerivAt (verticalReversePath y) + ((((4)/(2*Real.sqrt (y^2+4*s)) : ℝ) : ℂ)*I) s := by + convert! hd using 1 + simp only [Pi.add_apply, id_eq, zero_add, mul_one] + have he : reverseField y 0 (verticalReversePath y s) = + (((4)/(2*Real.sqrt (y^2+4*s)) : ℝ) : ℂ)*I := by + have hi : y ≤ (verticalReversePath y s).im := by + simpa [verticalReversePath] using hle + rw [reverseField,reverseCenter_eq hi,Complex.ofReal_zero,sub_zero] + dsimp [verticalReversePath] + have hc : (Real.sqrt (y^2+4*s) : ℂ) ≠ 0 := by exact_mod_cast hr.ne' + push_cast + field_simp [hc] + norm_num [Complex.I_sq] + exact he ▸ hd' + +theorem IsCapacityTwoWitness.inverse_vertical_approximation + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) {y : ℝ} (hy : 0 < y) + (δ : ℝ) (hU : ∀ t ∈ Icc 0 T, |U t| ≤ δ) : + dist (F T ((y:ℂ)*I)) (verticalReversePath y T) ≤ + gronwallBound 0 (4*y⁻¹^2) (2*y⁻¹^2*δ) T := by + apply h.inverse_approximation_bound T hy ((y:ℂ)*I) (by simp) + (fun _ => 0) δ (by simpa only [sub_zero] using hU) + (verticalReversePath y) (verticalReversePath_zero hy) + intro s hs + exact (verticalReversePath_derivative hy hs.1).hasDerivWithinAt + +theorem inverse_vertical_tendsto + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) + (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (h : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + (T : ℝ≥0) (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) + (hU : ∀ n t, t ∈ Icc 0 T → |U n t| ≤ δ n) + {y : ℝ} (hy : 0 < y) : + Tendsto (fun n => F n T ((y:ℂ)*I)) atTop (𝓝 (verticalReversePath y T)) := by + rw [tendsto_iff_dist_tendsto_zero] + have hb := fun n => (h n).inverse_vertical_approximation T hy (δ n) (hU n) + have ht := (gronwallBound_continuous_ε 0 (4*y⁻¹^2) T).continuousAt.tendsto.comp + (hδ.const_mul (2*y⁻¹^2)) + simp only [mul_zero,gronwallBound_ε0_δ0] at ht + exact squeeze_zero (fun _ => dist_nonneg) hb ht + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/MappedFutureArc.lean b/lean/OAI/Probability/SLE/Loewner/MappedFutureArc.lean index 7f5731e21f638e38eb00b17b6b9cff3bd472e6b8..416f2ea8c298cae084653607cf7f4607a09989dd 100644 --- a/lean/OAI/Probability/SLE/Loewner/MappedFutureArc.lean +++ b/lean/OAI/Probability/SLE/Loewner/MappedFutureArc.lean @@ -1,107 +1,107 @@ -import OAI.Probability.SLE.Loewner.InitialDrivingLimit -import OAI.Probability.SLE.Loewner.FillEndpoint -import OAI.Probability.SLE.Loewner.InitialChartIdentities - -namespace OAI - -/-! Each finite future subarc becomes a continuous attached arc in the -current half-plane chart. Its initial value is the proved driving limit. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma initialForwardChart_continuousAt (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range (initialSubarc p s)) : - ContinuousAt (initialForwardChart p hi h0 hH s) z := by - by_cases hs : s=0 - · simpa only [hs,initialForwardChart_zero] using continuousAt_id (x := z) - · rw [initialForwardChart,dite_eq_right hs] - exact ((initialSlitChart p hi h0 hH s hs).forward_differentiable.continuousOn z hz).continuousAt - ((initialSlitChart p hi h0 hH s hs).isOpen_domain.mem_nhds hz) - -def mappedFutureRaw (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (s t : unitInterval) (r : ℝ) : ℂ := - initialForwardChart p hi h0 hH s - (p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r))) - -lemma future_scaled_offset {s t : unitInterval} (hst : s p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r))) := - p.continuous.comp ((slitFutureParameter_continuous s).comp - (continuous_const.mul continuous_id)) - change ContinuousAt (fun r : ℝ => initialForwardChart p hi h0 hH s - (p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r)))) r - exact (initialForwardChart_continuousAt p hi h0 hH s hz).comp - (f := fun r : ℝ => p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r))) - (show ContinuousAt (fun r : ℝ => p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r))) r from - hc.continuousAt) - -lemma mappedFutureRaw_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s] (0:ℝ)) - (𝓝 ((initialDrivingValue p hi h0 hH s (hst.trans ht)):ℂ)) := by - have hd : 0<(t:ℝ)-(s:ℝ) := sub_pos.mpr hst - have hc : Tendsto (fun r : ℝ => ((t:ℝ)-(s:ℝ))*r) (𝓝[>] (0:ℝ)) (𝓝[>] (0:ℝ)) := by - apply tendsto_nhdsWithin_iff.mpr - constructor - · have hmul : Continuous (fun r : ℝ => ((t:ℝ)-(s:ℝ))*r) := - continuous_const.mul continuous_id - have hh : Tendsto (fun r : ℝ => ((t:ℝ)-(s:ℝ))*r) (𝓝 (0:ℝ)) (𝓝 0) := by - simpa only [mul_zero] using hmul.tendsto 0 - exact hh.mono_left nhdsWithin_le_nhds - · filter_upwards [self_mem_nhdsWithin] with r hr - exact mul_pos hd hr - exact (initialDrivingValue_spec p hi h0 hH s (hst.trans ht)).comp hc - -noncomputable def mappedFutureArc (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s mappedFutureRaw_continuousAt p hi h0 hH hst ht hr) - (mappedFutureRaw_tendsto p hi h0 hH hst ht)).comp_continuous continuous_subtype_val - (fun u => u.property) - -@[simp] lemma mappedFutureArc_zero (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s t : unitInterval} (hst : s p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r))) := + p.continuous.comp ((slitFutureParameter_continuous s).comp + (continuous_const.mul continuous_id)) + change ContinuousAt (fun r : ℝ => initialForwardChart p hi h0 hH s + (p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r)))) r + exact (initialForwardChart_continuousAt p hi h0 hH s hz).comp + (f := fun r : ℝ => p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r))) + (show ContinuousAt (fun r : ℝ => p (slitFutureParameter s (((t:ℝ)-(s:ℝ))*r))) r from + hc.continuousAt) + +lemma mappedFutureRaw_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s] (0:ℝ)) + (𝓝 ((initialDrivingValue p hi h0 hH s (hst.trans ht)):ℂ)) := by + have hd : 0<(t:ℝ)-(s:ℝ) := sub_pos.mpr hst + have hc : Tendsto (fun r : ℝ => ((t:ℝ)-(s:ℝ))*r) (𝓝[>] (0:ℝ)) (𝓝[>] (0:ℝ)) := by + apply tendsto_nhdsWithin_iff.mpr + constructor + · have hmul : Continuous (fun r : ℝ => ((t:ℝ)-(s:ℝ))*r) := + continuous_const.mul continuous_id + have hh : Tendsto (fun r : ℝ => ((t:ℝ)-(s:ℝ))*r) (𝓝 (0:ℝ)) (𝓝 0) := by + simpa only [mul_zero] using hmul.tendsto 0 + exact hh.mono_left nhdsWithin_le_nhds + · filter_upwards [self_mem_nhdsWithin] with r hr + exact mul_pos hd hr + exact (initialDrivingValue_spec p hi h0 hH s (hst.trans ht)).comp hc + +noncomputable def mappedFutureArc (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s mappedFutureRaw_continuousAt p hi h0 hH hst ht hr) + (mappedFutureRaw_tendsto p hi h0 hH hst ht)).comp_continuous continuous_subtype_val + (fun u => u.property) + +@[simp] lemma mappedFutureArc_zero (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s (a:ℝ)) (hi hp) - rw [slitFutureParameter_coe s (future_scaled_offset hst ht hur), - slitFutureParameter_coe s (future_scaled_offset hst ht hvr)] at hparam - apply Subtype.ext - have hd : (t:ℝ)-(s:ℝ)≠0 := ne_of_gt (sub_pos.mpr (show (s:ℝ)<(t:ℝ) from hst)) - exact mul_left_cancel₀ hd (add_left_cancel hparam) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.MappedFutureArc + +namespace OAI + +/-! The future chart does not identify distinct points of the prescribed +arc, including its newly filled real endpoint. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma mappedFutureArc_of_ne_zero (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s t : unitInterval} (hst : s (a:ℝ)) (hi hp) + rw [slitFutureParameter_coe s (future_scaled_offset hst ht hur), + slitFutureParameter_coe s (future_scaled_offset hst ht hvr)] at hparam + apply Subtype.ext + have hd : (t:ℝ)-(s:ℝ)≠0 := ne_of_gt (sub_pos.mpr (show (s:ℝ)<(t:ℝ) from hst)) + exact mul_left_cancel₀ hd (add_left_cancel hparam) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/MovingForwardBoundaryHeight.lean b/lean/OAI/Probability/SLE/Loewner/MovingForwardBoundaryHeight.lean index 709708393bd5828ef0f13e1e056cfa9d4a7aed5f..35248bab6795b74ab2fe59b4cfdf9927a6bac1d3 100644 --- a/lean/OAI/Probability/SLE/Loewner/MovingForwardBoundaryHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/MovingForwardBoundaryHeight.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Loewner.InitialInverseJointContinuity -import OAI.Probability.SLE.Loewner.SlitUniformForwardBound -import OAI.Probability.SLE.Loewner.FutureSlitDomain - -namespace OAI - -/-! Bounded moving forward coordinates can only approach the real line -when their physical point reaches the limiting slit boundary. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma initialForward_uniform_near (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 hp _) hz hzR - -theorem initialForward_boundary_height_zero (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {t : ℕ → unitInterval} {s : unitInterval} {z : ℕ → ℂ} {b : ℂ} - (ht : Tendsto t atTop (𝓝 s)) (hz : Tendsto z atTop (𝓝 b)) - (hzd : ∀ n, z n∈upperHalfPlane \ range (initialSubarc p (t n))) - (hb : b∉upperHalfPlane \ range (initialSubarc p s)) - {M : ℝ} (hM : ∀ n, ‖initialForwardChart p hi h0 hH (t n) (z n)‖≤M) : - Tendsto (fun n => (initialForwardChart p hi h0 hH (t n) (z n)).im) atTop (𝓝 0) := by - apply tendsto_of_subseq_tendsto - intro φ hφ - obtain ⟨w,_,ψ,hψ,hw⟩ := (isCompact_closedBall (0:ℂ) M).tendsto_subseq (fun n => by - rw [mem_closedBall_zero_iff] - exact hM (φ n)) - have hwim : 0≤w.im := ge_of_tendsto (continuous_im.continuousAt.tendsto.comp hw) - (Eventually.of_forall fun n => - (show 0<(initialForwardChart p hi h0 hH (t (φ (ψ n))) (z (φ (ψ n)))).im from - initialForwardChart_mem_upper p hi h0 hH _ (hzd _)).le) - have hwzero : w.im=0 := by - by_contra hne - have hwH : w∈upperHalfPlane := lt_of_le_of_ne hwim (Ne.symm hne) - have htu := (ht.comp hφ).comp hψ.tendsto_atTop - have hwu : Tendsto (fun n => (⟨initialForwardChart p hi h0 hH (t (φ (ψ n))) - (z (φ (ψ n))),initialForwardChart_mem_upper p hi h0 hH _ (hzd _)⟩ : upperHalfPlane)) - atTop (𝓝 (⟨w,hwH⟩ : upperHalfPlane)) := by - apply tendsto_subtype_rng.mpr - exact hw - have hback := ((initialInverseChart_joint_continuous p hi h0 hH).tendsto - (s,(⟨w,hwH⟩ : upperHalfPlane))).comp (htu.prodMk_nhds hwu) - have he : initialInverseChart p hi h0 hH s w=b := by - apply tendsto_nhds_unique hback - have hzu := (hz.comp hφ).comp hψ.tendsto_atTop - apply hzu.congr - intro n - exact (initialInverse_forward_at p hi h0 hH _ (hzd _)).symm - exact hb (he ▸ initialInverseChart_mem_domain p hi h0 hH s hwH) - refine ⟨ψ,?_⟩ - have hh := continuous_im.continuousAt.tendsto.comp hw - change Tendsto (fun n => (initialForwardChart p hi h0 hH (t (φ (ψ n))) - (z (φ (ψ n)))).im) atTop (𝓝 w.im) at hh - rwa [hwzero] at hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialInverseJointContinuity +import OAI.Probability.SLE.Loewner.SlitUniformForwardBound +import OAI.Probability.SLE.Loewner.FutureSlitDomain + +namespace OAI + +/-! Bounded moving forward coordinates can only approach the real line +when their physical point reaches the limiting slit boundary. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +lemma initialForward_uniform_near (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 hp _) hz hzR + +theorem initialForward_boundary_height_zero (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {t : ℕ → unitInterval} {s : unitInterval} {z : ℕ → ℂ} {b : ℂ} + (ht : Tendsto t atTop (𝓝 s)) (hz : Tendsto z atTop (𝓝 b)) + (hzd : ∀ n, z n∈upperHalfPlane \ range (initialSubarc p (t n))) + (hb : b∉upperHalfPlane \ range (initialSubarc p s)) + {M : ℝ} (hM : ∀ n, ‖initialForwardChart p hi h0 hH (t n) (z n)‖≤M) : + Tendsto (fun n => (initialForwardChart p hi h0 hH (t n) (z n)).im) atTop (𝓝 0) := by + apply tendsto_of_subseq_tendsto + intro φ hφ + obtain ⟨w,_,ψ,hψ,hw⟩ := (isCompact_closedBall (0:ℂ) M).tendsto_subseq (fun n => by + rw [mem_closedBall_zero_iff] + exact hM (φ n)) + have hwim : 0≤w.im := ge_of_tendsto (continuous_im.continuousAt.tendsto.comp hw) + (Eventually.of_forall fun n => + (show 0<(initialForwardChart p hi h0 hH (t (φ (ψ n))) (z (φ (ψ n)))).im from + initialForwardChart_mem_upper p hi h0 hH _ (hzd _)).le) + have hwzero : w.im=0 := by + by_contra hne + have hwH : w∈upperHalfPlane := lt_of_le_of_ne hwim (Ne.symm hne) + have htu := (ht.comp hφ).comp hψ.tendsto_atTop + have hwu : Tendsto (fun n => (⟨initialForwardChart p hi h0 hH (t (φ (ψ n))) + (z (φ (ψ n))),initialForwardChart_mem_upper p hi h0 hH _ (hzd _)⟩ : upperHalfPlane)) + atTop (𝓝 (⟨w,hwH⟩ : upperHalfPlane)) := by + apply tendsto_subtype_rng.mpr + exact hw + have hback := ((initialInverseChart_joint_continuous p hi h0 hH).tendsto + (s,(⟨w,hwH⟩ : upperHalfPlane))).comp (htu.prodMk_nhds hwu) + have he : initialInverseChart p hi h0 hH s w=b := by + apply tendsto_nhds_unique hback + have hzu := (hz.comp hφ).comp hψ.tendsto_atTop + apply hzu.congr + intro n + exact (initialInverse_forward_at p hi h0 hH _ (hzd _)).symm + exact hb (he ▸ initialInverseChart_mem_domain p hi h0 hH s hwH) + refine ⟨ψ,?_⟩ + have hh := continuous_im.continuousAt.tendsto.comp hw + change Tendsto (fun n => (initialForwardChart p hi h0 hH (t (φ (ψ n))) + (z (φ (ψ n)))).im) atTop (𝓝 w.im) at hh + rwa [hwzero] at hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/OpenSurvivalLifetime.lean b/lean/OAI/Probability/SLE/Loewner/OpenSurvivalLifetime.lean index ffa14bcd01fec6314751a915084a36b6c759b377..0f05892d285766033a948db95556764083f44e68 100644 --- a/lean/OAI/Probability/SLE/Loewner/OpenSurvivalLifetime.lean +++ b/lean/OAI/Probability/SLE/Loewner/OpenSurvivalLifetime.lean @@ -1,40 +1,40 @@ -import Mathlib - -namespace OAI - -/-! The finite lifetime of a downward-closed survival condition with -strict continuation at each surviving time. -/ -noncomputable section -open Set - -namespace SLEExactGauge - -lemma exists_open_survival_lifetime (p : ℝ → Prop) (hzero : p 0) - (hdown : ∀ s t, 0 ≤ s → s ≤ t → p t → p s) - (hext : ∀ t, 0 ≤ t → p t → ∃ s, t < s ∧ p s) - {T : ℝ} (hT : 0 ≤ T) (hdead : ¬p T) : - ∃ τ : ℝ, 0 < τ ∧ τ ≤ T ∧ ∀ t, 0 ≤ t → (p t ↔ t < τ) := by - let S : Set ℝ := {t | 0 ≤ t ∧ p t} - have hSne : S.Nonempty := ⟨0,le_rfl,hzero⟩ - have hbound (s : ℝ) (hs : s ∈ S) : s ≤ T := by - by_contra hn - exact hdead (hdown T s hT (lt_of_not_ge hn).le hs.2) - have hSb : BddAbove S := ⟨T,hbound⟩ - obtain ⟨s,hs,hsp⟩ := hext 0 le_rfl hzero - have hsS : s ∈ S := ⟨hs.le,hsp⟩ - have hτp : 0 < sSup S := hs.trans_le (le_csSup hSb hsS) - refine ⟨sSup S,hτp,csSup_le hSne hbound,?_⟩ - intro t ht - constructor - · intro hpt - obtain ⟨s,hts,hps⟩ := hext t ht hpt - exact hts.trans_le (le_csSup hSb ⟨ht.trans hts.le,hps⟩) - · intro hlt - obtain ⟨s,hsS,hts⟩ := exists_lt_of_lt_csSup hSne hlt - exact hdown t s ht hts.le hsS.2 - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! The finite lifetime of a downward-closed survival condition with +strict continuation at each surviving time. -/ +noncomputable section +open Set + +namespace SLEExactGauge + +lemma exists_open_survival_lifetime (p : ℝ → Prop) (hzero : p 0) + (hdown : ∀ s t, 0 ≤ s → s ≤ t → p t → p s) + (hext : ∀ t, 0 ≤ t → p t → ∃ s, t < s ∧ p s) + {T : ℝ} (hT : 0 ≤ T) (hdead : ¬p T) : + ∃ τ : ℝ, 0 < τ ∧ τ ≤ T ∧ ∀ t, 0 ≤ t → (p t ↔ t < τ) := by + let S : Set ℝ := {t | 0 ≤ t ∧ p t} + have hSne : S.Nonempty := ⟨0,le_rfl,hzero⟩ + have hbound (s : ℝ) (hs : s ∈ S) : s ≤ T := by + by_contra hn + exact hdead (hdown T s hT (lt_of_not_ge hn).le hs.2) + have hSb : BddAbove S := ⟨T,hbound⟩ + obtain ⟨s,hs,hsp⟩ := hext 0 le_rfl hzero + have hsS : s ∈ S := ⟨hs.le,hsp⟩ + have hτp : 0 < sSup S := hs.trans_le (le_csSup hSb hsS) + refine ⟨sSup S,hτp,csSup_le hSne hbound,?_⟩ + intro t ht + constructor + · intro hpt + obtain ⟨s,hts,hps⟩ := hext t ht hpt + exact hts.trans_le (le_csSup hSb ⟨ht.trans hts.le,hps⟩) + · intro hlt + obtain ⟨s,hsS,hts⟩ := exists_lt_of_lt_csSup hSne hlt + exact hdown t s ht hts.le hsS.2 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/PositiveReverseUniqueness.lean b/lean/OAI/Probability/SLE/Loewner/PositiveReverseUniqueness.lean index 613459d27da48c255fabd84174b5e7067555f3f1..53176b78eaa1389b3ed4f7ac112b56696af88cd2 100644 --- a/lean/OAI/Probability/SLE/Loewner/PositiveReverseUniqueness.lean +++ b/lean/OAI/Probability/SLE/Loewner/PositiveReverseUniqueness.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.LoewnerReflection - -namespace OAI - -/-! Ordinary uniqueness after both reverse trajectories have positive height. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -theorem positive_reverse_unique {a T : ℝ} (haT : a ≤ T) - (q r : ℝ → ℂ) (u : ℝ → ℝ) - (hqc : ContinuousOn q (Icc a T)) (hrc : ContinuousOn r (Icc a T)) - (hq : ∀ t ∈ Ico a T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Ici t) t) - (hr : ∀ t ∈ Ico a T, HasDerivWithinAt r - (-2/(r t-(u t : ℂ))) (Ici t) t) - (hqp : ∀ t ∈ Icc a T, 0 < (q t).im) - (hrp : ∀ t ∈ Icc a T, 0 < (r t).im) - (hinit : q a=r a) : ∀ t ∈ Icc a T, q t=r t := by - obtain ⟨s,hs,hsm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr haT) - (Complex.continuous_im.comp_continuousOn hqc) - obtain ⟨v,hv,hvm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr haT) - (Complex.continuous_im.comp_continuousOn hrc) - let η : ℝ := min (q s).im (r v).im - have hη : 0 < η := lt_min (hqp s hs) (hrp v hv) - have hQ (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (q t).im := - (min_le_left _ _).trans (hsm ht) - have hR (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (r t).im := - (min_le_right _ _).trans (hvm ht) - have hdQ (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt q - (reverseField η (u t) (q t)) (Ici t) t := by - rw [reverseField,reverseCenter_eq (hQ t (Ico_subset_Icc_self ht))] - simpa only [div_eq_mul_inv] using hq t ht - have hdR (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt r - (reverseField η (u t) (r t)) (Ici t) t := by - rw [reverseField,reverseCenter_eq (hR t (Ico_subset_Icc_self ht))] - simpa only [div_eq_mul_inv] using hr t ht - have hh := dist_le_of_trajectories_ODE (fun t => reverseField_lipschitz hη (u t)) - hqc hdQ hrc hdR (show dist (q a) (r a) ≤ 0 by rw [hinit]; simp) - intro t ht - exact dist_le_zero.mp (by simpa only [zero_mul] using hh t ht) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.LoewnerReflection + +namespace OAI + +/-! Ordinary uniqueness after both reverse trajectories have positive height. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +theorem positive_reverse_unique {a T : ℝ} (haT : a ≤ T) + (q r : ℝ → ℂ) (u : ℝ → ℝ) + (hqc : ContinuousOn q (Icc a T)) (hrc : ContinuousOn r (Icc a T)) + (hq : ∀ t ∈ Ico a T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Ici t) t) + (hr : ∀ t ∈ Ico a T, HasDerivWithinAt r + (-2/(r t-(u t : ℂ))) (Ici t) t) + (hqp : ∀ t ∈ Icc a T, 0 < (q t).im) + (hrp : ∀ t ∈ Icc a T, 0 < (r t).im) + (hinit : q a=r a) : ∀ t ∈ Icc a T, q t=r t := by + obtain ⟨s,hs,hsm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr haT) + (Complex.continuous_im.comp_continuousOn hqc) + obtain ⟨v,hv,hvm⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr haT) + (Complex.continuous_im.comp_continuousOn hrc) + let η : ℝ := min (q s).im (r v).im + have hη : 0 < η := lt_min (hqp s hs) (hrp v hv) + have hQ (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (q t).im := + (min_le_left _ _).trans (hsm ht) + have hR (t : ℝ) (ht : t ∈ Icc a T) : η ≤ (r t).im := + (min_le_right _ _).trans (hvm ht) + have hdQ (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt q + (reverseField η (u t) (q t)) (Ici t) t := by + rw [reverseField,reverseCenter_eq (hQ t (Ico_subset_Icc_self ht))] + simpa only [div_eq_mul_inv] using hq t ht + have hdR (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt r + (reverseField η (u t) (r t)) (Ici t) t := by + rw [reverseField,reverseCenter_eq (hR t (Ico_subset_Icc_self ht))] + simpa only [div_eq_mul_inv] using hr t ht + have hh := dist_le_of_trajectories_ODE (fun t => reverseField_lipschitz hη (u t)) + hqc hdQ hrc hdR (show dist (q a) (r a) ≤ 0 by rw [hinit]; simp) + intro t ht + exact dist_le_zero.mp (by simpa only [zero_mul] using hh t ht) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/PositiveSlitCapacity.lean b/lean/OAI/Probability/SLE/Loewner/PositiveSlitCapacity.lean index b87af3f52a55ad09114597f97d7287c868a436d0..9edbb0db88bac834d3d6d94826a1d0a6a46708f5 100644 --- a/lean/OAI/Probability/SLE/Loewner/PositiveSlitCapacity.lean +++ b/lean/OAI/Probability/SLE/Loewner/PositiveSlitCapacity.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Loewner.SlitCapacityChart -import OAI.Probability.SLE.Mass.HalfPlaneZeroMass - -namespace OAI - -/-! Every nontrivial simple attached slit has strictly positive capacity. -The proof rules out a zero-mass inverse displacement. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -theorem simple_slit_exists_positive_capacity_chart - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ∃ (G F : ℂ → ℂ) (c : ℝ), 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) ∧ - Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) ∧ - (∀ z ∈ upperHalfPlane \ range p, (G z).im ≤ z.im) := by - obtain ⟨G,F,c,hc,hG,hdG,hdF,hmF,hl,hr,hcoef,hn,hheight⟩ := - simple_slit_exists_nonnegative_capacity_chart p hi h0 hH - have hcpos : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) ∧ + Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) ∧ + (∀ z ∈ upperHalfPlane \ range p, (G z).im ≤ z.im) := by + obtain ⟨G,F,c,hc,hG,hdG,hdF,hmF,hl,hr,hcoef,hn,hheight⟩ := + simple_slit_exists_nonnegative_capacity_chart p hi h0 hH + have hcpos : 0 z/(R:ℂ)) (exteriorUpperRadius R) exteriorUpperDisk := by - have hR0 : (R:ℂ) ≠ 0 := ofReal_ne_zero.mpr hR.ne' - have hnorm : ‖(R:ℂ)‖=R := by simp only [Complex.norm_real,Real.norm_eq_abs,abs_of_pos hR] - refine ⟨?_,?_,?_⟩ - · intro z hz - refine ⟨?_,?_⟩ - · rw [norm_div,hnorm,lt_div_iff₀ hR,one_mul] - exact hz.1 - · change 0 < (z/(R:ℂ)).im - rw [div_ofReal_im] - exact div_pos hz.2 hR - · intro z _ w _ he - exact (div_left_inj' hR0).mp he - · intro z hz - refine ⟨(R:ℂ)*z,⟨?_,?_⟩,?_⟩ - · rw [norm_mul,hnorm] - nlinarith [hz.1] - · change 0 < ((R:ℂ)*z).im - simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos hR hz.2 - · field_simp - -lemma radiusHullMap_bijOn {R : ℝ} (hR : 0 z/(R:ℂ)) (exteriorUpperRadius R) := - differentiableOn_id.div_const _ - exact (exteriorJoukowski_differentiableOn.comp hd (divide_radius_bijOn hR).mapsTo).const_mul _ - -lemma radiusHullMap_coefficient {R : ℝ} (hR : 0 z*(radiusHullMap R z-z)) (cocompact ℂ) (𝓝 ((R:ℂ)^2)) := by - apply tendsto_const_nhds.congr' - filter_upwards [(isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact] with z hz - rw [radiusHullMap_formula hR,add_sub_cancel_left] - have hz0 : z ≠ 0 := hz - field_simp [hz0] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.PositiveSlitCapacity + +namespace OAI + +/-! The explicit normalized chart of an upper half disk. Comparing with +this chart bounds the capacity of a small slit. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +def exteriorUpperRadius (R : ℝ) : Set ℂ := {z | R<‖z‖ ∧ z ∈ upperHalfPlane} + +def radiusHullMap (R : ℝ) (z : ℂ) : ℂ := (R:ℂ)*exteriorJoukowski (z/(R:ℂ)) + +lemma divide_radius_bijOn {R : ℝ} (hR : 0 z/(R:ℂ)) (exteriorUpperRadius R) exteriorUpperDisk := by + have hR0 : (R:ℂ) ≠ 0 := ofReal_ne_zero.mpr hR.ne' + have hnorm : ‖(R:ℂ)‖=R := by simp only [Complex.norm_real,Real.norm_eq_abs,abs_of_pos hR] + refine ⟨?_,?_,?_⟩ + · intro z hz + refine ⟨?_,?_⟩ + · rw [norm_div,hnorm,lt_div_iff₀ hR,one_mul] + exact hz.1 + · change 0 < (z/(R:ℂ)).im + rw [div_ofReal_im] + exact div_pos hz.2 hR + · intro z _ w _ he + exact (div_left_inj' hR0).mp he + · intro z hz + refine ⟨(R:ℂ)*z,⟨?_,?_⟩,?_⟩ + · rw [norm_mul,hnorm] + nlinarith [hz.1] + · change 0 < ((R:ℂ)*z).im + simpa only [mul_im,ofReal_re,ofReal_im,zero_mul,add_zero] using mul_pos hR hz.2 + · field_simp + +lemma radiusHullMap_bijOn {R : ℝ} (hR : 0 z/(R:ℂ)) (exteriorUpperRadius R) := + differentiableOn_id.div_const _ + exact (exteriorJoukowski_differentiableOn.comp hd (divide_radius_bijOn hR).mapsTo).const_mul _ + +lemma radiusHullMap_coefficient {R : ℝ} (hR : 0 z*(radiusHullMap R z-z)) (cocompact ℂ) (𝓝 ((R:ℂ)^2)) := by + apply tendsto_const_nhds.congr' + filter_upwards [(isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact] with z hz + rw [radiusHullMap_formula hR,add_sub_cancel_left] + have hz0 : z ≠ 0 := hz + field_simp [hz0] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RadiusHullInverse.lean b/lean/OAI/Probability/SLE/Loewner/RadiusHullInverse.lean index 760f5a29c53903541d222aaa4d91970403be6054..c99af859ae7d20b739dce32d58c410c4998a8b5d 100644 --- a/lean/OAI/Probability/SLE/Loewner/RadiusHullInverse.lean +++ b/lean/OAI/Probability/SLE/Loewner/RadiusHullInverse.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Loewner.RadiusHalfPlaneHull - -namespace OAI - -/-! The inverse of the explicit half-disk chart is normalized at infinity -and stays within the radius of its argument. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -noncomputable def radiusHullInverse (R : ℝ) : ℂ → ℂ := - invFunOn (radiusHullMap R) (exteriorUpperRadius R) - -lemma radiusHullInverse_mapsTo {R : ℝ} (hR : 0 F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by - have hh : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by - simpa only [Metric.cobounded_eq_cocompact] using - (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) - exact hh.mono_left inf_le_left - have hbound : ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, - ‖(F z-z)/z‖≤R*‖z⁻¹‖ := by - rw [eventually_inf_principal] - apply Eventually.of_forall - intro z hz - rw [norm_div,norm_inv,div_eq_mul_inv] - exact mul_le_mul_of_nonneg_right (hF z hz) (inv_nonneg.mpr (norm_nonneg _)) - have hlim : Tendsto (fun z : ℂ => R*‖z⁻¹‖) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by - simpa only [norm_zero,mul_zero] using hi.norm.const_mul R - have hn := squeeze_zero' (Eventually.of_forall (fun z : ℂ => norm_nonneg ((F z-z)/z))) hbound hlim - have hz : Tendsto (fun z : ℂ => (F z-z)/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := tendsto_zero_iff_norm_tendsto_zero.mpr hn - have hh := hz.const_add 1 - simp only [add_zero] at hh - apply hh.congr' - change ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, 1 + (F z-z)/z = F z/z - rw [eventually_inf_principal] - apply Eventually.of_forall - intro z hzH - have hz0 : z ≠ 0 := by - intro he - simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hzH - field_simp [hz0] - ring - -lemma radiusHullInverse_normalized {R : ℝ} (hR : 0 radiusHullInverse R z/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := - ratio_one_of_bounded_displacement (fun _ hz => radiusHullInverse_displacement hR hz) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RadiusHalfPlaneHull + +namespace OAI + +/-! The inverse of the explicit half-disk chart is normalized at infinity +and stays within the radius of its argument. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +noncomputable def radiusHullInverse (R : ℝ) : ℂ → ℂ := + invFunOn (radiusHullMap R) (exteriorUpperRadius R) + +lemma radiusHullInverse_mapsTo {R : ℝ} (hR : 0 F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by + have hh : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by + simpa only [Metric.cobounded_eq_cocompact] using + (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) + exact hh.mono_left inf_le_left + have hbound : ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, + ‖(F z-z)/z‖≤R*‖z⁻¹‖ := by + rw [eventually_inf_principal] + apply Eventually.of_forall + intro z hz + rw [norm_div,norm_inv,div_eq_mul_inv] + exact mul_le_mul_of_nonneg_right (hF z hz) (inv_nonneg.mpr (norm_nonneg _)) + have hlim : Tendsto (fun z : ℂ => R*‖z⁻¹‖) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by + simpa only [norm_zero,mul_zero] using hi.norm.const_mul R + have hn := squeeze_zero' (Eventually.of_forall (fun z : ℂ => norm_nonneg ((F z-z)/z))) hbound hlim + have hz : Tendsto (fun z : ℂ => (F z-z)/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := tendsto_zero_iff_norm_tendsto_zero.mpr hn + have hh := hz.const_add 1 + simp only [add_zero] at hh + apply hh.congr' + change ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, 1 + (F z-z)/z = F z/z + rw [eventually_inf_principal] + apply Eventually.of_forall + intro z hzH + have hz0 : z ≠ 0 := by + intro he + simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hzH + field_simp [hz0] + ring + +lemma radiusHullInverse_normalized {R : ℝ} (hR : 0 radiusHullInverse R z/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := + ratio_one_of_bounded_displacement (fun _ hz => radiusHullInverse_displacement hR hz) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawBoundaryReverseTrace.lean b/lean/OAI/Probability/SLE/Loewner/RawBoundaryReverseTrace.lean index 8082ea32f723a264b1badc92c5eb928bae6e452a..bcdd7329270d6cc77c8df7bb43d4feb790bb828b 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawBoundaryReverseTrace.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawBoundaryReverseTrace.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Loewner.RawBoundaryTraceIdentification -import OAI.Probability.SLE.Loewner.InitialConeFromInterval - -namespace OAI - -/-! Every continuous positive reverse solution issuing from the driver -ends at the prescribed raw trace. The cone estimate is derived. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem boundary_reverse_eq_trace - (b : Driver) (T : ℝ≥0) (hT : 0 < T) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) - (q : ℝ → ℂ) (hqc : ContinuousOn q (Icc 0 (T : ℝ))) - (hq0 : q 0=(b T : ℂ)) - (hq : ∀ t ∈ Ioc 0 (T : ℝ), HasDerivWithinAt q - (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici t) t) - (hqp : ∀ t ∈ Ioc 0 (T : ℝ), 0 < (q t).im) : - q T=trace 1 b T := by - let u (s : ℝ) := driverValue b ((T : ℝ)-s) - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := - ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 - simp only [mul_neg,mul_one] - have hzero : q 0=(u 0 : ℂ) := by simpa [u,driverValue] using hq0 - obtain ⟨c,hc,hcT,hcone⟩ := exists_initial_reverse_cone_of_interval_derivative - (show 0 < (T : ℝ) from hT) hL q u (fun t => -b' ((T : ℝ)-t)) hqc hu hzero hq hqp - (fun t ht => by simpa only [abs_neg] using hb _ (hreflect t ht)) - obtain ⟨a,ha,hac,hsmall⟩ := exists_smooth_window c hc L - exact boundary_reverse_eq_trace_of_cone b T b' hd hL hb ha (hac.trans hcT) hsmall - q hqc hq0 hq hqp (fun t ht => hcone t ⟨ht.1,ht.2.trans hac⟩) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawBoundaryTraceIdentification +import OAI.Probability.SLE.Loewner.InitialConeFromInterval + +namespace OAI + +/-! Every continuous positive reverse solution issuing from the driver +ends at the prescribed raw trace. The cone estimate is derived. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem boundary_reverse_eq_trace + (b : Driver) (T : ℝ≥0) (hT : 0 < T) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) + (q : ℝ → ℂ) (hqc : ContinuousOn q (Icc 0 (T : ℝ))) + (hq0 : q 0=(b T : ℂ)) + (hq : ∀ t ∈ Ioc 0 (T : ℝ), HasDerivWithinAt q + (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici t) t) + (hqp : ∀ t ∈ Ioc 0 (T : ℝ), 0 < (q t).im) : + q T=trace 1 b T := by + let u (s : ℝ) := driverValue b ((T : ℝ)-s) + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := + ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 + simp only [mul_neg,mul_one] + have hzero : q 0=(u 0 : ℂ) := by simpa [u,driverValue] using hq0 + obtain ⟨c,hc,hcT,hcone⟩ := exists_initial_reverse_cone_of_interval_derivative + (show 0 < (T : ℝ) from hT) hL q u (fun t => -b' ((T : ℝ)-t)) hqc hu hzero hq hqp + (fun t ht => by simpa only [abs_neg] using hb _ (hreflect t ht)) + obtain ⟨a,ha,hac,hsmall⟩ := exists_smooth_window c hc L + exact boundary_reverse_eq_trace_of_cone b T b' hd hL hb ha (hac.trans hcT) hsmall + q hqc hq0 hq hqp (fun t ht => hcone t ⟨ht.1,ht.2.trans hac⟩) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawBoundaryTraceIdentification.lean b/lean/OAI/Probability/SLE/Loewner/RawBoundaryTraceIdentification.lean index 972b71ea263cce35cbcf7a3e6318eb84249b834a..089e2f7c1967a3cdc85f490e147c6ed8f16f233f 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawBoundaryTraceIdentification.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawBoundaryTraceIdentification.lean @@ -1,69 +1,69 @@ -import OAI.Probability.SLE.Geometry.BoundaryRadialDistance -import OAI.Probability.SLE.Loewner.SmoothRawTrace - -namespace OAI - -/-! A positive reverse solution that starts at the driver and satisfies -the local cone estimate ends at the exact raw trace. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem boundary_reverse_eq_trace_of_cone - (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L a : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) - (ha : 0 < a) (haT : a ≤ T) (hsmall : L^2*a ≤ 1) - (q : ℝ → ℂ) (hqc : ContinuousOn q (Icc 0 (T : ℝ))) - (hq0 : q 0=(b T : ℂ)) - (hq : ∀ t ∈ Ioc 0 (T : ℝ), HasDerivWithinAt q - (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici t) t) - (hqp : ∀ t ∈ Ioc 0 (T : ℝ), 0 < (q t).im) - (hcone : ∀ t ∈ Icc 0 a, - ((q t).re-driverValue b ((T : ℝ)-t))^2 ≤ t ∧ 3*t ≤ (q t).im^2) : - q T=trace 1 b T := by - let u (s : ℝ) := driverValue b ((T : ℝ)-s) - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := - ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 - simp only [mul_neg,mul_one] - obtain ⟨K,_hK,hbound⟩ := exists_boundary_radial_distance_constant ha haT hL hsmall - q u (fun t => -b' ((T : ℝ)-t)) hqc hq hqp hcone hu - (fun t ht => by simpa only [abs_neg] using hb _ (hreflect t ht)) - have hdist (y : ℝ) (hy : 0 < y) : - dist (q T) (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) ≤ y*K := by - let w : ℂ := (b T : ℂ)+(y : ℂ)*I - have hw : w ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy - obtain ⟨r,hr0,hrp,hrd,hrT⟩ := exists_raw_reverse_trajectory b T w hw - have he := hbound r (fun t ht => hrd t ht.1) - (fun t ht => hy.trans_le (by simpa [w] using hrp t ht.1)) - (by simp [hr0,w,u,driverValue]) - have hzero : dist (q 0) (r 0)=y := by - rw [hq0,hr0,dist_comm,dist_eq_norm] - simp [w,abs_of_pos hy] - rw [hzero,← hrT] at he - exact he - have hlim := tendsto_inverseFlow_smooth_trace b T b' hd hL hb - have hdistlim : Tendsto - (fun y : ℝ => dist (q T) (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I))) - (𝓝[>] 0) (𝓝 (dist (q T) (trace 1 b T))) := tendsto_const_nhds.dist hlim - have hzero : Tendsto (fun y : ℝ => y*K) (𝓝[>] 0) (𝓝 (0 : ℝ)) := by - simpa only [id_eq,zero_mul] using - ((tendsto_id : Tendsto (id : ℝ → ℝ) (𝓝 0) (𝓝 0)).mono_left nhdsWithin_le_nhds).mul_const K - apply dist_le_zero.mp - exact le_of_tendsto_of_tendsto hdistlim hzero - (by filter_upwards [self_mem_nhdsWithin] with y hy; exact hdist y hy) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Geometry.BoundaryRadialDistance +import OAI.Probability.SLE.Loewner.SmoothRawTrace + +namespace OAI + +/-! A positive reverse solution that starts at the driver and satisfies +the local cone estimate ends at the exact raw trace. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem boundary_reverse_eq_trace_of_cone + (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L a : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) + (ha : 0 < a) (haT : a ≤ T) (hsmall : L^2*a ≤ 1) + (q : ℝ → ℂ) (hqc : ContinuousOn q (Icc 0 (T : ℝ))) + (hq0 : q 0=(b T : ℂ)) + (hq : ∀ t ∈ Ioc 0 (T : ℝ), HasDerivWithinAt q + (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici t) t) + (hqp : ∀ t ∈ Ioc 0 (T : ℝ), 0 < (q t).im) + (hcone : ∀ t ∈ Icc 0 a, + ((q t).re-driverValue b ((T : ℝ)-t))^2 ≤ t ∧ 3*t ≤ (q t).im^2) : + q T=trace 1 b T := by + let u (s : ℝ) := driverValue b ((T : ℝ)-s) + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := + ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 + simp only [mul_neg,mul_one] + obtain ⟨K,_hK,hbound⟩ := exists_boundary_radial_distance_constant ha haT hL hsmall + q u (fun t => -b' ((T : ℝ)-t)) hqc hq hqp hcone hu + (fun t ht => by simpa only [abs_neg] using hb _ (hreflect t ht)) + have hdist (y : ℝ) (hy : 0 < y) : + dist (q T) (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) ≤ y*K := by + let w : ℂ := (b T : ℂ)+(y : ℂ)*I + have hw : w ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy + obtain ⟨r,hr0,hrp,hrd,hrT⟩ := exists_raw_reverse_trajectory b T w hw + have he := hbound r (fun t ht => hrd t ht.1) + (fun t ht => hy.trans_le (by simpa [w] using hrp t ht.1)) + (by simp [hr0,w,u,driverValue]) + have hzero : dist (q 0) (r 0)=y := by + rw [hq0,hr0,dist_comm,dist_eq_norm] + simp [w,abs_of_pos hy] + rw [hzero,← hrT] at he + exact he + have hlim := tendsto_inverseFlow_smooth_trace b T b' hd hL hb + have hdistlim : Tendsto + (fun y : ℝ => dist (q T) (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I))) + (𝓝[>] 0) (𝓝 (dist (q T) (trace 1 b T))) := tendsto_const_nhds.dist hlim + have hzero : Tendsto (fun y : ℝ => y*K) (𝓝[>] 0) (𝓝 (0 : ℝ)) := by + simpa only [id_eq,zero_mul] using + ((tendsto_id : Tendsto (id : ℝ → ℝ) (𝓝 0) (𝓝 0)).mono_left nhdsWithin_le_nhds).mul_const K + apply dist_le_zero.mp + exact le_of_tendsto_of_tendsto hdistlim hzero + (by filter_upwards [self_mem_nhdsWithin] with y hy; exact hdist y hy) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawConstantTailInverse.lean b/lean/OAI/Probability/SLE/Loewner/RawConstantTailInverse.lean index ca5e45edcf2df8122f9392ed43c170b94f79f31c..57a441835634ae8a52e35e99b5b5ab18ff960e99 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawConstantTailInverse.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawConstantTailInverse.lean @@ -1,103 +1,103 @@ -import OAI.Probability.SLE.Loewner.RawSurvivingDerivative -import OAI.Probability.SLE.Loewner.RawSurvivingMonotone -import OAI.Probability.SLE.Loewner.SlitTimeGluing -import OAI.Probability.SLE.Loewner.LinearReverseTrajectory - -namespace OAI - -/-! The inverse after a constant-driver tail is the translated vertical -inverse followed by the inverse at the joining time. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem inverseFlow_constant_tail (b : Driver) (A S : ℝ≥0) (u : ℝ) - (hb : ∀ t : ℝ≥0, A ≤ t → b t=u) {w : ℂ} (hw : w∈upperHalfPlane) : - inverseFlow 1 b (A+S) w= - inverseFlow 1 b A ((u:ℂ)+inverseFlow 1 (linearDriver 0) S (w-(u:ℂ))) := by - let c := linearDriver 0 - let v := inverseFlow 1 c S (w-(u:ℂ)) - have hw' : w-(u:ℂ)∈upperHalfPlane := by simpa [upperHalfPlane] using hw - have hvS : Surviving 1 c v S := inverseFlow_surviving 1 (by norm_num) c S _ hw' - have hvH : v∈upperHalfPlane := surviving_mem_upperHalfPlane 1 c v S S.coe_nonneg hvS - have hvend : flow 1 c v S=w-(u:ℂ) := flow_inverseFlow 1 (by norm_num) c S _ hw' - let z := inverseFlow 1 b A ((u:ℂ)+v) - have hqH : (u:ℂ)+v∈upperHalfPlane := by simpa [upperHalfPlane] using hvH - have hzA : Surviving 1 b z A := inverseFlow_surviving 1 (by norm_num) b A _ hqH - have hzH : z∈upperHalfPlane := surviving_mem_upperHalfPlane 1 b z A A.coe_nonneg hzA - have hzjoin : flow 1 b z A=(u:ℂ)+v := flow_inverseFlow 1 (by norm_num) b A _ hqH - let f := flow 1 b z - let g := fun t : ℝ => (u:ℂ)+flow 1 c v (t-(A:ℝ)) - let Y := fun t : ℝ => if t ≤ (A:ℝ) then f t else g t - have hjoin : f A=g A := by - dsimp [f,g] - rw [sub_self,flow_zero 1 c v hvH] - exact hzjoin - have hleft (t : ℝ) (ht : t ≤ A) : Y t=f t := ite_eq_left ht - have hright (t : ℝ) (ht : A ≤ t) : Y t=g t := by - by_cases ht' : t ≤ A - · have he : t=A := le_antisymm ht' ht - simpa only [he,Y,ite_eq_left le_rfl] using hjoin - · exact ite_eq_right ht' - have hd (t : ℝ) (ht : t∈Icc 0 ((A+S:ℝ≥0):ℝ)) : - HasDerivWithinAt Y (2/(Y t-(b t.toNNReal:ℂ))) (Ici 0) t := by - apply hasDerivWithinAt_time_splice f g A t _ hjoin - · intro htA - have hs := surviving_of_le_time 1 b z ht.1 htA hzA - have hh := flow_hasDerivWithinAt_of_survival b z ht.1 hs - simpa only [←hleft t htA,f] using hh - · intro hAt - have hts : 0 ≤ t-(A:ℝ) := sub_nonneg.mpr hAt - have htsS : t-(A:ℝ) ≤ S := by - have hh : t ≤ (A:ℝ)+(S:ℝ) := by simpa only [NNReal.coe_add] using ht.2 - linarith - have hs := surviving_of_le_time 1 c v hts htsS hvS - have hh := flow_hasDerivWithinAt_of_survival c v hts hs - have hlin : HasDerivWithinAt (fun t : ℝ => t-(A:ℝ)) 1 (Ici (A:ℝ)) t := - ((hasDerivAt_id t).sub_const (A:ℝ)).hasDerivWithinAt - have hm : MapsTo (fun t : ℝ => t-(A:ℝ)) (Ici (A:ℝ)) (Ici 0) := - fun t ht => show 0 ≤ t-(A:ℝ) from sub_nonneg.mpr ht - have hh' := (hh.scomp t hlin hm).const_add (u:ℂ) - have hbt : b t.toNNReal=u := hb _ (by rwa [Real.le_toNNReal_iff_coe_le ht.1]) - have hct : c (t-(A:ℝ)).toNNReal=0 := by simp [c] - simp only [one_smul,hct] at hh' - have hhg : HasDerivWithinAt g (2/(flow 1 c v (t-(A:ℝ))-(0:ℂ))) (Ici (A:ℝ)) t := by - simpa only [Function.comp_def,ofReal_zero,g] using hh' - apply hhg.congr_deriv - rw [hright t hAt,hbt] - dsimp only [g] - ring - have hYc : ContinuousOn Y (Icc 0 ((A+S:ℝ≥0):ℝ)) := - fun t ht => ((hd t ht).mono Icc_subset_Ici_self).continuousWithinAt - have hYpos (t : ℝ) (ht : t∈Icc 0 ((A+S:ℝ≥0):ℝ)) : 0<(Y t).im := by - by_cases htA : t ≤ A - · rw [hleft t htA] - exact surviving_flow_mem_upperHalfPlane 1 b z t ht.1 - (surviving_of_le_time 1 b z ht.1 htA hzA) - · have hAt : A ≤ t := (le_of_not_ge htA) - rw [hright t hAt] - have hts : 0 ≤ t-(A:ℝ) := sub_nonneg.mpr hAt - have htsS : t-(A:ℝ) ≤ S := by - have hh : t ≤ (A:ℝ)+(S:ℝ) := by simpa only [NNReal.coe_add] using ht.2 - linarith - have hp := surviving_flow_mem_upperHalfPlane 1 c v _ hts - (surviving_of_le_time 1 c v hts htsS hvS) - simpa only [g,add_im,ofReal_im,zero_add] using (show 0<(flow 1 c v (t-(A:ℝ))).im from hp) - obtain ⟨r,hr,hrmin⟩ := isCompact_Icc.exists_isMinOn - (nonempty_Icc.mpr (A+S).coe_nonneg) (Complex.continuous_im.comp_continuousOn hYc) - have hY0 : Y 0=z := by rw [hleft 0 A.coe_nonneg]; exact flow_zero 1 b z hzH - have hraw := flow_eq_of_positive_trajectory 1 b z (A+S) (A+S).coe_nonneg Y hYc hY0 - (Y r).im (hYpos r hr) (fun t ht => hrmin ht) (fun t ht => by - simpa only [a,div_one,Complex.ofReal_ofNat,driverValue] using - (hd t (Ico_subset_Icc_self ht)).mono (Ici_subset_Ici.mpr ht.1)) - apply inverseFlow_eq_of_surviving 1 b (A+S) (A+S).coe_nonneg z w hraw.1 - rw [hraw.2,hright _ (by simp)] - simp only [g,add_sub_cancel_left,hvend,add_sub_cancel] - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawSurvivingDerivative +import OAI.Probability.SLE.Loewner.RawSurvivingMonotone +import OAI.Probability.SLE.Loewner.SlitTimeGluing +import OAI.Probability.SLE.Loewner.LinearReverseTrajectory + +namespace OAI + +/-! The inverse after a constant-driver tail is the translated vertical +inverse followed by the inverse at the joining time. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem inverseFlow_constant_tail (b : Driver) (A S : ℝ≥0) (u : ℝ) + (hb : ∀ t : ℝ≥0, A ≤ t → b t=u) {w : ℂ} (hw : w∈upperHalfPlane) : + inverseFlow 1 b (A+S) w= + inverseFlow 1 b A ((u:ℂ)+inverseFlow 1 (linearDriver 0) S (w-(u:ℂ))) := by + let c := linearDriver 0 + let v := inverseFlow 1 c S (w-(u:ℂ)) + have hw' : w-(u:ℂ)∈upperHalfPlane := by simpa [upperHalfPlane] using hw + have hvS : Surviving 1 c v S := inverseFlow_surviving 1 (by norm_num) c S _ hw' + have hvH : v∈upperHalfPlane := surviving_mem_upperHalfPlane 1 c v S S.coe_nonneg hvS + have hvend : flow 1 c v S=w-(u:ℂ) := flow_inverseFlow 1 (by norm_num) c S _ hw' + let z := inverseFlow 1 b A ((u:ℂ)+v) + have hqH : (u:ℂ)+v∈upperHalfPlane := by simpa [upperHalfPlane] using hvH + have hzA : Surviving 1 b z A := inverseFlow_surviving 1 (by norm_num) b A _ hqH + have hzH : z∈upperHalfPlane := surviving_mem_upperHalfPlane 1 b z A A.coe_nonneg hzA + have hzjoin : flow 1 b z A=(u:ℂ)+v := flow_inverseFlow 1 (by norm_num) b A _ hqH + let f := flow 1 b z + let g := fun t : ℝ => (u:ℂ)+flow 1 c v (t-(A:ℝ)) + let Y := fun t : ℝ => if t ≤ (A:ℝ) then f t else g t + have hjoin : f A=g A := by + dsimp [f,g] + rw [sub_self,flow_zero 1 c v hvH] + exact hzjoin + have hleft (t : ℝ) (ht : t ≤ A) : Y t=f t := ite_eq_left ht + have hright (t : ℝ) (ht : A ≤ t) : Y t=g t := by + by_cases ht' : t ≤ A + · have he : t=A := le_antisymm ht' ht + simpa only [he,Y,ite_eq_left le_rfl] using hjoin + · exact ite_eq_right ht' + have hd (t : ℝ) (ht : t∈Icc 0 ((A+S:ℝ≥0):ℝ)) : + HasDerivWithinAt Y (2/(Y t-(b t.toNNReal:ℂ))) (Ici 0) t := by + apply hasDerivWithinAt_time_splice f g A t _ hjoin + · intro htA + have hs := surviving_of_le_time 1 b z ht.1 htA hzA + have hh := flow_hasDerivWithinAt_of_survival b z ht.1 hs + simpa only [←hleft t htA,f] using hh + · intro hAt + have hts : 0 ≤ t-(A:ℝ) := sub_nonneg.mpr hAt + have htsS : t-(A:ℝ) ≤ S := by + have hh : t ≤ (A:ℝ)+(S:ℝ) := by simpa only [NNReal.coe_add] using ht.2 + linarith + have hs := surviving_of_le_time 1 c v hts htsS hvS + have hh := flow_hasDerivWithinAt_of_survival c v hts hs + have hlin : HasDerivWithinAt (fun t : ℝ => t-(A:ℝ)) 1 (Ici (A:ℝ)) t := + ((hasDerivAt_id t).sub_const (A:ℝ)).hasDerivWithinAt + have hm : MapsTo (fun t : ℝ => t-(A:ℝ)) (Ici (A:ℝ)) (Ici 0) := + fun t ht => show 0 ≤ t-(A:ℝ) from sub_nonneg.mpr ht + have hh' := (hh.scomp t hlin hm).const_add (u:ℂ) + have hbt : b t.toNNReal=u := hb _ (by rwa [Real.le_toNNReal_iff_coe_le ht.1]) + have hct : c (t-(A:ℝ)).toNNReal=0 := by simp [c] + simp only [one_smul,hct] at hh' + have hhg : HasDerivWithinAt g (2/(flow 1 c v (t-(A:ℝ))-(0:ℂ))) (Ici (A:ℝ)) t := by + simpa only [Function.comp_def,ofReal_zero,g] using hh' + apply hhg.congr_deriv + rw [hright t hAt,hbt] + dsimp only [g] + ring + have hYc : ContinuousOn Y (Icc 0 ((A+S:ℝ≥0):ℝ)) := + fun t ht => ((hd t ht).mono Icc_subset_Ici_self).continuousWithinAt + have hYpos (t : ℝ) (ht : t∈Icc 0 ((A+S:ℝ≥0):ℝ)) : 0<(Y t).im := by + by_cases htA : t ≤ A + · rw [hleft t htA] + exact surviving_flow_mem_upperHalfPlane 1 b z t ht.1 + (surviving_of_le_time 1 b z ht.1 htA hzA) + · have hAt : A ≤ t := (le_of_not_ge htA) + rw [hright t hAt] + have hts : 0 ≤ t-(A:ℝ) := sub_nonneg.mpr hAt + have htsS : t-(A:ℝ) ≤ S := by + have hh : t ≤ (A:ℝ)+(S:ℝ) := by simpa only [NNReal.coe_add] using ht.2 + linarith + have hp := surviving_flow_mem_upperHalfPlane 1 c v _ hts + (surviving_of_le_time 1 c v hts htsS hvS) + simpa only [g,add_im,ofReal_im,zero_add] using (show 0<(flow 1 c v (t-(A:ℝ))).im from hp) + obtain ⟨r,hr,hrmin⟩ := isCompact_Icc.exists_isMinOn + (nonempty_Icc.mpr (A+S).coe_nonneg) (Complex.continuous_im.comp_continuousOn hYc) + have hY0 : Y 0=z := by rw [hleft 0 A.coe_nonneg]; exact flow_zero 1 b z hzH + have hraw := flow_eq_of_positive_trajectory 1 b z (A+S) (A+S).coe_nonneg Y hYc hY0 + (Y r).im (hYpos r hr) (fun t ht => hrmin ht) (fun t ht => by + simpa only [a,div_one,Complex.ofReal_ofNat,driverValue] using + (hd t (Ico_subset_Icc_self ht)).mono (Ici_subset_Ici.mpr ht.1)) + apply inverseFlow_eq_of_surviving 1 b (A+S) (A+S).coe_nonneg z w hraw.1 + rw [hraw.2,hright _ (by simp)] + simp only [g,add_sub_cancel_left,hvend,add_sub_cancel] + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawConstantTailTrace.lean b/lean/OAI/Probability/SLE/Loewner/RawConstantTailTrace.lean index da6acb67ef05122bc16083bbd685978506e8873a..de6e5c5f0464bc867f3efcba87338e2dabb74e4d 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawConstantTailTrace.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawConstantTailTrace.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Loewner.RawConstantTailInverse -import OAI.Probability.SLE.Loewner.SmoothRawTraceInterior -import OAI.Probability.SLE.Loewner.SmoothRawTrace - -namespace OAI - -/-! A constant driving tail produces the vertical slit transported by -the inverse chart at its starting time. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem trace_constant_tail (b : Driver) (A S : ℝ≥0) (hS : 0 (linearDriver_derivative 0 t ht.1).mono Icc_subset_Ici_self - have hlim := tendsto_inverseFlow_smooth_trace (linearDriver 0) S (fun _ => 0) hd - (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) - have hpos : trace 1 (linearDriver 0) S∈upperHalfPlane := - smooth_raw_trace_mem_upperHalfPlane (linearDriver 0) S hS (fun _ => 0) hd - (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) - have hq : (u:ℂ)+trace 1 (linearDriver 0) S∈upperHalfPlane := by - change 0<((u:ℂ)+trace 1 (linearDriver 0) S).im - simpa only [add_im,ofReal_im,zero_add] using - (show 0<(trace 1 (linearDriver 0) S).im from hpos) - have hF : ContinuousAt (inverseFlow 1 b A) ((u:ℂ)+trace 1 (linearDriver 0) S) := - (continuousOn_inverseFlow 1 (by norm_num) b A).continuousAt (isOpen_upperHalfPlane.mem_nhds hq) - have hright := hF.tendsto.comp (tendsto_const_nhds.add hlim) - have hleft : Tendsto (fun y : ℝ => inverseFlow 1 b (A+S) ((u:ℂ)+(y:ℂ)*I)) - (𝓝[>] (0:ℝ)) (𝓝 (inverseFlow 1 b A ((u:ℂ)+trace 1 (linearDriver 0) S))) := by - apply hright.congr' - filter_upwards [self_mem_nhdsWithin] with y hy - have hw : (u:ℂ)+(y:ℂ)*I∈upperHalfPlane := by simpa [upperHalfPlane] using hy - rw [inverseFlow_constant_tail b A S u hb hw] - simp only [linearDriver_apply,zero_mul,ofReal_zero,zero_add,add_sub_cancel_left,Function.comp_def] - have he : b (A+S)=u := hb _ (le_add_of_nonneg_right zero_le) - simpa only [trace,one_mul,driverValue,←NNReal.coe_add,Real.toNNReal_coe,he] using hleft.limUnder_eq - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawConstantTailInverse +import OAI.Probability.SLE.Loewner.SmoothRawTraceInterior +import OAI.Probability.SLE.Loewner.SmoothRawTrace + +namespace OAI + +/-! A constant driving tail produces the vertical slit transported by +the inverse chart at its starting time. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem trace_constant_tail (b : Driver) (A S : ℝ≥0) (hS : 0 (linearDriver_derivative 0 t ht.1).mono Icc_subset_Ici_self + have hlim := tendsto_inverseFlow_smooth_trace (linearDriver 0) S (fun _ => 0) hd + (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) + have hpos : trace 1 (linearDriver 0) S∈upperHalfPlane := + smooth_raw_trace_mem_upperHalfPlane (linearDriver 0) S hS (fun _ => 0) hd + (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) + have hq : (u:ℂ)+trace 1 (linearDriver 0) S∈upperHalfPlane := by + change 0<((u:ℂ)+trace 1 (linearDriver 0) S).im + simpa only [add_im,ofReal_im,zero_add] using + (show 0<(trace 1 (linearDriver 0) S).im from hpos) + have hF : ContinuousAt (inverseFlow 1 b A) ((u:ℂ)+trace 1 (linearDriver 0) S) := + (continuousOn_inverseFlow 1 (by norm_num) b A).continuousAt (isOpen_upperHalfPlane.mem_nhds hq) + have hright := hF.tendsto.comp (tendsto_const_nhds.add hlim) + have hleft : Tendsto (fun y : ℝ => inverseFlow 1 b (A+S) ((u:ℂ)+(y:ℂ)*I)) + (𝓝[>] (0:ℝ)) (𝓝 (inverseFlow 1 b A ((u:ℂ)+trace 1 (linearDriver 0) S))) := by + apply hright.congr' + filter_upwards [self_mem_nhdsWithin] with y hy + have hw : (u:ℂ)+(y:ℂ)*I∈upperHalfPlane := by simpa [upperHalfPlane] using hy + rw [inverseFlow_constant_tail b A S u hb hw] + simp only [linearDriver_apply,zero_mul,ofReal_zero,zero_add,add_sub_cancel_left,Function.comp_def] + have he : b (A+S)=u := hb _ (le_add_of_nonneg_right zero_le) + simpa only [trace,one_mul,driverValue,←NNReal.coe_add,Real.toNNReal_coe,he] using hleft.limUnder_eq + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawDiskAngleTail.lean b/lean/OAI/Probability/SLE/Loewner/RawDiskAngleTail.lean index 6212f745a29c43f1d5c15cd3e7ff8e9881a071a2..dda57d9d0e5129e813aed69d70e774f8dad7cd4d 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawDiskAngleTail.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawDiskAngleTail.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Loewner.RawDiskRadiusLaw - -namespace OAI - -/-! The finite centered disk success probability is a measurable function -of its initial angle, so it can be tested against the actual angle law. -/ -noncomputable section -open Set _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint -local instance angleTailMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance angleTailBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -def RawCanonicalMass.angleDiskTail (R : RawCanonicalMass) - (ν : Measure C(ℝ≥0,ℝ)) (κ b v θ : ℝ) : ℝ≥0∞ := - (ν.withDensity (fun p => canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p))) - {p | b ≤ R.diskAtRadius κ θ v p} - -lemma RawCanonicalMass.angleDiskTail_measurable (R : RawCanonicalMass) - (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν] (κ b v : ℝ) : - Measurable (R.angleDiskTail ν κ b v) := by - let E : Set (ℝ × C(ℝ≥0,ℝ)) := {q | b ≤ R.diskAtRadius κ q.1 v q.2} - let D : ℝ × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun q => - canonicalFiniteDensity κ (v,Target.diskMapInv q.1 0,q.2) - have hE : MeasurableSet E := measurableSet_le measurable_const - (R.diskAtRadius_measurable κ Prod.fst (fun _ => v) measurable_fst measurable_const - Prod.snd measurable_snd) - have hz : Measurable (fun θ => Target.diskMapInv θ 0) := by - unfold Target.diskMapInv - fun_prop - have hD : Measurable D := (canonicalFiniteDensity_measurable κ).comp - (measurable_const.prodMk ((hz.comp measurable_fst).prodMk measurable_snd)) - have hK := (hD.indicator hE).lintegral_prod_right' (ν := ν) - have he (θ : ℝ) : (∫⁻ p, E.indicator D (θ,p) ∂ν)=R.angleDiskTail ν κ b v θ := by - have hEθ : MeasurableSet {p | b ≤ R.diskAtRadius κ θ v p} := - measurableSet_le measurable_const - (R.diskAtRadius_measurable κ (fun _ => θ) (fun _ => v) - measurable_const measurable_const id measurable_id) - change (∫⁻ p, {p | b ≤ R.diskAtRadius κ θ v p}.indicator - (fun p => D (θ,p)) p ∂ν)=_ - rw [lintegral_indicator hEθ,← withDensity_apply _ hEθ] - rfl - simpa only [he] using hK - -lemma RawCanonicalMass.angleDiskTail_average (R : RawCanonicalMass) - (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν] (κ : ℝ) (hκ : 0 < κ) (b v : ℝ) : - (∫⁻ θ, R.angleDiskTail ν κ b v θ ∂Target.angleLaw κ)= - rawCenteredDiskLaw ν κ v {q | b ≤ R.diskAtRadius κ q.1 v q.2} := by - symm - exact measure_joint_tail_of_lintegral (rawCenteredDiskLaw ν κ v) (Target.angleLaw κ) - (fun θ => ν.withDensity (fun p => canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p))) - (rawCenteredDiskLaw_lintegral ν κ hκ v) _ - (R.diskAtRadius_measurable κ Prod.fst (fun _ => v) measurable_fst measurable_const - Prod.snd measurable_snd) b - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawDiskRadiusLaw + +namespace OAI + +/-! The finite centered disk success probability is a measurable function +of its initial angle, so it can be tested against the actual angle law. -/ +noncomputable section +open Set _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint +local instance angleTailMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance angleTailBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +def RawCanonicalMass.angleDiskTail (R : RawCanonicalMass) + (ν : Measure C(ℝ≥0,ℝ)) (κ b v θ : ℝ) : ℝ≥0∞ := + (ν.withDensity (fun p => canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p))) + {p | b ≤ R.diskAtRadius κ θ v p} + +lemma RawCanonicalMass.angleDiskTail_measurable (R : RawCanonicalMass) + (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν] (κ b v : ℝ) : + Measurable (R.angleDiskTail ν κ b v) := by + let E : Set (ℝ × C(ℝ≥0,ℝ)) := {q | b ≤ R.diskAtRadius κ q.1 v q.2} + let D : ℝ × C(ℝ≥0,ℝ) → ℝ≥0∞ := fun q => + canonicalFiniteDensity κ (v,Target.diskMapInv q.1 0,q.2) + have hE : MeasurableSet E := measurableSet_le measurable_const + (R.diskAtRadius_measurable κ Prod.fst (fun _ => v) measurable_fst measurable_const + Prod.snd measurable_snd) + have hz : Measurable (fun θ => Target.diskMapInv θ 0) := by + unfold Target.diskMapInv + fun_prop + have hD : Measurable D := (canonicalFiniteDensity_measurable κ).comp + (measurable_const.prodMk ((hz.comp measurable_fst).prodMk measurable_snd)) + have hK := (hD.indicator hE).lintegral_prod_right' (ν := ν) + have he (θ : ℝ) : (∫⁻ p, E.indicator D (θ,p) ∂ν)=R.angleDiskTail ν κ b v θ := by + have hEθ : MeasurableSet {p | b ≤ R.diskAtRadius κ θ v p} := + measurableSet_le measurable_const + (R.diskAtRadius_measurable κ (fun _ => θ) (fun _ => v) + measurable_const measurable_const id measurable_id) + change (∫⁻ p, {p | b ≤ R.diskAtRadius κ θ v p}.indicator + (fun p => D (θ,p)) p ∂ν)=_ + rw [lintegral_indicator hEθ,← withDensity_apply _ hEθ] + rfl + simpa only [he] using hK + +lemma RawCanonicalMass.angleDiskTail_average (R : RawCanonicalMass) + (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν] (κ : ℝ) (hκ : 0 < κ) (b v : ℝ) : + (∫⁻ θ, R.angleDiskTail ν κ b v θ ∂Target.angleLaw κ)= + rawCenteredDiskLaw ν κ v {q | b ≤ R.diskAtRadius κ q.1 v q.2} := by + symm + exact measure_joint_tail_of_lintegral (rawCenteredDiskLaw ν κ v) (Target.angleLaw κ) + (fun θ => ν.withDensity (fun p => canonicalFiniteDensity κ (v,Target.diskMapInv θ 0,p))) + (rawCenteredDiskLaw_lintegral ν κ hκ v) _ + (R.diskAtRadius_measurable κ Prod.fst (fun _ => v) measurable_fst measurable_const + Prod.snd measurable_snd) b + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawDiskRadiusLaw.lean b/lean/OAI/Probability/SLE/Loewner/RawDiskRadiusLaw.lean index a4849cdc36c2cf088530566deccc599bffbf1377..a69a24dfdb0ab4ea96e046bbe95c6bc0e9b8edba 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawDiskRadiusLaw.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawDiskRadiusLaw.lean @@ -1,142 +1,142 @@ -import OAI.Probability.SLE.Geometry.CanonicalDiskTails -import OAI.Probability.SLE.Gauge.CanonicalDenseSections -import OAI.Probability.SLE.Geometry.ProductDensity - -namespace OAI - -/-! The disk variable on raw driver space has exactly the finite centered -law used in the prescribed moment coefficients and tail budget. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint -local instance rawDiskRadiusMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawDiskRadiusBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -def RawCanonicalMass.diskAtRadius (R : RawCanonicalMass) (κ θ u : ℝ) - (p : C(ℝ≥0,ℝ)) : ℝ := - (diskMassValue κ θ (extendedCanonicalMass R.process R.terminal - (canonicalRadiusLevel κ (u,Target.diskMapInv θ 0,p)) p)).toReal - -lemma RawCanonicalMass.diskAtRadius_nonneg (R : RawCanonicalMass) (κ θ u : ℝ) - (p : C(ℝ≥0,ℝ)) : 0 ≤ R.diskAtRadius κ θ u p := ENNReal.toReal_nonneg - -lemma RawCanonicalMass.diskAtRadius_measurable - {Ξ : Type*} [MeasurableSpace Ξ] (R : RawCanonicalMass) (κ : ℝ) - (θ u : Ξ → ℝ) (hθ : Measurable θ) (hu : Measurable u) - (Y : Ξ → C(ℝ≥0,ℝ)) (hY : Measurable Y) : - Measurable (fun x => R.diskAtRadius κ (θ x) (u x) (Y x)) := by - have hz : Measurable (fun x => Target.diskMapInv (θ x) 0) := by - unfold Target.diskMapInv - fun_prop - exact (R.disk_at_parameter_measurable κ θ hθ Y hY _ - (measurable_canonicalRadiusLevel_random_state κ hY hu hz)).ennreal_toReal - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - -include C hc F hw R he in -lemma CanonicalMass.raw_diskAtRadius_eq : - ∀ᵐ ω ∂P, ∀ θ u : ℝ, 0 ≤ u → - R.diskAtRadius κ θ u (sourceDriverPath B hc ω)=C.diskAtRadius θ u ω := by - filter_upwards [he] with ω hω - intro θ u hu - unfold RawCanonicalMass.diskAtRadius CanonicalMass.diskAtRadius - rw [canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) u hu] - unfold CanonicalMass.atTime extendedCanonicalMass - split_ifs <;> simp only [hω.1,hω.2] - -include C hc F hw R he in -lemma CanonicalMass.raw_diskAtRadius_finiteCentered_tail [SFinite P] - (θ u b : ℝ) (hu : 0 ≤ u) : - ((P.map (sourceDriverPath B hc)).withDensity - (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv θ 0,p))) - {p | b ≤ R.diskAtRadius κ θ u p} = - finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u - {ω | b ≤ C.diskAtRadius θ u ω} := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - rw [← canonical_finiteCenteredLaw_map P κ B hm hc γ G F (Eventually.of_forall hw) u hu] - have hA : MeasurableSet {p | b ≤ R.diskAtRadius κ θ u p} := by - exact measurableSet_le measurable_const - (R.diskAtRadius_measurable κ (fun _ => θ) (fun _ => u) - measurable_const measurable_const id measurable_id) - rw [Measure.map_apply (source_path_measurable _ hm) hA] - apply measure_congr - have habs : (finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u).AbsolutelyContinuous P := - withDensity_absolutelyContinuous P _ - filter_upwards [habs.ae_le (C.raw_diskAtRadius_eq hc F hw R he)] with ω hω - exact propext (by rw [mem_preimage,mem_ofPred_eq,hω θ u hu]) - -end SLEExactGauge.Multipoint - -namespace SLEExactGauge -local instance rawCenteredDiskMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawCenteredDiskBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -def rawCenteredDiskLaw (ν : Measure C(ℝ≥0,ℝ)) (κ u : ℝ) : Measure (ℝ × C(ℝ≥0,ℝ)) := - ((Target.angleLaw κ).prod ν).withDensity - (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv p.1 0,p.2)) - -lemma rawCenteredDiskLaw_lintegral - (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν] (κ : ℝ) (hκ : 0 < κ) (u : ℝ) - (f : ℝ × C(ℝ≥0,ℝ) → ℝ≥0∞) (hf : Measurable f) : - (∫⁻ p, f p ∂rawCenteredDiskLaw ν κ u) = - ∫⁻ θ, ∫⁻ p, f (θ,p) ∂ν.withDensity - (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv θ 0,p)) ∂Target.angleLaw κ := by - let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ - have hd : Measurable (fun p : ℝ × C(ℝ≥0,ℝ) => - canonicalFiniteDensity κ (u,Target.diskMapInv p.1 0,p.2)) := by - apply (canonicalFiniteDensity_measurable κ).comp - apply measurable_const.prodMk - apply Measurable.prodMk - · unfold Target.diskMapInv - fun_prop - · exact measurable_snd - exact product_withDensity_lintegral (Target.angleLaw κ) ν _ hd f hf - - -namespace Multipoint.CanonicalMass -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -lemma raw_centeredDiskTail_eq - (C : CanonicalMass P κ B hBm γ G) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (u b : ℝ) (hu : 0 ≤ u) : - rawCenteredDiskLaw (P.map (sourceDriverPath B hc)) κ u - {p | b ≤ R.diskAtRadius κ p.1 u p.2} = C.centeredDiskTail hc b u := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) := - (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance - rw [measure_joint_tail_of_lintegral _ (Target.angleLaw κ) - (fun θ => (P.map (sourceDriverPath B hc)).withDensity - (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv θ 0,p))) - (rawCenteredDiskLaw_lintegral _ κ hκ u) _ - (R.diskAtRadius_measurable κ Prod.fst (fun _ => u) measurable_fst measurable_const - Prod.snd measurable_snd) b, - C.centeredDiskTail_eq_integral hc hκ F hw b u hu] - apply lintegral_congr - intro θ - exact C.raw_diskAtRadius_finiteCentered_tail hc F hw R he θ u b hu - -end Multipoint.CanonicalMass -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalDiskTails +import OAI.Probability.SLE.Gauge.CanonicalDenseSections +import OAI.Probability.SLE.Geometry.ProductDensity + +namespace OAI + +/-! The disk variable on raw driver space has exactly the finite centered +law used in the prescribed moment coefficients and tail budget. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint +local instance rawDiskRadiusMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawDiskRadiusBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +def RawCanonicalMass.diskAtRadius (R : RawCanonicalMass) (κ θ u : ℝ) + (p : C(ℝ≥0,ℝ)) : ℝ := + (diskMassValue κ θ (extendedCanonicalMass R.process R.terminal + (canonicalRadiusLevel κ (u,Target.diskMapInv θ 0,p)) p)).toReal + +lemma RawCanonicalMass.diskAtRadius_nonneg (R : RawCanonicalMass) (κ θ u : ℝ) + (p : C(ℝ≥0,ℝ)) : 0 ≤ R.diskAtRadius κ θ u p := ENNReal.toReal_nonneg + +lemma RawCanonicalMass.diskAtRadius_measurable + {Ξ : Type*} [MeasurableSpace Ξ] (R : RawCanonicalMass) (κ : ℝ) + (θ u : Ξ → ℝ) (hθ : Measurable θ) (hu : Measurable u) + (Y : Ξ → C(ℝ≥0,ℝ)) (hY : Measurable Y) : + Measurable (fun x => R.diskAtRadius κ (θ x) (u x) (Y x)) := by + have hz : Measurable (fun x => Target.diskMapInv (θ x) 0) := by + unfold Target.diskMapInv + fun_prop + exact (R.disk_at_parameter_measurable κ θ hθ Y hY _ + (measurable_canonicalRadiusLevel_random_state κ hY hu hz)).ennreal_toReal + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + +include C hc F hw R he in +lemma CanonicalMass.raw_diskAtRadius_eq : + ∀ᵐ ω ∂P, ∀ θ u : ℝ, 0 ≤ u → + R.diskAtRadius κ θ u (sourceDriverPath B hc ω)=C.diskAtRadius θ u ω := by + filter_upwards [he] with ω hω + intro θ u hu + unfold RawCanonicalMass.diskAtRadius CanonicalMass.diskAtRadius + rw [canonicalRadiusLevel_eq κ (sourceDriverPath B hc ω) (hw ω) u hu] + unfold CanonicalMass.atTime extendedCanonicalMass + split_ifs <;> simp only [hω.1,hω.2] + +include C hc F hw R he in +lemma CanonicalMass.raw_diskAtRadius_finiteCentered_tail [SFinite P] + (θ u b : ℝ) (hu : 0 ≤ u) : + ((P.map (sourceDriverPath B hc)).withDensity + (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv θ 0,p))) + {p | b ≤ R.diskAtRadius κ θ u p} = + finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u + {ω | b ≤ C.diskAtRadius θ u ω} := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + rw [← canonical_finiteCenteredLaw_map P κ B hm hc γ G F (Eventually.of_forall hw) u hu] + have hA : MeasurableSet {p | b ≤ R.diskAtRadius κ θ u p} := by + exact measurableSet_le measurable_const + (R.diskAtRadius_measurable κ (fun _ => θ) (fun _ => u) + measurable_const measurable_const id measurable_id) + rw [Measure.map_apply (source_path_measurable _ hm) hA] + apply measure_congr + have habs : (finiteCenteredLaw P κ B γ G (Target.diskMapInv θ 0) u).AbsolutelyContinuous P := + withDensity_absolutelyContinuous P _ + filter_upwards [habs.ae_le (C.raw_diskAtRadius_eq hc F hw R he)] with ω hω + exact propext (by rw [mem_preimage,mem_ofPred_eq,hω θ u hu]) + +end SLEExactGauge.Multipoint + +namespace SLEExactGauge +local instance rawCenteredDiskMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawCenteredDiskBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +def rawCenteredDiskLaw (ν : Measure C(ℝ≥0,ℝ)) (κ u : ℝ) : Measure (ℝ × C(ℝ≥0,ℝ)) := + ((Target.angleLaw κ).prod ν).withDensity + (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv p.1 0,p.2)) + +lemma rawCenteredDiskLaw_lintegral + (ν : Measure C(ℝ≥0,ℝ)) [SFinite ν] (κ : ℝ) (hκ : 0 < κ) (u : ℝ) + (f : ℝ × C(ℝ≥0,ℝ) → ℝ≥0∞) (hf : Measurable f) : + (∫⁻ p, f p ∂rawCenteredDiskLaw ν κ u) = + ∫⁻ θ, ∫⁻ p, f (θ,p) ∂ν.withDensity + (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv θ 0,p)) ∂Target.angleLaw κ := by + let : IsProbabilityMeasure (Target.angleLaw κ) := Target.angleLaw_isProbability hκ + have hd : Measurable (fun p : ℝ × C(ℝ≥0,ℝ) => + canonicalFiniteDensity κ (u,Target.diskMapInv p.1 0,p.2)) := by + apply (canonicalFiniteDensity_measurable κ).comp + apply measurable_const.prodMk + apply Measurable.prodMk + · unfold Target.diskMapInv + fun_prop + · exact measurable_snd + exact product_withDensity_lintegral (Target.angleLaw κ) ν _ hd f hf + + +namespace Multipoint.CanonicalMass +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +lemma raw_centeredDiskTail_eq + (C : CanonicalMass P κ B hBm γ G) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (u b : ℝ) (hu : 0 ≤ u) : + rawCenteredDiskLaw (P.map (sourceDriverPath B hc)) κ u + {p | b ≤ R.diskAtRadius κ p.1 u p.2} = C.centeredDiskTail hc b u := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + let : IsProbabilityMeasure (P.map (sourceDriverPath B hc)) := + (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance + rw [measure_joint_tail_of_lintegral _ (Target.angleLaw κ) + (fun θ => (P.map (sourceDriverPath B hc)).withDensity + (fun p => canonicalFiniteDensity κ (u,Target.diskMapInv θ 0,p))) + (rawCenteredDiskLaw_lintegral _ κ hκ u) _ + (R.diskAtRadius_measurable κ Prod.fst (fun _ => u) measurable_fst measurable_const + Prod.snd measurable_snd) b, + C.centeredDiskTail_eq_integral hc hκ F hw b u hu] + apply lintegral_congr + intro θ + exact C.raw_diskAtRadius_finiteCentered_tail hc F hw R he θ u b hu + +end Multipoint.CanonicalMass +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawFixedRestart.lean b/lean/OAI/Probability/SLE/Loewner/RawFixedRestart.lean index 5a404632c7319e1325e020f7dfde531f9db1bec4..5c03329ef9a25e679c980d86b1c75f189ac27bc1 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawFixedRestart.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawFixedRestart.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Mass.PositiveMassContinuity -import OAI.Probability.SLE.Mass.CanonicalMassRestart -import OAI.Probability.SLE.Mass.RawScaledMassPaths - -namespace OAI - -/-! Any fixed raw realization with the full-law path certificate restarts -through the actual inverse map. This keeps the raw functional used by scaling. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge.Multipoint -local instance rawFixedRestartMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawFixedRestartBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma CanonicalMass.raw_fixed_restart - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) - (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈E) - (hEp : ∀ p∈E, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) : - ∀ᵐ ω ∂P, ∀ t, C.process (τ ω+t) ω-C.process (τ ω) ω = - stoppedFreshMass B κ γ G F hw τ hc R t ω := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hFresh := fresh_capacity_witness_exists hB hm hc hpublished τ hτ - have hfin := C.stoppedFreshTerminalMass_ae_finite_on_compacts hB hc hκ F hw τ hτ R - (he.mono fun _ h => h.2) hpublished - have hraw := sourceRestart_mem_full_driver_set hB hm hc τ hτ E hE hEf - have hp := stoppedFreshMass_paths B κ γ G F hw τ hc R hm hτ hκ - (hraw.mono fun _ h => by - have hr := hEp _ h - exact ⟨hr.zero,hr.monotone,hr.continuous_real_tests,hr.dominated⟩) hfin - exact C.stoppedFreshMass_indistinguishable hB hc hκ F hw R he τ hτ hFresh hp - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.PositiveMassContinuity +import OAI.Probability.SLE.Mass.CanonicalMassRestart +import OAI.Probability.SLE.Mass.RawScaledMassPaths + +namespace OAI + +/-! Any fixed raw realization with the full-law path certificate restarts +through the actual inverse map. This keeps the raw functional used by scaling. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge.Multipoint +local instance rawFixedRestartMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawFixedRestartBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma CanonicalMass.raw_fixed_restart + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) + (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈E) + (hEp : ∀ p∈E, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) : + ∀ᵐ ω ∂P, ∀ t, C.process (τ ω+t) ω-C.process (τ ω) ω = + stoppedFreshMass B κ γ G F hw τ hc R t ω := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hFresh := fresh_capacity_witness_exists hB hm hc hpublished τ hτ + have hfin := C.stoppedFreshTerminalMass_ae_finite_on_compacts hB hc hκ F hw τ hτ R + (he.mono fun _ h => h.2) hpublished + have hraw := sourceRestart_mem_full_driver_set hB hm hc τ hτ E hE hEf + have hp := stoppedFreshMass_paths B κ γ G F hw τ hc R hm hτ hκ + (hraw.mono fun _ h => by + have hr := hEp _ h + exact ⟨hr.zero,hr.monotone,hr.continuous_real_tests,hr.dominated⟩) hfin + exact C.stoppedFreshMass_indistinguishable hB hc hκ F hw R he τ hτ hFresh hp + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawForwardCoordinates.lean b/lean/OAI/Probability/SLE/Loewner/RawForwardCoordinates.lean index 9eabad818e8a54d067aa7cfd01e0d39d1529c9c6..5be24bdcebdbcce31bcb2487145978732d16f8eb 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawForwardCoordinates.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawForwardCoordinates.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Loewner.RawForwardHeight - -namespace OAI - -/-! Exact real coordinate equations before a raw survival lifetime. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge.Target - -lemma flow_coordinate_derivatives (b : Driver) (z : ℂ) {t S : ℝ} - (ht : 0 ≤ t) (htS : t < S) (hs : Surviving 1 b z S) : - HasDerivWithinAt (fun s => (flow 1 b z s).re) - (2*((flow 1 b z t).re-driverValue b t)/ - (((flow 1 b z t).re-driverValue b t)^2+(flow 1 b z t).im^2)) (Ici t) t ∧ - HasDerivWithinAt (fun s => (flow 1 b z s).im) - (-2*(flow 1 b z t).im/ - (((flow 1 b z t).re-driverValue b t)^2+(flow 1 b z t).im^2)) (Ici t) t := by - have hd := (flow_hasDerivWithinAt_of_later_survival 1 b z t S ht htS hs).mono - (Ici_subset_Ici.mpr ht) - have hr := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hd - have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hd - change HasDerivWithinAt (fun s => (flow 1 b z s).re) - (((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).re) (Ici t) t at hr - change HasDerivWithinAt (fun s => (flow 1 b z s).im) - (((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).im) (Ici t) t at hi - constructor - · convert! hr using 1 - simp [a,Complex.div_re,Complex.normSq_apply,pow_two] - · convert! hi using 1 - simp [a,Complex.div_im,Complex.normSq_apply,pow_two] - ring - -lemma flow_continuousOn_before_lifetime (b : Driver) (z : ℂ) {τ : ℝ} - (hs : ∀ t ∈ Ico 0 τ, Surviving 1 b z t) : - ContinuousOn (flow 1 b z) (Ico 0 τ) := by - intro t ht - let S : ℝ := (t+τ)/2 - have htS : t < S := by dsimp [S]; linarith [ht.2] - have hSτ : S < τ := by dsimp [S]; linarith [ht.2] - exact (flow_hasDerivWithinAt_of_later_survival 1 b z t S ht.1 htS - (hs S ⟨ht.1.trans htS.le,hSτ⟩)).continuousWithinAt.mono Ico_subset_Ici_self - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawForwardHeight + +namespace OAI + +/-! Exact real coordinate equations before a raw survival lifetime. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge.Target + +lemma flow_coordinate_derivatives (b : Driver) (z : ℂ) {t S : ℝ} + (ht : 0 ≤ t) (htS : t < S) (hs : Surviving 1 b z S) : + HasDerivWithinAt (fun s => (flow 1 b z s).re) + (2*((flow 1 b z t).re-driverValue b t)/ + (((flow 1 b z t).re-driverValue b t)^2+(flow 1 b z t).im^2)) (Ici t) t ∧ + HasDerivWithinAt (fun s => (flow 1 b z s).im) + (-2*(flow 1 b z t).im/ + (((flow 1 b z t).re-driverValue b t)^2+(flow 1 b z t).im^2)) (Ici t) t := by + have hd := (flow_hasDerivWithinAt_of_later_survival 1 b z t S ht htS hs).mono + (Ici_subset_Ici.mpr ht) + have hr := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hd + have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t hd + change HasDerivWithinAt (fun s => (flow 1 b z s).re) + (((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).re) (Ici t) t at hr + change HasDerivWithinAt (fun s => (flow 1 b z s).im) + (((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).im) (Ici t) t at hi + constructor + · convert! hr using 1 + simp [a,Complex.div_re,Complex.normSq_apply,pow_two] + · convert! hi using 1 + simp [a,Complex.div_im,Complex.normSq_apply,pow_two] + ring + +lemma flow_continuousOn_before_lifetime (b : Driver) (z : ℂ) {τ : ℝ} + (hs : ∀ t ∈ Ico 0 τ, Surviving 1 b z t) : + ContinuousOn (flow 1 b z) (Ico 0 τ) := by + intro t ht + let S : ℝ := (t+τ)/2 + have htS : t < S := by dsimp [S]; linarith [ht.2] + have hSτ : S < τ := by dsimp [S]; linarith [ht.2] + exact (flow_hasDerivWithinAt_of_later_survival 1 b z t S ht.1 htS + (hs S ⟨ht.1.trans htS.le,hSτ⟩)).continuousWithinAt.mono Ico_subset_Ici_self + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawForwardHeight.lean b/lean/OAI/Probability/SLE/Loewner/RawForwardHeight.lean index 32e70ff46bdf427293737632be4f6c8ab7779bc3..6a36f831053d26306d2eed70dda59299e9f01b25 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawForwardHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawForwardHeight.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Angle.TargetFlowEvolution -import OAI.Probability.SLE.Loewner.RawSurvivingMonotone - -namespace OAI - -/-! Continuity and height monotonicity of the exact raw forward flow on -each surviving interval, without a geometric trace witness. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge.Target - -lemma flow_continuousOn_of_survival (κ : ℝ) (b : Driver) (z : ℂ) - {T : ℝ} (hT : 0 ≤ T) (hs : Surviving κ b z T) : - ContinuousOn (flow κ b z) (Icc 0 T) := by - obtain ⟨j,hj,hmin⟩ := hs - apply ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self).congr - intro t ht - exact flow_eq_cutoff_on_interval κ b z T hT j hj hmin t ht - -lemma flow_height_antitone_of_survival (b : Driver) (z : ℂ) - {T : ℝ} (hT : 0 ≤ T) (hs : Surviving 1 b z T) : - AntitoneOn (fun t => (flow 1 b z t).im) (Icc 0 T) := by - have hc := Complex.continuous_im.comp_continuousOn (flow_continuousOn_of_survival 1 b z hT hs) - have hd (t : ℝ) (ht : t ∈ interior (Icc 0 T)) : - HasDerivWithinAt (fun s => (flow 1 b z s).im) - (-2*(flow 1 b z t).im/Complex.normSq (flow 1 b z t-(driverValue b t : ℂ))) - (interior (Icc 0 T)) t := by - have hti : t ∈ Ioo 0 T := by simpa only [interior_Icc] using ht - have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t - (flow_hasDerivWithinAt_of_later_survival 1 b z t T hti.1.le hti.2 hs) - change HasDerivWithinAt (fun s => (flow 1 b z s).im) - (((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).im) (Ici 0) t at hh - have he : ((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).im = - -2*(flow 1 b z t).im/Complex.normSq (flow 1 b z t-(driverValue b t : ℂ)) := by - simp [a,Complex.div_im] - ring - rw [he] at hh - exact hh.mono (interior_subset.trans Icc_subset_Ici_self) - apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _) hc hd - intro t ht - have hti : t ∈ Icc 0 T := interior_subset ht - have hp := surviving_flow_mem_upperHalfPlane 1 b z t hti.1 - (surviving_of_le_time 1 b z hti.1 hti.2 hs) - exact div_nonpos_of_nonpos_of_nonneg - (mul_nonpos_of_nonpos_of_nonneg (by norm_num) hp.le) (Complex.normSq_nonneg _) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetFlowEvolution +import OAI.Probability.SLE.Loewner.RawSurvivingMonotone + +namespace OAI + +/-! Continuity and height monotonicity of the exact raw forward flow on +each surviving interval, without a geometric trace witness. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge.Target + +lemma flow_continuousOn_of_survival (κ : ℝ) (b : Driver) (z : ℂ) + {T : ℝ} (hT : 0 ≤ T) (hs : Surviving κ b z T) : + ContinuousOn (flow κ b z) (Icc 0 T) := by + obtain ⟨j,hj,hmin⟩ := hs + apply ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self).congr + intro t ht + exact flow_eq_cutoff_on_interval κ b z T hT j hj hmin t ht + +lemma flow_height_antitone_of_survival (b : Driver) (z : ℂ) + {T : ℝ} (hT : 0 ≤ T) (hs : Surviving 1 b z T) : + AntitoneOn (fun t => (flow 1 b z t).im) (Icc 0 T) := by + have hc := Complex.continuous_im.comp_continuousOn (flow_continuousOn_of_survival 1 b z hT hs) + have hd (t : ℝ) (ht : t ∈ interior (Icc 0 T)) : + HasDerivWithinAt (fun s => (flow 1 b z s).im) + (-2*(flow 1 b z t).im/Complex.normSq (flow 1 b z t-(driverValue b t : ℂ))) + (interior (Icc 0 T)) t := by + have hti : t ∈ Ioo 0 T := by simpa only [interior_Icc] using ht + have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t + (flow_hasDerivWithinAt_of_later_survival 1 b z t T hti.1.le hti.2 hs) + change HasDerivWithinAt (fun s => (flow 1 b z s).im) + (((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).im) (Ici 0) t at hh + have he : ((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))).im = + -2*(flow 1 b z t).im/Complex.normSq (flow 1 b z t-(driverValue b t : ℂ)) := by + simp [a,Complex.div_im] + ring + rw [he] at hh + exact hh.mono (interior_subset.trans Icc_subset_Ici_self) + apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _) hc hd + intro t ht + have hti : t ∈ Icc 0 T := interior_subset ht + have hp := surviving_flow_mem_upperHalfPlane 1 b z t hti.1 + (surviving_of_le_time 1 b z hti.1 hti.2 hs) + exact div_nonpos_of_nonpos_of_nonneg + (mul_nonpos_of_nonpos_of_nonneg (by norm_num) hp.le) (Complex.normSq_nonneg _) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawForwardInfinity.lean b/lean/OAI/Probability/SLE/Loewner/RawForwardInfinity.lean index f1492404842fbe7e1aeb3acd5a231f3f46e8dd75..3df404d1d547ae9edb08bde8d3e9dfb4e53bbfb4 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawForwardInfinity.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawForwardInfinity.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Loewner.ForwardCapacityError -import OAI.Probability.SLE.Loewner.RawForwardHeight - -namespace OAI - -/-! Hydrodynamic normalization of the raw forward map follows from its -ODE on surviving trajectories, without a prior geometric witness. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem flow_hydrodynamic_normalization (b : Driver) (T : ℝ≥0) : - Tendsto (fun z : ℂ => z*(flow 1 b z T-z)) - (cocompact ℂ ⊓ 𝓟 {z | Surviving 1 b z T}) (𝓝 (2*(T : ℂ))) := by - have hc : Continuous (driverValue b) := b.continuous.comp continuous_real_toNNReal - obtain ⟨M,hM,hMb⟩ := (isCompact_Icc.image hc).isBounded.exists_pos_norm_lt - have hbound (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : |driverValue b t| ≤ M := by - simpa only [Real.norm_eq_abs] using (hMb _ (mem_image_of_mem _ ht)).le - have hfar (z : ℂ) (hz : Surviving 1 b z T) - (hlarge : 8*(M+(T : ℝ)+1) ≤ ‖z‖) : - ‖z*(flow 1 b z T-z)-2*(T : ℂ)‖ ≤ 4*(T : ℝ)*(M+T)/‖z‖ := by - have hzH := surviving_mem_upperHalfPlane 1 b z T T.coe_nonneg hz - apply forward_far_capacity_error T.coe_nonneg hM.le - (flow 1 b z) (driverValue b) z - (flow_continuousOn_of_survival 1 b z T.coe_nonneg hz) (flow_zero 1 b z hzH) - (fun t ht => ?_) hbound hlarge - let S : ℝ := (t+T)/2 - have htS : t < S := by dsimp [S]; linarith [ht.2] - have hST : S ≤ T := by dsimp [S]; linarith [ht.2] - have hS : 0 ≤ S := ht.1.trans htS.le - have hh := (flow_hasDerivWithinAt_of_later_survival 1 b z t S ht.1 htS - (surviving_of_le_time 1 b z hS hST hz)).mono (Ici_subset_Ici.mpr ht.1) - simpa only [a,div_one,ofReal_ofNat] using hh - let l := cocompact ℂ ⊓ 𝓟 {z | Surviving 1 b z T} - have herr : Tendsto (fun z : ℂ => z*(flow 1 b z T-z)-2*(T : ℂ)) l (𝓝 0) := by - apply squeeze_zero_norm' - · change ∀ᶠ z in cocompact ℂ ⊓ 𝓟 {z | Surviving 1 b z T}, - ‖z*(flow 1 b z T-z)-2*(T : ℂ)‖ ≤ 4*(T : ℝ)*(M+T)/‖z‖ - rw [eventually_inf_principal] - filter_upwards [tendsto_norm_cocompact_atTop.eventually - (eventually_ge_atTop (8*(M+(T : ℝ)+1)))] with z hz - exact fun hsurvive => hfar z hsurvive hz - · exact tendsto_const_nhds.div_atTop - (tendsto_norm_cocompact_atTop.mono_left inf_le_left) - simpa only [sub_add_cancel,zero_add] using herr.add_const (2*(T : ℂ)) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.ForwardCapacityError +import OAI.Probability.SLE.Loewner.RawForwardHeight + +namespace OAI + +/-! Hydrodynamic normalization of the raw forward map follows from its +ODE on surviving trajectories, without a prior geometric witness. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem flow_hydrodynamic_normalization (b : Driver) (T : ℝ≥0) : + Tendsto (fun z : ℂ => z*(flow 1 b z T-z)) + (cocompact ℂ ⊓ 𝓟 {z | Surviving 1 b z T}) (𝓝 (2*(T : ℂ))) := by + have hc : Continuous (driverValue b) := b.continuous.comp continuous_real_toNNReal + obtain ⟨M,hM,hMb⟩ := (isCompact_Icc.image hc).isBounded.exists_pos_norm_lt + have hbound (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : |driverValue b t| ≤ M := by + simpa only [Real.norm_eq_abs] using (hMb _ (mem_image_of_mem _ ht)).le + have hfar (z : ℂ) (hz : Surviving 1 b z T) + (hlarge : 8*(M+(T : ℝ)+1) ≤ ‖z‖) : + ‖z*(flow 1 b z T-z)-2*(T : ℂ)‖ ≤ 4*(T : ℝ)*(M+T)/‖z‖ := by + have hzH := surviving_mem_upperHalfPlane 1 b z T T.coe_nonneg hz + apply forward_far_capacity_error T.coe_nonneg hM.le + (flow 1 b z) (driverValue b) z + (flow_continuousOn_of_survival 1 b z T.coe_nonneg hz) (flow_zero 1 b z hzH) + (fun t ht => ?_) hbound hlarge + let S : ℝ := (t+T)/2 + have htS : t < S := by dsimp [S]; linarith [ht.2] + have hST : S ≤ T := by dsimp [S]; linarith [ht.2] + have hS : 0 ≤ S := ht.1.trans htS.le + have hh := (flow_hasDerivWithinAt_of_later_survival 1 b z t S ht.1 htS + (surviving_of_le_time 1 b z hS hST hz)).mono (Ici_subset_Ici.mpr ht.1) + simpa only [a,div_one,ofReal_ofNat] using hh + let l := cocompact ℂ ⊓ 𝓟 {z | Surviving 1 b z T} + have herr : Tendsto (fun z : ℂ => z*(flow 1 b z T-z)-2*(T : ℂ)) l (𝓝 0) := by + apply squeeze_zero_norm' + · change ∀ᶠ z in cocompact ℂ ⊓ 𝓟 {z | Surviving 1 b z T}, + ‖z*(flow 1 b z T-z)-2*(T : ℂ)‖ ≤ 4*(T : ℝ)*(M+T)/‖z‖ + rw [eventually_inf_principal] + filter_upwards [tendsto_norm_cocompact_atTop.eventually + (eventually_ge_atTop (8*(M+(T : ℝ)+1)))] with z hz + exact fun hsurvive => hfar z hsurvive hz + · exact tendsto_const_nhds.div_atTop + (tendsto_norm_cocompact_atTop.mono_left inf_le_left) + simpa only [sub_add_cancel,zero_add] using herr.add_const (2*(T : ℂ)) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawHeightContinuation.lean b/lean/OAI/Probability/SLE/Loewner/RawHeightContinuation.lean index cd748d2763a00d4ee7490a1df3f8cfbda2a648c6..206549a5fd57955e7dea16a3a1d1adaa2ed2d367 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawHeightContinuation.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawHeightContinuation.lean @@ -1,72 +1,72 @@ -import OAI.Probability.SLE.Loewner.RawForwardHeight -import OAI.Probability.SLE.Geometry.ComplexLipschitzExtension - -namespace OAI - -/-! A raw forward flow whose height stays uniformly positive survives -the limiting finite time. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge.Target - -theorem survives_of_uniform_height_before (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) - {T m : ℝ} (hT : 0 < T) (hm : 0 < m) - (hs : ∀ t ∈ Ico 0 T, Surviving 1 b z t) - (hh : ∀ t ∈ Ico 0 T, m ≤ (flow 1 b z t).im) : - Surviving 1 b z T := by - have hlater (t : ℝ) (ht : t ∈ Ico 0 T) : - HasDerivWithinAt (flow 1 b z) - ((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))) (Ici 0) t := by - let S : ℝ := (t+T)/2 - have htS : t < S := by dsimp [S]; linarith [ht.2] - have hST : S < T := by dsimp [S]; linarith [ht.2] - exact flow_hasDerivWithinAt_of_later_survival 1 b z t S ht.1 htS - (hs S ⟨ht.1.trans htS.le,hST⟩) - have hd (t : ℝ) (ht : t ∈ Ico 0 T) : HasDerivWithinAt (flow 1 b z) - ((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))) (Ico 0 T) t := - (hlater t ht).mono Ico_subset_Ici_self - have hnorm (t : ℝ) (ht : t ∈ Ico 0 T) : - ‖(a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))‖ ≤ 2/m := by - have hden : m ≤ ‖flow 1 b z t-(driverValue b t : ℂ)‖ := by - calc - m ≤ (flow 1 b z t).im := hh t ht - _ ≤ |(flow 1 b z t-(driverValue b t : ℂ)).im| := by - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] - exact le_abs_self _ - _ ≤ _ := Complex.abs_im_le_norm _ - calc - _ = 2/‖flow 1 b z t-(driverValue b t : ℂ)‖ := by norm_num [norm_div,a] - _ ≤ 2/m := div_le_div_of_nonneg_left (by norm_num) hm hden - have hLip : LipschitzOnWith ⟨2/m,by positivity⟩ (flow 1 b z) (Ico 0 T) := by - apply (convex_Ico (0 : ℝ) T).lipschitzOnWith_of_nnnorm_hasDerivWithin_le hd - intro t ht - exact_mod_cast hnorm t ht - obtain ⟨g,hg,hge⟩ := exists_continuous_complex_extension (flow 1 b z) hLip - have hheight : ∀ t ∈ Icc 0 T, m ≤ (g t).im := by - have hc : IsClosed {t : ℝ | m ≤ (g t).im} := - isClosed_le continuous_const (Complex.continuous_im.comp hg) - have hsub : Ico 0 T ⊆ {t : ℝ | m ≤ (g t).im} := by - intro t ht - change m ≤ (g t).im - rw [hge ht] - exact hh t ht - have hcsub := closure_minimal hsub hc - rw [closure_Ico hT.ne] at hcsub - exact fun t ht => hcsub ht - have hzero : g 0=z := (hge ⟨le_rfl,hT⟩).trans (flow_zero 1 b z hz) - apply surviving_of_positive_trajectory 1 b z T hT.le g hg.continuousOn hzero m hm hheight - intro t ht - have he : g =ᶠ[𝓝[Ici t] t] flow 1 b z := by - filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (gt_mem_nhds ht.2)] with s hs hsT - exact hge ⟨ht.1.trans hs,hsT⟩ - have hv := hge ht - have hh' := ((hlater t ht).mono (Ici_subset_Ici.mpr ht.1)).congr_of_eventuallyEq he hv - simpa only [← hv] using hh' - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawForwardHeight +import OAI.Probability.SLE.Geometry.ComplexLipschitzExtension + +namespace OAI + +/-! A raw forward flow whose height stays uniformly positive survives +the limiting finite time. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge.Target + +theorem survives_of_uniform_height_before (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) + {T m : ℝ} (hT : 0 < T) (hm : 0 < m) + (hs : ∀ t ∈ Ico 0 T, Surviving 1 b z t) + (hh : ∀ t ∈ Ico 0 T, m ≤ (flow 1 b z t).im) : + Surviving 1 b z T := by + have hlater (t : ℝ) (ht : t ∈ Ico 0 T) : + HasDerivWithinAt (flow 1 b z) + ((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))) (Ici 0) t := by + let S : ℝ := (t+T)/2 + have htS : t < S := by dsimp [S]; linarith [ht.2] + have hST : S < T := by dsimp [S]; linarith [ht.2] + exact flow_hasDerivWithinAt_of_later_survival 1 b z t S ht.1 htS + (hs S ⟨ht.1.trans htS.le,hST⟩) + have hd (t : ℝ) (ht : t ∈ Ico 0 T) : HasDerivWithinAt (flow 1 b z) + ((a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))) (Ico 0 T) t := + (hlater t ht).mono Ico_subset_Ici_self + have hnorm (t : ℝ) (ht : t ∈ Ico 0 T) : + ‖(a 1 : ℂ)/(flow 1 b z t-(driverValue b t : ℂ))‖ ≤ 2/m := by + have hden : m ≤ ‖flow 1 b z t-(driverValue b t : ℂ)‖ := by + calc + m ≤ (flow 1 b z t).im := hh t ht + _ ≤ |(flow 1 b z t-(driverValue b t : ℂ)).im| := by + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] + exact le_abs_self _ + _ ≤ _ := Complex.abs_im_le_norm _ + calc + _ = 2/‖flow 1 b z t-(driverValue b t : ℂ)‖ := by norm_num [norm_div,a] + _ ≤ 2/m := div_le_div_of_nonneg_left (by norm_num) hm hden + have hLip : LipschitzOnWith ⟨2/m,by positivity⟩ (flow 1 b z) (Ico 0 T) := by + apply (convex_Ico (0 : ℝ) T).lipschitzOnWith_of_nnnorm_hasDerivWithin_le hd + intro t ht + exact_mod_cast hnorm t ht + obtain ⟨g,hg,hge⟩ := exists_continuous_complex_extension (flow 1 b z) hLip + have hheight : ∀ t ∈ Icc 0 T, m ≤ (g t).im := by + have hc : IsClosed {t : ℝ | m ≤ (g t).im} := + isClosed_le continuous_const (Complex.continuous_im.comp hg) + have hsub : Ico 0 T ⊆ {t : ℝ | m ≤ (g t).im} := by + intro t ht + change m ≤ (g t).im + rw [hge ht] + exact hh t ht + have hcsub := closure_minimal hsub hc + rw [closure_Ico hT.ne] at hcsub + exact fun t ht => hcsub ht + have hzero : g 0=z := (hge ⟨le_rfl,hT⟩).trans (flow_zero 1 b z hz) + apply surviving_of_positive_trajectory 1 b z T hT.le g hg.continuousOn hzero m hm hheight + intro t ht + have he : g =ᶠ[𝓝[Ici t] t] flow 1 b z := by + filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (gt_mem_nhds ht.2)] with s hs hsT + exact hge ⟨ht.1.trans hs,hsT⟩ + have hv := hge ht + have hh' := ((hlater t ht).mono (Ici_subset_Ici.mpr ht.1)).congr_of_eventuallyEq he hv + simpa only [← hv] using hh' + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawInverseDriverApproximation.lean b/lean/OAI/Probability/SLE/Loewner/RawInverseDriverApproximation.lean index 015f08bb22514c175ab797b1702c809e942dc6f5..35a54322a4ef333ecef0564c7057550ddd92b5f5 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawInverseDriverApproximation.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawInverseDriverApproximation.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Loewner.SmoothRawInverse -import OAI.Probability.SLE.Loewner.LoewnerInverseStability - -namespace OAI - -/-! A geometric inverse witness approaches the raw inverse under a -uniform driver perturbation. The reference driver needs no trace. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem IsCapacityTwoWitness.raw_inverse_driver_distance - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (b : Target.Driver) (T : ℝ≥0) - {a : ℝ} (ha : 0 < a) (w : ℂ) (haw : a ≤ w.im) - (δ : ℝ) (hUb : ∀ t ∈ Icc 0 T, |U t-b t| ≤ δ) : - dist (F T w) (Target.inverseFlow 1 b T w) ≤ - gronwallBound 0 (4*a⁻¹^2) (2*a⁻¹^2*δ) T := by - obtain ⟨q,hq0,hheight,hqd,he⟩ := Target.exists_raw_reverse_trajectory b T w (ha.trans_le haw) - rw [he] - apply h.inverse_approximation_bound T ha w haw b δ hUb q hq0 - intro s hs - have hh := (hqd s hs.1).mono (show Icc 0 (T : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) - have hr : reverseField a (b ((T : ℝ)-s).toNNReal) (q s) = - -2/(q s-(Target.driverValue b ((T : ℝ)-s) : ℂ)) := by - rw [reverseField,reverseCenter_eq (haw.trans (hheight s hs.1))] - rfl - rw [hr] - exact hh - -theorem raw_inverse_limit_of_witness_drivers - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (hw : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - (b : Target.Driver) (T : ℝ≥0) (δ : ℕ → ℝ) - (hδ : Tendsto δ atTop (𝓝 0)) - (hUb : ∀ n t, t ∈ Icc 0 T → |U n t-b t| ≤ δ n) - {w : ℂ} (hwH : w ∈ upperHalfPlane) : - Tendsto (fun n => F n T w) atTop (𝓝 (Target.inverseFlow 1 b T w)) := by - rw [tendsto_iff_dist_tendsto_zero] - have hb := fun n => (hw n).raw_inverse_driver_distance b T hwH w le_rfl (δ n) (hUb n) - have ht := (gronwallBound_continuous_ε 0 (4*w.im⁻¹^2) T).continuousAt.tendsto.comp - (hδ.const_mul (2*w.im⁻¹^2)) - simp only [Function.comp_def,mul_zero,gronwallBound_ε0_δ0] at ht - exact squeeze_zero (fun _ => dist_nonneg) hb ht - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawInverse +import OAI.Probability.SLE.Loewner.LoewnerInverseStability + +namespace OAI + +/-! A geometric inverse witness approaches the raw inverse under a +uniform driver perturbation. The reference driver needs no trace. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem IsCapacityTwoWitness.raw_inverse_driver_distance + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (b : Target.Driver) (T : ℝ≥0) + {a : ℝ} (ha : 0 < a) (w : ℂ) (haw : a ≤ w.im) + (δ : ℝ) (hUb : ∀ t ∈ Icc 0 T, |U t-b t| ≤ δ) : + dist (F T w) (Target.inverseFlow 1 b T w) ≤ + gronwallBound 0 (4*a⁻¹^2) (2*a⁻¹^2*δ) T := by + obtain ⟨q,hq0,hheight,hqd,he⟩ := Target.exists_raw_reverse_trajectory b T w (ha.trans_le haw) + rw [he] + apply h.inverse_approximation_bound T ha w haw b δ hUb q hq0 + intro s hs + have hh := (hqd s hs.1).mono (show Icc 0 (T : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) + have hr : reverseField a (b ((T : ℝ)-s).toNNReal) (q s) = + -2/(q s-(Target.driverValue b ((T : ℝ)-s) : ℂ)) := by + rw [reverseField,reverseCenter_eq (haw.trans (hheight s hs.1))] + rfl + rw [hr] + exact hh + +theorem raw_inverse_limit_of_witness_drivers + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (hw : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + (b : Target.Driver) (T : ℝ≥0) (δ : ℕ → ℝ) + (hδ : Tendsto δ atTop (𝓝 0)) + (hUb : ∀ n t, t ∈ Icc 0 T → |U n t-b t| ≤ δ n) + {w : ℂ} (hwH : w ∈ upperHalfPlane) : + Tendsto (fun n => F n T w) atTop (𝓝 (Target.inverseFlow 1 b T w)) := by + rw [tendsto_iff_dist_tendsto_zero] + have hb := fun n => (hw n).raw_inverse_driver_distance b T hwH w le_rfl (δ n) (hUb n) + have ht := (gronwallBound_continuous_ε 0 (4*w.im⁻¹^2) T).continuousAt.tendsto.comp + (hδ.const_mul (2*w.im⁻¹^2)) + simp only [Function.comp_def,mul_zero,gronwallBound_ε0_δ0] at ht + exact squeeze_zero (fun _ => dist_nonneg) hb ht + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawInverseHolomorphic.lean b/lean/OAI/Probability/SLE/Loewner/RawInverseHolomorphic.lean index fc9e4801dcc1020fb43481dc572c2ad7092ca13a..f956b4b0963a953a0234e7499f77a4498309eacb 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawInverseHolomorphic.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawInverseHolomorphic.lean @@ -1,68 +1,68 @@ -import OAI.Probability.SLE.Loewner.RawInverseDriverApproximation -import OAI.Probability.SLE.Loewner.LoewnerInverseCompactness -import OAI.Probability.SLE.Geometry.ActualCompactAvoidance -import OAI.Probability.SLE.Loewner.UnivalentInverseAnalytic - -namespace OAI - -/-! Holomorphicity of every continuous-driver raw inverse follows from -interior approximation by actual SLE witnesses. The argument uses the -finite Brownian driver tubes, not geometric SLE support. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem raw_inverse_holomorphic_of_witness_approximation - (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) - (hw : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) - (b : Target.Driver) (T : ℝ≥0) (δ : ℕ → ℝ) - (hδ : Tendsto δ atTop (𝓝 0)) - (hUb : ∀ n t, t ∈ Icc 0 T → |U n t-b t| ≤ δ n) : - DifferentiableOn ℂ (Target.inverseFlow 1 b T) upperHalfPlane := by - obtain ⟨σ,f,hσ,hf,hconv⟩ := inverse_family_subsequence U γ G F hw T - have he (w : ℂ) (hwH : w ∈ upperHalfPlane) : f w=Target.inverseFlow 1 b T w := - tendsto_nhds_unique (hconv.tendsto_at hwH) - ((raw_inverse_limit_of_witness_drivers U γ G F hw b T δ hδ hUb hwH).comp hσ.tendsto_atTop) - exact hf.congr (fun w hwH => (he w hwH).symm) - -theorem RegularSLEPresentation.raw_inverse_holomorphic - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) : - DifferentiableOn ℂ (Target.inverseFlow 1 b T) upperHalfPlane := by - let δ (n : ℕ) : ℝ := 1/((n : ℝ)+1) - have hδ (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity - have hex (n : ℕ) : ∃ ω : a.samples, - ∀ t ∈ Icc 0 (T+1), |Real.sqrt κ*a.B t ω-b t| < δ n := by - have hp := regular_scaled_driver_tube_positive a hκ b.toContinuousMap - (show b.toContinuousMap 0=0 from map_zero b) (T+1) (by positivity) (hδ n) - exact nonempty_of_measure_ne_zero hp.ne' - choose ω hω using hex - exact raw_inverse_holomorphic_of_witness_approximation - (fun n t => Real.sqrt κ*a.B t (ω n)) (fun n => γ (ω n)) - (fun n => a.G (ω n)) (fun n => a.F (ω n)) (fun n => a.witness (ω n)) b T δ - tendsto_one_div_add_atTop_nhds_zero_nat (fun n t ht => (hω n t - ⟨ht.1,ht.2.trans (le_add_of_nonneg_right zero_le)⟩).le) - -lemma Target.inverseFlow_injOn (b : Driver) (T : ℝ≥0) : - InjOn (inverseFlow 1 b T) upperHalfPlane := by - intro z hz w hw he - have hh := congrArg (fun q => flow 1 b q T) he - rwa [flow_inverseFlow 1 (by norm_num) b T z hz, - flow_inverseFlow 1 (by norm_num) b T w hw] at hh - -theorem RegularSLEPresentation.raw_inverse_deriv_ne_zero - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) {z : ℂ} (hz : z ∈ upperHalfPlane) : - deriv (Target.inverseFlow 1 b T) z ≠ 0 := - univalent_deriv_ne_zero isOpen_upperHalfPlane (a.raw_inverse_holomorphic hκ b T) - (Target.inverseFlow_injOn b T) hz - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawInverseDriverApproximation +import OAI.Probability.SLE.Loewner.LoewnerInverseCompactness +import OAI.Probability.SLE.Geometry.ActualCompactAvoidance +import OAI.Probability.SLE.Loewner.UnivalentInverseAnalytic + +namespace OAI + +/-! Holomorphicity of every continuous-driver raw inverse follows from +interior approximation by actual SLE witnesses. The argument uses the +finite Brownian driver tubes, not geometric SLE support. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem raw_inverse_holomorphic_of_witness_approximation + (U : ℕ → ℝ≥0 → ℝ) (γ : ℕ → ℝ≥0 → ℂ) (G F : ℕ → ℝ≥0 → ℂ → ℂ) + (hw : ∀ n, IsCapacityTwoWitness (U n) (γ n) (G n) (F n)) + (b : Target.Driver) (T : ℝ≥0) (δ : ℕ → ℝ) + (hδ : Tendsto δ atTop (𝓝 0)) + (hUb : ∀ n t, t ∈ Icc 0 T → |U n t-b t| ≤ δ n) : + DifferentiableOn ℂ (Target.inverseFlow 1 b T) upperHalfPlane := by + obtain ⟨σ,f,hσ,hf,hconv⟩ := inverse_family_subsequence U γ G F hw T + have he (w : ℂ) (hwH : w ∈ upperHalfPlane) : f w=Target.inverseFlow 1 b T w := + tendsto_nhds_unique (hconv.tendsto_at hwH) + ((raw_inverse_limit_of_witness_drivers U γ G F hw b T δ hδ hUb hwH).comp hσ.tendsto_atTop) + exact hf.congr (fun w hwH => (he w hwH).symm) + +theorem RegularSLEPresentation.raw_inverse_holomorphic + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) : + DifferentiableOn ℂ (Target.inverseFlow 1 b T) upperHalfPlane := by + let δ (n : ℕ) : ℝ := 1/((n : ℝ)+1) + have hδ (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity + have hex (n : ℕ) : ∃ ω : a.samples, + ∀ t ∈ Icc 0 (T+1), |Real.sqrt κ*a.B t ω-b t| < δ n := by + have hp := regular_scaled_driver_tube_positive a hκ b.toContinuousMap + (show b.toContinuousMap 0=0 from map_zero b) (T+1) (by positivity) (hδ n) + exact nonempty_of_measure_ne_zero hp.ne' + choose ω hω using hex + exact raw_inverse_holomorphic_of_witness_approximation + (fun n t => Real.sqrt κ*a.B t (ω n)) (fun n => γ (ω n)) + (fun n => a.G (ω n)) (fun n => a.F (ω n)) (fun n => a.witness (ω n)) b T δ + tendsto_one_div_add_atTop_nhds_zero_nat (fun n t ht => (hω n t + ⟨ht.1,ht.2.trans (le_add_of_nonneg_right zero_le)⟩).le) + +lemma Target.inverseFlow_injOn (b : Driver) (T : ℝ≥0) : + InjOn (inverseFlow 1 b T) upperHalfPlane := by + intro z hz w hw he + have hh := congrArg (fun q => flow 1 b q T) he + rwa [flow_inverseFlow 1 (by norm_num) b T z hz, + flow_inverseFlow 1 (by norm_num) b T w hw] at hh + +theorem RegularSLEPresentation.raw_inverse_deriv_ne_zero + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) {z : ℂ} (hz : z ∈ upperHalfPlane) : + deriv (Target.inverseFlow 1 b T) z ≠ 0 := + univalent_deriv_ne_zero isOpen_upperHalfPlane (a.raw_inverse_holomorphic hκ b T) + (Target.inverseFlow_injOn b T) hz + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawLifetimeFlowLimit.lean b/lean/OAI/Probability/SLE/Loewner/RawLifetimeFlowLimit.lean index 685a677cb11c0afc681541db165325aa133d114d..21d8567406667015a5ba0798c1044fbeeb28111e 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawLifetimeFlowLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawLifetimeFlowLimit.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Loewner.RawLifetimeHeightLimit -import OAI.Probability.SLE.Loewner.RawForwardCoordinates -import OAI.Probability.SLE.Loewner.VanishingHeightRealLimit - -namespace OAI - -/-! Both coordinates of the raw forward flow approach the driver at a -finite lifetime. Updating the terminal value gives a continuous path. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge.Target - -theorem flow_tendsto_driver_at_lifetime - (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) - (hs : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) - (hdead : ¬Surviving 1 b z τ) : - Tendsto (flow 1 b z) (𝓝[<] τ) (𝓝 (driverValue b τ : ℂ)) := by - have hc := flow_continuousOn_before_lifetime b z (fun t ht => hs t ht.1 ht.2) - have hy := flow_height_tendsto_zero_at_lifetime b z hz hτ hs hdead - have hd (t : ℝ) (ht : t ∈ Ico 0 τ) := - flow_coordinate_derivatives b z ht.1 - (show t < (t+τ)/2 by linarith [ht.2]) - (hs ((t+τ)/2) (by linarith [ht.1,ht.2]) (by linarith [ht.2])) - have hx := forward_real_tendsto_driver_of_height_zero hτ - (fun t => (flow 1 b z t).re) (fun t => (flow 1 b z t).im) (driverValue b) - (Complex.continuous_re.comp_continuousOn hc) - (Complex.continuous_im.comp_continuousOn hc) - (fun t ht => (hd t ht).1) (fun t ht => (hd t ht).2) - (fun t ht => surviving_flow_mem_upperHalfPlane 1 b z t ht.1 (hs t ht.1 ht.2)) - (b.continuous.comp continuous_real_toNNReal).continuousAt hy - have hh := hx.ofReal.add (hy.ofReal.mul_const I) - simpa only [ofReal_zero,zero_mul,add_zero,re_add_im] using hh - -theorem continuousOn_flow_with_terminal_driver - (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) - (hs : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) - (hdead : ¬Surviving 1 b z τ) : - ContinuousOn (Function.update (flow 1 b z) τ (driverValue b τ : ℂ)) (Icc 0 τ) := by - have heq : Icc 0 τ \ {τ} = Ico 0 τ := by - ext t - simp only [Set.mem_sdiff,mem_Icc,mem_singleton_iff,mem_Ico] - constructor - · rintro ⟨⟨ht0,htτ⟩,hne⟩ - exact ⟨ht0,lt_of_le_of_ne htτ hne⟩ - · rintro ⟨ht0,htτ⟩ - exact ⟨⟨ht0,htτ.le⟩,htτ.ne⟩ - apply continuousOn_update_iff.mpr - rw [heq] - refine ⟨flow_continuousOn_before_lifetime b z (fun t ht => hs t ht.1 ht.2),?_⟩ - intro _ - exact (flow_tendsto_driver_at_lifetime b z hz hτ hs hdead).mono_left - (nhdsWithin_mono _ Ico_subset_Iio_self) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawLifetimeHeightLimit +import OAI.Probability.SLE.Loewner.RawForwardCoordinates +import OAI.Probability.SLE.Loewner.VanishingHeightRealLimit + +namespace OAI + +/-! Both coordinates of the raw forward flow approach the driver at a +finite lifetime. Updating the terminal value gives a continuous path. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge.Target + +theorem flow_tendsto_driver_at_lifetime + (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) + (hs : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) + (hdead : ¬Surviving 1 b z τ) : + Tendsto (flow 1 b z) (𝓝[<] τ) (𝓝 (driverValue b τ : ℂ)) := by + have hc := flow_continuousOn_before_lifetime b z (fun t ht => hs t ht.1 ht.2) + have hy := flow_height_tendsto_zero_at_lifetime b z hz hτ hs hdead + have hd (t : ℝ) (ht : t ∈ Ico 0 τ) := + flow_coordinate_derivatives b z ht.1 + (show t < (t+τ)/2 by linarith [ht.2]) + (hs ((t+τ)/2) (by linarith [ht.1,ht.2]) (by linarith [ht.2])) + have hx := forward_real_tendsto_driver_of_height_zero hτ + (fun t => (flow 1 b z t).re) (fun t => (flow 1 b z t).im) (driverValue b) + (Complex.continuous_re.comp_continuousOn hc) + (Complex.continuous_im.comp_continuousOn hc) + (fun t ht => (hd t ht).1) (fun t ht => (hd t ht).2) + (fun t ht => surviving_flow_mem_upperHalfPlane 1 b z t ht.1 (hs t ht.1 ht.2)) + (b.continuous.comp continuous_real_toNNReal).continuousAt hy + have hh := hx.ofReal.add (hy.ofReal.mul_const I) + simpa only [ofReal_zero,zero_mul,add_zero,re_add_im] using hh + +theorem continuousOn_flow_with_terminal_driver + (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) + (hs : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) + (hdead : ¬Surviving 1 b z τ) : + ContinuousOn (Function.update (flow 1 b z) τ (driverValue b τ : ℂ)) (Icc 0 τ) := by + have heq : Icc 0 τ \ {τ} = Ico 0 τ := by + ext t + simp only [Set.mem_sdiff,mem_Icc,mem_singleton_iff,mem_Ico] + constructor + · rintro ⟨⟨ht0,htτ⟩,hne⟩ + exact ⟨ht0,lt_of_le_of_ne htτ hne⟩ + · rintro ⟨ht0,htτ⟩ + exact ⟨⟨ht0,htτ.le⟩,htτ.ne⟩ + apply continuousOn_update_iff.mpr + rw [heq] + refine ⟨flow_continuousOn_before_lifetime b z (fun t ht => hs t ht.1 ht.2),?_⟩ + intro _ + exact (flow_tendsto_driver_at_lifetime b z hz hτ hs hdead).mono_left + (nhdsWithin_mono _ Ico_subset_Iio_self) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawLifetimeHeightLimit.lean b/lean/OAI/Probability/SLE/Loewner/RawLifetimeHeightLimit.lean index 5c660b262d3de5decae19761efac85245335af80..34c7db58dd967df7e002bf28c37b068fadd40fd9 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawLifetimeHeightLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawLifetimeHeightLimit.lean @@ -1,59 +1,59 @@ -import OAI.Probability.SLE.Loewner.RawHeightContinuation - -namespace OAI - -/-! The height of a raw point tends to zero at its finite lifetime. -A positive limiting height would continue the exact cutoff trajectory. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge.Target - -theorem flow_height_tendsto_zero_at_lifetime - (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) - (hsurvive : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) - (hdead : ¬Surviving 1 b z τ) : - Tendsto (fun t => (flow 1 b z t).im) (𝓝[<] τ) (𝓝 0) := by - let f (t : ℝ) := (flow 1 b z t).im - have hp (t : ℝ) (ht : t ∈ Ioo 0 τ) : 0 < f t := - surviving_flow_mem_upperHalfPlane 1 b z t ht.1.le (hsurvive t ht.1.le ht.2) - have hmono : AntitoneOn f (Ioo 0 τ) := by - intro s hs t ht hst - exact flow_height_antitone_of_survival b z ht.1.le (hsurvive t ht.1.le ht.2) - ⟨hs.1.le,hst⟩ ⟨ht.1.le,le_rfl⟩ hst - have hne : (Ioo (0 : ℝ) τ).Nonempty := nonempty_Ioo.mpr hτ - have himage : (f '' Ioo 0 τ).Nonempty := hne.image f - have hbelow : BddBelow (f '' Ioo 0 τ) := by - refine ⟨0,?_⟩ - rintro y ⟨t,ht,rfl⟩ - exact (hp t ht).le - have hy : 0 ≤ sInf (f '' Ioo 0 τ) := by - apply le_csInf himage - rintro y ⟨t,ht,rfl⟩ - exact (hp t ht).le - have hmin (t : ℝ) (ht : t ∈ Ioo 0 τ) : sInf (f '' Ioo 0 τ) ≤ f t := - csInf_le hbelow ⟨t,ht,rfl⟩ - have hzero : sInf (f '' Ioo 0 τ)=0 := by - apply le_antisymm _ hy - by_contra hn - have hyp : 0 < sInf (f '' Ioo 0 τ) := lt_of_not_ge hn - apply hdead - apply survives_of_uniform_height_before b z hz hτ hyp - (fun t ht => hsurvive t ht.1 ht.2) - intro t ht - by_cases ht0 : t=0 - · subst t - have hhalf : τ/2 ∈ Ioo (0 : ℝ) τ := ⟨half_pos hτ,half_lt_self hτ⟩ - exact (hmin _ hhalf).trans - (flow_height_antitone_of_survival b z (half_pos hτ).le - (hsurvive _ (half_pos hτ).le (half_lt_self hτ)) - ⟨le_rfl,(half_pos hτ).le⟩ ⟨(half_pos hτ).le,le_rfl⟩ (half_pos hτ).le) - · exact hmin t ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0),ht.2⟩ - simpa only [hzero] using hmono.tendsto_nhdsWithin_Ioo_left hne hbelow - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawHeightContinuation + +namespace OAI + +/-! The height of a raw point tends to zero at its finite lifetime. +A positive limiting height would continue the exact cutoff trajectory. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge.Target + +theorem flow_height_tendsto_zero_at_lifetime + (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) + (hsurvive : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) + (hdead : ¬Surviving 1 b z τ) : + Tendsto (fun t => (flow 1 b z t).im) (𝓝[<] τ) (𝓝 0) := by + let f (t : ℝ) := (flow 1 b z t).im + have hp (t : ℝ) (ht : t ∈ Ioo 0 τ) : 0 < f t := + surviving_flow_mem_upperHalfPlane 1 b z t ht.1.le (hsurvive t ht.1.le ht.2) + have hmono : AntitoneOn f (Ioo 0 τ) := by + intro s hs t ht hst + exact flow_height_antitone_of_survival b z ht.1.le (hsurvive t ht.1.le ht.2) + ⟨hs.1.le,hst⟩ ⟨ht.1.le,le_rfl⟩ hst + have hne : (Ioo (0 : ℝ) τ).Nonempty := nonempty_Ioo.mpr hτ + have himage : (f '' Ioo 0 τ).Nonempty := hne.image f + have hbelow : BddBelow (f '' Ioo 0 τ) := by + refine ⟨0,?_⟩ + rintro y ⟨t,ht,rfl⟩ + exact (hp t ht).le + have hy : 0 ≤ sInf (f '' Ioo 0 τ) := by + apply le_csInf himage + rintro y ⟨t,ht,rfl⟩ + exact (hp t ht).le + have hmin (t : ℝ) (ht : t ∈ Ioo 0 τ) : sInf (f '' Ioo 0 τ) ≤ f t := + csInf_le hbelow ⟨t,ht,rfl⟩ + have hzero : sInf (f '' Ioo 0 τ)=0 := by + apply le_antisymm _ hy + by_contra hn + have hyp : 0 < sInf (f '' Ioo 0 τ) := lt_of_not_ge hn + apply hdead + apply survives_of_uniform_height_before b z hz hτ hyp + (fun t ht => hsurvive t ht.1 ht.2) + intro t ht + by_cases ht0 : t=0 + · subst t + have hhalf : τ/2 ∈ Ioo (0 : ℝ) τ := ⟨half_pos hτ,half_lt_self hτ⟩ + exact (hmin _ hhalf).trans + (flow_height_antitone_of_survival b z (half_pos hτ).le + (hsurvive _ (half_pos hτ).le (half_lt_self hτ)) + ⟨le_rfl,(half_pos hτ).le⟩ ⟨(half_pos hτ).le,le_rfl⟩ (half_pos hτ).le) + · exact hmin t ⟨lt_of_le_of_ne ht.1 (Ne.symm ht0),ht.2⟩ + simpa only [hzero] using hmono.tendsto_nhdsWithin_Ioo_left hne hbelow + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawLifetimeTrace.lean b/lean/OAI/Probability/SLE/Loewner/RawLifetimeTrace.lean index f53cacb149f786128c249fc22849299f17729a43..f17d9e45e7cb86a28eaef032f91fab6e82caf07a 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawLifetimeTrace.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawLifetimeTrace.lean @@ -1,67 +1,67 @@ -import OAI.Probability.SLE.Loewner.RawLifetimeFlowLimit -import OAI.Probability.SLE.Geometry.OpenBoundaryTrace - -namespace OAI - -/-! A point lost by the smooth raw flow is exactly the trace tip at its -finite lifetime. The reverse path is the forward path read backwards. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem eq_trace_at_lifetime - (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) - (hs : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) - (hdead : ¬Surviving 1 b z τ) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 τ, HasDerivWithinAt (driverValue b) (b' t) (Icc 0 τ) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 τ, |b' t| ≤ L) : - z=trace 1 b τ := by - let f := Function.update (flow 1 b z) τ (driverValue b τ : ℂ) - let q (t : ℝ) := f (τ-t) - have hqeq (t : ℝ) (ht : 0 < t) : q t=flow 1 b z (τ-t) := by - exact Function.update_of_ne (by linarith : τ-t ≠ τ) _ _ - have hqc : ContinuousOn q (Icc 0 τ) := by - apply (continuousOn_flow_with_terminal_driver b z hz hτ hs hdead).comp - (continuous_const.sub continuous_id).continuousOn - intro t ht - exact ⟨sub_nonneg.mpr ht.2,sub_le_self _ ht.1⟩ - have hq0 : q 0=(b (⟨τ,hτ.le⟩ : ℝ≥0) : ℂ) := by - simp only [q,f,sub_zero,Function.update_self] - change (b τ.toNNReal : ℂ) = _ - have ht : τ.toNNReal = (⟨τ,hτ.le⟩ : ℝ≥0) := by - apply Subtype.ext - exact Real.coe_toNNReal τ hτ.le - rw [ht] - have hqp (t : ℝ) (ht : t ∈ Ioc 0 τ) : 0 < (q t).im := by - rw [hqeq t ht.1] - exact surviving_flow_mem_upperHalfPlane 1 b z (τ-t) (sub_nonneg.mpr ht.2) - (hs _ (sub_nonneg.mpr ht.2) (by linarith [ht.1])) - have hqd (t : ℝ) (ht : t ∈ Ioo 0 τ) : HasDerivWithinAt q - (-2/(q t-(driverValue b (τ-t) : ℂ))) (Ici t) t := by - have hx : 0 < τ-t := sub_pos.mpr ht.2 - let S : ℝ := ((τ-t)+τ)/2 - have hxS : τ-t < S := by dsimp [S]; linarith [ht.1] - have hSτ : S < τ := by dsimp [S]; linarith [ht.1] - have hflow := (flow_hasDerivWithinAt_of_later_survival 1 b z (τ-t) S hx.le hxS - (hs S (hx.le.trans hxS.le) hSτ)).hasDerivAt (Ici_mem_nhds hx) - have hh := hflow.scomp t ((hasDerivAt_id t).const_sub τ) - have hevent : q =ᶠ[𝓝[Ici t] t] (fun s : ℝ => flow 1 b z (τ-s)) := by - filter_upwards [mem_nhdsWithin_of_mem_nhds (lt_mem_nhds ht.1)] with s hs' - exact hqeq s hs' - have hh' := hh.hasDerivWithinAt.congr_of_eventuallyEq hevent (hqeq t ht.1) - convert! hh' using 1 - rw [hqeq t ht.1] - simp [a,neg_div] - have hfinal : q τ=z := by - rw [hqeq τ hτ,sub_self,flow_zero 1 b z hz] - have htip := boundary_reverse_eq_trace_of_open_derivative b ⟨τ,hτ.le⟩ hτ b' - hd hL hb q hqc hq0 hqd hqp - exact hfinal.symm.trans htip - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawLifetimeFlowLimit +import OAI.Probability.SLE.Geometry.OpenBoundaryTrace + +namespace OAI + +/-! A point lost by the smooth raw flow is exactly the trace tip at its +finite lifetime. The reverse path is the forward path read backwards. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem eq_trace_at_lifetime + (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) {τ : ℝ} (hτ : 0 < τ) + (hs : ∀ t, 0 ≤ t → t < τ → Surviving 1 b z t) + (hdead : ¬Surviving 1 b z τ) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 τ, HasDerivWithinAt (driverValue b) (b' t) (Icc 0 τ) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 τ, |b' t| ≤ L) : + z=trace 1 b τ := by + let f := Function.update (flow 1 b z) τ (driverValue b τ : ℂ) + let q (t : ℝ) := f (τ-t) + have hqeq (t : ℝ) (ht : 0 < t) : q t=flow 1 b z (τ-t) := by + exact Function.update_of_ne (by linarith : τ-t ≠ τ) _ _ + have hqc : ContinuousOn q (Icc 0 τ) := by + apply (continuousOn_flow_with_terminal_driver b z hz hτ hs hdead).comp + (continuous_const.sub continuous_id).continuousOn + intro t ht + exact ⟨sub_nonneg.mpr ht.2,sub_le_self _ ht.1⟩ + have hq0 : q 0=(b (⟨τ,hτ.le⟩ : ℝ≥0) : ℂ) := by + simp only [q,f,sub_zero,Function.update_self] + change (b τ.toNNReal : ℂ) = _ + have ht : τ.toNNReal = (⟨τ,hτ.le⟩ : ℝ≥0) := by + apply Subtype.ext + exact Real.coe_toNNReal τ hτ.le + rw [ht] + have hqp (t : ℝ) (ht : t ∈ Ioc 0 τ) : 0 < (q t).im := by + rw [hqeq t ht.1] + exact surviving_flow_mem_upperHalfPlane 1 b z (τ-t) (sub_nonneg.mpr ht.2) + (hs _ (sub_nonneg.mpr ht.2) (by linarith [ht.1])) + have hqd (t : ℝ) (ht : t ∈ Ioo 0 τ) : HasDerivWithinAt q + (-2/(q t-(driverValue b (τ-t) : ℂ))) (Ici t) t := by + have hx : 0 < τ-t := sub_pos.mpr ht.2 + let S : ℝ := ((τ-t)+τ)/2 + have hxS : τ-t < S := by dsimp [S]; linarith [ht.1] + have hSτ : S < τ := by dsimp [S]; linarith [ht.1] + have hflow := (flow_hasDerivWithinAt_of_later_survival 1 b z (τ-t) S hx.le hxS + (hs S (hx.le.trans hxS.le) hSτ)).hasDerivAt (Ici_mem_nhds hx) + have hh := hflow.scomp t ((hasDerivAt_id t).const_sub τ) + have hevent : q =ᶠ[𝓝[Ici t] t] (fun s : ℝ => flow 1 b z (τ-s)) := by + filter_upwards [mem_nhdsWithin_of_mem_nhds (lt_mem_nhds ht.1)] with s hs' + exact hqeq s hs' + have hh' := hh.hasDerivWithinAt.congr_of_eventuallyEq hevent (hqeq t ht.1) + convert! hh' using 1 + rw [hqeq t ht.1] + simp [a,neg_div] + have hfinal : q τ=z := by + rw [hqeq τ hτ,sub_self,flow_zero 1 b z hz] + have htip := boundary_reverse_eq_trace_of_open_derivative b ⟨τ,hτ.le⟩ hτ b' + hd hL hb q hqc hq0 hqd hqp + exact hfinal.symm.trans htip + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawRadialTipBoundary.lean b/lean/OAI/Probability/SLE/Loewner/RawRadialTipBoundary.lean index 0437986adaf377f75eb9231e152c44e91ac78224..cc336c75672aded52cdcceb6f3144e5abd01f112 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawRadialTipBoundary.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawRadialTipBoundary.lean @@ -1,44 +1,44 @@ -import OAI.Probability.SLE.Loewner.RawSurvivingAnalytic - -namespace OAI - -/-! A radial inverse limit lies outside the open raw survival domain. -This uses only the stated limit, with no regularity of the driver beyond -its defining continuity. -/ -noncomputable section -open Set Filter Complex _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem RegularSLEPresentation.raw_radial_tip_not_surviving - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) - (b : Target.Driver) (T : ℝ≥0) (z : ℂ) - (hl : Tendsto (fun y : ℝ => Target.inverseFlow 1 b T ((b T:ℂ)+(y:ℂ)*I)) - (𝓝[>] 0) (𝓝 z)) : ¬Target.Surviving 1 b z T := by - intro hs - have hc := ((a.raw_forward_holomorphic hκ b T).differentiableAt - ((a.raw_surviving_isOpen hκ b T).mem_nhds hs)).continuousAt - have ht := hc.tendsto.comp hl - have he : (fun y : ℝ => Target.flow 1 b - (Target.inverseFlow 1 b T ((b T:ℂ)+(y:ℂ)*I)) T) - =ᶠ[𝓝[>] (0:ℝ)] (fun y => (b T:ℂ)+(y:ℂ)*I) := by - filter_upwards [self_mem_nhdsWithin] with y hy - exact Target.flow_inverseFlow 1 (by norm_num) b T _ - (by simpa [upperHalfPlane] using hy) - have hv : Tendsto (fun y : ℝ => (b T:ℂ)+(y:ℂ)*I) - (𝓝[>] 0) (𝓝 (b T:ℂ)) := by - have h : Continuous (fun y : ℝ => (b T:ℂ)+(y:ℂ)*I) := by fun_prop - simpa only [ofReal_zero,zero_mul,add_zero] using - (h.tendsto 0).mono_left nhdsWithin_le_nhds - have hval := tendsto_nhds_unique ht (hv.congr' he.symm) - have hp := Target.surviving_flow_mem_upperHalfPlane 1 b z T T.coe_nonneg hs - rw [hval] at hp - simp [upperHalfPlane] at hp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawSurvivingAnalytic + +namespace OAI + +/-! A radial inverse limit lies outside the open raw survival domain. +This uses only the stated limit, with no regularity of the driver beyond +its defining continuity. -/ +noncomputable section +open Set Filter Complex _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem RegularSLEPresentation.raw_radial_tip_not_surviving + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) + (b : Target.Driver) (T : ℝ≥0) (z : ℂ) + (hl : Tendsto (fun y : ℝ => Target.inverseFlow 1 b T ((b T:ℂ)+(y:ℂ)*I)) + (𝓝[>] 0) (𝓝 z)) : ¬Target.Surviving 1 b z T := by + intro hs + have hc := ((a.raw_forward_holomorphic hκ b T).differentiableAt + ((a.raw_surviving_isOpen hκ b T).mem_nhds hs)).continuousAt + have ht := hc.tendsto.comp hl + have he : (fun y : ℝ => Target.flow 1 b + (Target.inverseFlow 1 b T ((b T:ℂ)+(y:ℂ)*I)) T) + =ᶠ[𝓝[>] (0:ℝ)] (fun y => (b T:ℂ)+(y:ℂ)*I) := by + filter_upwards [self_mem_nhdsWithin] with y hy + exact Target.flow_inverseFlow 1 (by norm_num) b T _ + (by simpa [upperHalfPlane] using hy) + have hv : Tendsto (fun y : ℝ => (b T:ℂ)+(y:ℂ)*I) + (𝓝[>] 0) (𝓝 (b T:ℂ)) := by + have h : Continuous (fun y : ℝ => (b T:ℂ)+(y:ℂ)*I) := by fun_prop + simpa only [ofReal_zero,zero_mul,add_zero] using + (h.tendsto 0).mono_left nhdsWithin_le_nhds + have hval := tendsto_nhds_unique ht (hv.congr' he.symm) + have hp := Target.surviving_flow_mem_upperHalfPlane 1 b z T T.coe_nonneg hs + rw [hval] at hp + simp [upperHalfPlane] at hp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawRestartAdaptation.lean b/lean/OAI/Probability/SLE/Loewner/RawRestartAdaptation.lean index 9c0f8b4e3d3075db1c4ed437faf2ab0ccbe1c3e8..f453ec4af418a7cbcb9996c59100af92283b4323 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawRestartAdaptation.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawRestartAdaptation.lean @@ -1,123 +1,123 @@ -import OAI.Probability.SLE.Mass.RawMassPathLaw -import OAI.Probability.SLE.Stochastic.ShiftedStoppingFiltration - -namespace OAI - -/-! The fixed raw mass applied to the fresh driver is progressive in the -original filtration shifted by the stopping time. Its continuous increasing -paths are transferred on the explicit measurable good set. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology CompactlySupported NNReal ENNReal - -namespace SLEExactGauge - -local instance rawRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma driverPath_measurable_past_of_adapted - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (Y : Ω → C(ℝ≥0,ℝ)) (hY : Adapted ℱ (fun t ω => Y ω t)) (t : ℝ≥0) : - Measurable[ℱ t,driverPathFiltration t] Y := by - rw [driverPathFiltration,Filtration.natural_eq_comap] - let : MeasurableSpace Ω := ℱ t - apply measurable_comap_iff.mpr - apply measurable_pi_iff.mpr - intro s - exact (hY s).mono (ℱ.mono s.property) le_rfl - -theorem raw_process_comp_progressive - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (Y : Ω → C(ℝ≥0,ℝ)) (hY : Adapted ℱ (fun t ω => Y ω t)) - (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) - (hR : IsProgressive driverPathFiltration R) : - IsProgressive ℱ (fun t ω => R t (Y ω)) := by - intro t - exact (hR t).comp (measurable_fst.prodMk - ((driverPath_measurable_past_of_adapted ℱ Y hY t).comp measurable_snd)) - -lemma sourceRestartDriverPath_shifted_adapted - {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) : - Adapted (shiftedStoppingFiltration hτ) - (fun t ω => sourceRestartDriverPath B hc τ ω t) := by - have hX := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc - have hs := shiftedStopping_adapted hτ B hX - have h0 : StronglyMeasurable[(shiftedStoppingFiltration hτ) 0] - (fun ω => B (τ ω) ω) := by - simpa only [add_zero] using hs 0 - intro t - have hd := ((hs t).sub (h0.mono ((shiftedStoppingFiltration hτ).mono (show 0 ≤ t from zero_le)))).measurable - exact hd - -theorem sourceRestart_raw_process_progressive - {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) - (hR : IsProgressive driverPathFiltration R) : - IsProgressive (shiftedStoppingFiltration hτ) - (fun t ω => R t (sourceRestartDriverPath B hc τ ω)) := - raw_process_comp_progressive _ _ (sourceRestartDriverPath_shifted_adapted B hm hc τ hτ) R hR - -theorem sourceRestart_mem_full_driver_set - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) - (hfull : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E) : - ∀ᵐ ω ∂P, sourceRestartDriverPath B hc τ ω ∈ E := by - have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _ - (fun t => (sourceRestartDriverPath_shifted_adapted B hm hc τ hτ).measurable (i := t)) - have hlaw : P.map (sourceRestartDriverPath B hc τ)=P.map (sourceDriverPath B hc) := by - simpa only [Measure.restrict_univ,measure_univ,one_smul] using - source_restart_continuous_driver_law hB hm hc τ hτ (MeasurableSet.univ : MeasurableSet[hτ.measurableSpace] univ) - apply (ae_map_iff hY.aemeasurable hE).mp - rwa [hlaw] - -namespace Multipoint - -theorem CanonicalMass.exists_raw_with_restarted_paths - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hBc : ∀ ω, Continuous (fun t => B t ω)) : - ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane, - IsProgressive driverPathFiltration R ∧ - (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧ - (∀ t p r, (∀ s ≤ t, p s=r s) → R t p=R t r) ∧ - (∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω)=C.process t ω) ∧ - ∀ (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))), - IsProgressive (shiftedStoppingFiltration hτ) - (fun t ω => R t (sourceRestartDriverPath B hBc τ ω)) ∧ - ∀ᵐ ω ∂P, R 0 (sourceRestartDriverPath B hBc τ ω)=0 ∧ - Monotone (fun t => R t (sourceRestartDriverPath B hBc τ ω)) ∧ - ∀ f : C_c(upperHalfPlane,ℝ), - Continuous (fun t => ∫ z, f z ∂(R t (sourceRestartDriverPath B hBc τ ω))) := by - obtain ⟨R,E,hRp,hRr,hRn,hE,hfull,hgood,he⟩ := C.exists_raw_nonanticipating_with_good_set hBc - refine ⟨R,hRp,hRr,hRn,he,?_⟩ - intro τ hτ - have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - refine ⟨sourceRestart_raw_process_progressive B hm hBc τ hτ R hRp,?_⟩ - filter_upwards [sourceRestart_mem_full_driver_set hB hm hBc τ hτ E hE hfull] with ω hω - exact hgood _ hω - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RawMassPathLaw +import OAI.Probability.SLE.Stochastic.ShiftedStoppingFiltration + +namespace OAI + +/-! The fixed raw mass applied to the fresh driver is progressive in the +original filtration shifted by the stopping time. Its continuous increasing +paths are transferred on the explicit measurable good set. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology CompactlySupported NNReal ENNReal + +namespace SLEExactGauge + +local instance rawRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma driverPath_measurable_past_of_adapted + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (Y : Ω → C(ℝ≥0,ℝ)) (hY : Adapted ℱ (fun t ω => Y ω t)) (t : ℝ≥0) : + Measurable[ℱ t,driverPathFiltration t] Y := by + rw [driverPathFiltration,Filtration.natural_eq_comap] + let : MeasurableSpace Ω := ℱ t + apply measurable_comap_iff.mpr + apply measurable_pi_iff.mpr + intro s + exact (hY s).mono (ℱ.mono s.property) le_rfl + +theorem raw_process_comp_progressive + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (Y : Ω → C(ℝ≥0,ℝ)) (hY : Adapted ℱ (fun t ω => Y ω t)) + (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) + (hR : IsProgressive driverPathFiltration R) : + IsProgressive ℱ (fun t ω => R t (Y ω)) := by + intro t + exact (hR t).comp (measurable_fst.prodMk + ((driverPath_measurable_past_of_adapted ℱ Y hY t).comp measurable_snd)) + +lemma sourceRestartDriverPath_shifted_adapted + {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) : + Adapted (shiftedStoppingFiltration hτ) + (fun t ω => sourceRestartDriverPath B hc τ ω t) := by + have hX := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc + have hs := shiftedStopping_adapted hτ B hX + have h0 : StronglyMeasurable[(shiftedStoppingFiltration hτ) 0] + (fun ω => B (τ ω) ω) := by + simpa only [add_zero] using hs 0 + intro t + have hd := ((hs t).sub (h0.mono ((shiftedStoppingFiltration hτ).mono (show 0 ≤ t from zero_le)))).measurable + exact hd + +theorem sourceRestart_raw_process_progressive + {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) + (hR : IsProgressive driverPathFiltration R) : + IsProgressive (shiftedStoppingFiltration hτ) + (fun t ω => R t (sourceRestartDriverPath B hc τ ω)) := + raw_process_comp_progressive _ _ (sourceRestartDriverPath_shifted_adapted B hm hc τ hτ) R hR + +theorem sourceRestart_mem_full_driver_set + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) + (hfull : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E) : + ∀ᵐ ω ∂P, sourceRestartDriverPath B hc τ ω ∈ E := by + have hY : Measurable (sourceRestartDriverPath B hc τ) := source_path_measurable _ + (fun t => (sourceRestartDriverPath_shifted_adapted B hm hc τ hτ).measurable (i := t)) + have hlaw : P.map (sourceRestartDriverPath B hc τ)=P.map (sourceDriverPath B hc) := by + simpa only [Measure.restrict_univ,measure_univ,one_smul] using + source_restart_continuous_driver_law hB hm hc τ hτ (MeasurableSet.univ : MeasurableSet[hτ.measurableSpace] univ) + apply (ae_map_iff hY.aemeasurable hE).mp + rwa [hlaw] + +namespace Multipoint + +theorem CanonicalMass.exists_raw_with_restarted_paths + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hBc : ∀ ω, Continuous (fun t => B t ω)) : + ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane, + IsProgressive driverPathFiltration R ∧ + (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧ + (∀ t p r, (∀ s ≤ t, p s=r s) → R t p=R t r) ∧ + (∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω)=C.process t ω) ∧ + ∀ (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))), + IsProgressive (shiftedStoppingFiltration hτ) + (fun t ω => R t (sourceRestartDriverPath B hBc τ ω)) ∧ + ∀ᵐ ω ∂P, R 0 (sourceRestartDriverPath B hBc τ ω)=0 ∧ + Monotone (fun t => R t (sourceRestartDriverPath B hBc τ ω)) ∧ + ∀ f : C_c(upperHalfPlane,ℝ), + Continuous (fun t => ∫ z, f z ∂(R t (sourceRestartDriverPath B hBc τ ω))) := by + obtain ⟨R,E,hRp,hRr,hRn,hE,hfull,hgood,he⟩ := C.exists_raw_nonanticipating_with_good_set hBc + refine ⟨R,hRp,hRr,hRn,he,?_⟩ + intro τ hτ + have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + refine ⟨sourceRestart_raw_process_progressive B hm hBc τ hτ R hRp,?_⟩ + filter_upwards [sourceRestart_mem_full_driver_set hB hm hBc τ hτ E hE hfull] with ω hω + exact hgood _ hω + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawRestartCanonical.lean b/lean/OAI/Probability/SLE/Loewner/RawRestartCanonical.lean index 1fb026d5365672df862fbf86b1b5330bd192c439..0ce98b7f7fb1fbaeb3d04b30eb270564f752d349 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawRestartCanonical.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawRestartCanonical.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.Loewner.RawRestartAdaptation -import OAI.Probability.SLE.Loewner.RawTerminalExhaustion - -namespace OAI - -/-! One fixed raw canonical mass works at every finite Brownian restart. -The finite process is progressive in the shifted past and locally exhausts -its own terminal measure on one full event for each restart. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology CompactlySupported NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -local instance rawRestartCanonicalMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawRestartCanonicalBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem CanonicalMass.exists_raw_restarted_canonical - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hBc : ∀ ω, Continuous (fun t => B t ω)) : - ∃ R : RawCanonicalMass, - (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hBc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hBc ω)=C.rational.terminal ω) ∧ - ∀ (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))), - IsProgressive (shiftedStoppingFiltration hτ) - (fun t ω => R.process t (sourceRestartDriverPath B hBc τ ω)) ∧ - ∀ᵐ ω ∂P, - R.process 0 (sourceRestartDriverPath B hBc τ ω)=0 ∧ - Monotone (fun t => R.process t (sourceRestartDriverPath B hBc τ ω)) ∧ - (∀ f : C_c(upperHalfPlane,ℝ), Continuous - (fun t => ∫ z, f z ∂(R.process t (sourceRestartDriverPath B hBc τ ω)))) ∧ - (∀ t, R.process t (sourceRestartDriverPath B hBc τ ω) ≤ - R.terminal (sourceRestartDriverPath B hBc τ ω)) ∧ - ∀ K : Set upperHalfPlane, IsCompact K → Tendsto - (fun t : ℝ≥0 => (R.terminal (sourceRestartDriverPath B hBc τ ω)- - R.process t (sourceRestartDriverPath B hBc τ ω)) K) atTop (𝓝 0) := by - obtain ⟨N,hNp,hNr,hNn,he,hfresh⟩ := C.exists_raw_with_restarted_paths hB hBc - let R : RawCanonicalMass := ⟨N,rawTerminalMass N,hNp,hNr, - rawTerminalMass_measurable N hNp hNr,rawTerminalMass_radon N,hNn⟩ - have hterm := C.rawTerminal_realizes hBc N he - obtain ⟨hE,hEf,hEg⟩ := C.raw_terminal_good_set hBc N hNp hNr he - refine ⟨R,?_,?_⟩ - · filter_upwards [he,hterm] with ω hf ht - exact ⟨hf,ht⟩ - · intro τ hτ - obtain ⟨hprog,hpath⟩ := hfresh τ hτ - refine ⟨hprog,?_⟩ - have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - have hset := sourceRestart_mem_full_driver_set hB hm hBc τ hτ _ hE hEf - filter_upwards [hpath,hset] with ω hp hs - obtain ⟨hdom,hex⟩ := hEg _ hs hp.2.1 - exact ⟨hp.1,hp.2.1,hp.2.2,hdom,hex⟩ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawRestartAdaptation +import OAI.Probability.SLE.Loewner.RawTerminalExhaustion + +namespace OAI + +/-! One fixed raw canonical mass works at every finite Brownian restart. +The finite process is progressive in the shifted past and locally exhausts +its own terminal measure on one full event for each restart. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology CompactlySupported NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +local instance rawRestartCanonicalMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawRestartCanonicalBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem CanonicalMass.exists_raw_restarted_canonical + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hBc : ∀ ω, Continuous (fun t => B t ω)) : + ∃ R : RawCanonicalMass, + (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hBc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hBc ω)=C.rational.terminal ω) ∧ + ∀ (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))), + IsProgressive (shiftedStoppingFiltration hτ) + (fun t ω => R.process t (sourceRestartDriverPath B hBc τ ω)) ∧ + ∀ᵐ ω ∂P, + R.process 0 (sourceRestartDriverPath B hBc τ ω)=0 ∧ + Monotone (fun t => R.process t (sourceRestartDriverPath B hBc τ ω)) ∧ + (∀ f : C_c(upperHalfPlane,ℝ), Continuous + (fun t => ∫ z, f z ∂(R.process t (sourceRestartDriverPath B hBc τ ω)))) ∧ + (∀ t, R.process t (sourceRestartDriverPath B hBc τ ω) ≤ + R.terminal (sourceRestartDriverPath B hBc τ ω)) ∧ + ∀ K : Set upperHalfPlane, IsCompact K → Tendsto + (fun t : ℝ≥0 => (R.terminal (sourceRestartDriverPath B hBc τ ω)- + R.process t (sourceRestartDriverPath B hBc τ ω)) K) atTop (𝓝 0) := by + obtain ⟨N,hNp,hNr,hNn,he,hfresh⟩ := C.exists_raw_with_restarted_paths hB hBc + let R : RawCanonicalMass := ⟨N,rawTerminalMass N,hNp,hNr, + rawTerminalMass_measurable N hNp hNr,rawTerminalMass_radon N,hNn⟩ + have hterm := C.rawTerminal_realizes hBc N he + obtain ⟨hE,hEf,hEg⟩ := C.raw_terminal_good_set hBc N hNp hNr he + refine ⟨R,?_,?_⟩ + · filter_upwards [he,hterm] with ω hf ht + exact ⟨hf,ht⟩ + · intro τ hτ + obtain ⟨hprog,hpath⟩ := hfresh τ hτ + refine ⟨hprog,?_⟩ + have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + have hset := sourceRestart_mem_full_driver_set hB hm hBc τ hτ _ hE hEf + filter_upwards [hpath,hset] with ω hp hs + obtain ⟨hdom,hex⟩ := hEg _ hs hp.2.1 + exact ⟨hp.1,hp.2.1,hp.2.2,hdom,hex⟩ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawRestartGreenKernel.lean b/lean/OAI/Probability/SLE/Loewner/RawRestartGreenKernel.lean index 696ae6515ecf81e582f3370f327f175914be38b1..a1d54c6b34c91b2ed6602e8aaec9df615818361a 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawRestartGreenKernel.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawRestartGreenKernel.lean @@ -1,108 +1,108 @@ -import OAI.Probability.SLE.Geometry.CanonicalGreenPast -import OAI.Probability.SLE.Geometry.CanonicalRawTerminal -import OAI.Probability.SLE.Geometry.RestartContinuationKernel - -namespace OAI - -/-! The canonical mass's Green decomposition transfers to the full -original past at every time after a finite restart. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance rawGreenKernelDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawGreenKernelDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma bind_kernel_add {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (N M : Ω → Measure upperHalfPlane) (hN : Measurable N) (hM : Measurable M) : - P.bind (fun ω => N ω+M ω)=P.bind N+P.bind M := by - have hsum : Measurable (fun ω => N ω+M ω) := hN.add hM - apply Measure.ext - intro A hA - rw [Measure.bind_apply hA hsum.aemeasurable,Measure.add_apply, - Measure.bind_apply hA hN.aemeasurable,Measure.bind_apply hA hM.aemeasurable] - simp only [Measure.add_apply] - exact lintegral_add_left ((Measure.measurable_coe hA).comp hN) _ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - -include C hc F hw R he in -lemma CanonicalMass.raw_green_kernel (t : ℝ≥0) (A : Set Ω) - (hA : MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] A) : - (P.restrict A).bind (fun ω => R.terminal (sourceDriverPath B hc ω)) = - (P.restrict A).bind (fun ω => R.process t (sourceDriverPath B hc ω)+ - canonicalGreenMass κ t (sourceDriverPath B hc ω)) := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hG (ω : Ω) : canonicalGreenMass κ t (sourceDriverPath B hc ω)= - halfPlaneGreenMeasure κ B γ G t ω := by - simpa only [sourceDriverPath,ContinuousMap.coe_mk,halfPlaneGreenMeasure] using - canonicalGreenMass_eq κ (sourceDriverPath B hc ω) (hw ω) t - have hleft : (P.restrict A).bind (fun ω => R.terminal (sourceDriverPath B hc ω)) = - (P.restrict A).bind C.rational.terminal := - Measure.bind_congr_right (ae_restrict_of_ae (he.mono fun _ h => h.2)) - have hright : (P.restrict A).bind (fun ω => R.process t (sourceDriverPath B hc ω)+ - canonicalGreenMass κ t (sourceDriverPath B hc ω)) = - (P.restrict A).bind (fun ω => C.process t ω+halfPlaneGreenMeasure κ B γ G t ω) := by - apply Measure.bind_congr_right - filter_upwards [ae_restrict_of_ae he] with ω hω - rw [hω.1 t,hG ω] - have hs := isStoppingTime_const (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t - have hAs : MeasurableSet[hs.measurableSpace] A := by - have hspace : hs.measurableSpace = (Filtration.natural B - (fun s => (hm s).stronglyMeasurable)) t := - IsStoppingTime.measurableSpace_const _ t - rw [hspace] - exact hA - have h := C.kernel (fun _ => (t:WithTop ℝ≥0)) hs A hAs - change (P.restrict A).bind C.rational.terminal = - (P.restrict A).bind (C.process t) + - (P.restrict A).bind (halfPlaneGreenMeasure κ B γ G t) at h - calc - _ = (P.restrict A).bind C.rational.terminal := hleft - _ = (P.restrict A).bind (C.process t) + - (P.restrict A).bind (halfPlaneGreenMeasure κ B γ G t) := h - _ = (P.restrict A).bind (fun ω => C.process t ω+halfPlaneGreenMeasure κ B γ G t ω) := - (bind_kernel_add _ _ _ (C.progressive.adapted.measurable (i := t)) - (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t)).symm - _ = _ := hright.symm - -include C hc F hw R he in -theorem CanonicalMass.raw_restarted_green_kernel [IsProbabilityMeasure P] - (hB : IsBrownianReal B P) (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : - (P.restrict E).bind (fun ω => R.terminal (sourceRestartDriverPath B hc τ ω)) = - (P.restrict E).bind (fun ω => R.process t (sourceRestartDriverPath B hc τ ω))+ - (P.restrict E).bind (fun ω => canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hK : Measurable[driverPathFiltration t] - (fun p => R.process t p+canonicalGreenMass κ t p) := - (R.progressive.adapted t).add (canonicalGreenMass_measurable_past κ t) - have h := source_restart_bind_of_original_kernel hB hm hc t R.terminal - (fun p => R.process t p+canonicalGreenMass κ t p) R.terminal_measurable hK - (fun A hA => C.raw_green_kernel hc F hw R he t A hA) τ hτ E hE - rw [h] - have hY := sourceRestartDriverPath_measurable B hm hc τ hτ - exact bind_kernel_add _ _ _ ((R.progressive.adapted.measurable (i := t)).comp hY) - ((canonicalGreenMass_measurable κ t).comp hY) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalGreenPast +import OAI.Probability.SLE.Geometry.CanonicalRawTerminal +import OAI.Probability.SLE.Geometry.RestartContinuationKernel + +namespace OAI + +/-! The canonical mass's Green decomposition transfers to the full +original past at every time after a finite restart. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance rawGreenKernelDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawGreenKernelDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma bind_kernel_add {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (N M : Ω → Measure upperHalfPlane) (hN : Measurable N) (hM : Measurable M) : + P.bind (fun ω => N ω+M ω)=P.bind N+P.bind M := by + have hsum : Measurable (fun ω => N ω+M ω) := hN.add hM + apply Measure.ext + intro A hA + rw [Measure.bind_apply hA hsum.aemeasurable,Measure.add_apply, + Measure.bind_apply hA hN.aemeasurable,Measure.bind_apply hA hM.aemeasurable] + simp only [Measure.add_apply] + exact lintegral_add_left ((Measure.measurable_coe hA).comp hN) _ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + +include C hc F hw R he in +lemma CanonicalMass.raw_green_kernel (t : ℝ≥0) (A : Set Ω) + (hA : MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] A) : + (P.restrict A).bind (fun ω => R.terminal (sourceDriverPath B hc ω)) = + (P.restrict A).bind (fun ω => R.process t (sourceDriverPath B hc ω)+ + canonicalGreenMass κ t (sourceDriverPath B hc ω)) := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hG (ω : Ω) : canonicalGreenMass κ t (sourceDriverPath B hc ω)= + halfPlaneGreenMeasure κ B γ G t ω := by + simpa only [sourceDriverPath,ContinuousMap.coe_mk,halfPlaneGreenMeasure] using + canonicalGreenMass_eq κ (sourceDriverPath B hc ω) (hw ω) t + have hleft : (P.restrict A).bind (fun ω => R.terminal (sourceDriverPath B hc ω)) = + (P.restrict A).bind C.rational.terminal := + Measure.bind_congr_right (ae_restrict_of_ae (he.mono fun _ h => h.2)) + have hright : (P.restrict A).bind (fun ω => R.process t (sourceDriverPath B hc ω)+ + canonicalGreenMass κ t (sourceDriverPath B hc ω)) = + (P.restrict A).bind (fun ω => C.process t ω+halfPlaneGreenMeasure κ B γ G t ω) := by + apply Measure.bind_congr_right + filter_upwards [ae_restrict_of_ae he] with ω hω + rw [hω.1 t,hG ω] + have hs := isStoppingTime_const (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) t + have hAs : MeasurableSet[hs.measurableSpace] A := by + have hspace : hs.measurableSpace = (Filtration.natural B + (fun s => (hm s).stronglyMeasurable)) t := + IsStoppingTime.measurableSpace_const _ t + rw [hspace] + exact hA + have h := C.kernel (fun _ => (t:WithTop ℝ≥0)) hs A hAs + change (P.restrict A).bind C.rational.terminal = + (P.restrict A).bind (C.process t) + + (P.restrict A).bind (halfPlaneGreenMeasure κ B γ G t) at h + calc + _ = (P.restrict A).bind C.rational.terminal := hleft + _ = (P.restrict A).bind (C.process t) + + (P.restrict A).bind (halfPlaneGreenMeasure κ B γ G t) := h + _ = (P.restrict A).bind (fun ω => C.process t ω+halfPlaneGreenMeasure κ B γ G t ω) := + (bind_kernel_add _ _ _ (C.progressive.adapted.measurable (i := t)) + (halfPlaneGreenMeasure_measurable hBm hc κ γ G F hw t)).symm + _ = _ := hright.symm + +include C hc F hw R he in +theorem CanonicalMass.raw_restarted_green_kernel [IsProbabilityMeasure P] + (hB : IsBrownianReal B P) (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : + (P.restrict E).bind (fun ω => R.terminal (sourceRestartDriverPath B hc τ ω)) = + (P.restrict E).bind (fun ω => R.process t (sourceRestartDriverPath B hc τ ω))+ + (P.restrict E).bind (fun ω => canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hK : Measurable[driverPathFiltration t] + (fun p => R.process t p+canonicalGreenMass κ t p) := + (R.progressive.adapted t).add (canonicalGreenMass_measurable_past κ t) + have h := source_restart_bind_of_original_kernel hB hm hc t R.terminal + (fun p => R.process t p+canonicalGreenMass κ t p) R.terminal_measurable hK + (fun A hA => C.raw_green_kernel hc F hw R he t A hA) τ hτ E hE + rw [h] + have hY := sourceRestartDriverPath_measurable B hm hc τ hτ + exact bind_kernel_add _ _ _ ((R.progressive.adapted.measurable (i := t)).comp hY) + ((canonicalGreenMass_measurable κ t).comp hY) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawRestartRandomTests.lean b/lean/OAI/Probability/SLE/Loewner/RawRestartRandomTests.lean index 2256e9509aaeb9323a24b4d053499976fa8fa7b5..8f0f2f7d9dd15043c91ab8e82cf428f5faa8cad8 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawRestartRandomTests.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawRestartRandomTests.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Loewner.RawRestartGreenKernel -import OAI.Probability.SLE.Geometry.CanonicalRawMean -import OAI.Probability.SLE.Mass.ConditionalRadonRandomTests - -namespace OAI - -/-! The restarted Green kernel accepts arbitrary random spatial tests -measurable in the full original past at the later time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance rawRestartTestsDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawRestartTestsDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - -include C hB hc F hw R he τ hτ in -theorem CanonicalMass.raw_restarted_random_green_test - (t : ℝ≥0) (f : Ω × upperHalfPlane → ℝ≥0∞) - (hf : Measurable[(shiftedStoppingFiltration hτ t).prod inferInstance] f) : - (∫⁻ ω, ∫⁻ z, f (ω,z) ∂(R.terminal (sourceRestartDriverPath B hc τ ω)) ∂P) = - ∫⁻ ω, ∫⁻ z, f (ω,z) ∂(R.process t (sourceRestartDriverPath B hc τ ω)+ - canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) ∂P := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hY := sourceRestartDriverPath_measurable B hm hc τ hτ - let μ (ω : Ω) := R.terminal (sourceRestartDriverPath B hc τ ω) - let ν (ω : Ω) := R.process t (sourceRestartDriverPath B hc τ ω)+ - canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω) - have hμ : Measurable μ := R.terminal_measurable.comp hY - have hR : Measurable (fun ω => R.process t (sourceRestartDriverPath B hc τ ω)) := - (R.progressive.adapted.measurable (i := t)).comp hY - have hG : Measurable (fun ω => canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) := - (canonicalGreenMass_measurable κ t).comp hY - have hν : Measurable ν := hR.add hG - have hmean : P.bind μ=initialGreenMeasure κ := - C.restarted_raw_terminal_mean hB hc R (he.mono fun _ h => h.2) τ hτ - have hevent (A : Set Ω) (hA : MeasurableSet[shiftedStoppingFiltration hτ t] A) : - (P.restrict A).bind μ=(P.restrict A).bind ν := by - have h := C.raw_restarted_green_kernel hc F hw R he hB τ hτ t A hA - exact h.trans (bind_kernel_add _ _ _ hR hG).symm - have htotal : P.bind μ=P.bind ν := by - simpa only [Measure.restrict_univ] using hevent univ MeasurableSet.univ - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hμmean (K : Set upperHalfPlane) (hK : IsCompact K) : P.bind μ K < ∞ := by - rw [hmean] - exact hK.measure_lt_top - have hνmean (K : Set upperHalfPlane) (hK : IsCompact K) : P.bind ν K < ∞ := by - rw [← htotal] - exact hμmean K hK - exact conditional_halfPlane_random_test (shiftedStoppingFiltration hτ t) - ((shiftedStoppingFiltration hτ).le t) P μ ν hμ hν - (ae_of_all _ (fun ω => (R.terminal_radon _).2)) - (ae_finite_on_compacts_of_compact_mean P ν hν hνmean) hμmean hevent f hf - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawRestartGreenKernel +import OAI.Probability.SLE.Geometry.CanonicalRawMean +import OAI.Probability.SLE.Mass.ConditionalRadonRandomTests + +namespace OAI + +/-! The restarted Green kernel accepts arbitrary random spatial tests +measurable in the full original past at the later time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance rawRestartTestsDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawRestartTestsDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + +include C hB hc F hw R he τ hτ in +theorem CanonicalMass.raw_restarted_random_green_test + (t : ℝ≥0) (f : Ω × upperHalfPlane → ℝ≥0∞) + (hf : Measurable[(shiftedStoppingFiltration hτ t).prod inferInstance] f) : + (∫⁻ ω, ∫⁻ z, f (ω,z) ∂(R.terminal (sourceRestartDriverPath B hc τ ω)) ∂P) = + ∫⁻ ω, ∫⁻ z, f (ω,z) ∂(R.process t (sourceRestartDriverPath B hc τ ω)+ + canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) ∂P := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hY := sourceRestartDriverPath_measurable B hm hc τ hτ + let μ (ω : Ω) := R.terminal (sourceRestartDriverPath B hc τ ω) + let ν (ω : Ω) := R.process t (sourceRestartDriverPath B hc τ ω)+ + canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω) + have hμ : Measurable μ := R.terminal_measurable.comp hY + have hR : Measurable (fun ω => R.process t (sourceRestartDriverPath B hc τ ω)) := + (R.progressive.adapted.measurable (i := t)).comp hY + have hG : Measurable (fun ω => canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) := + (canonicalGreenMass_measurable κ t).comp hY + have hν : Measurable ν := hR.add hG + have hmean : P.bind μ=initialGreenMeasure κ := + C.restarted_raw_terminal_mean hB hc R (he.mono fun _ h => h.2) τ hτ + have hevent (A : Set Ω) (hA : MeasurableSet[shiftedStoppingFiltration hτ t] A) : + (P.restrict A).bind μ=(P.restrict A).bind ν := by + have h := C.raw_restarted_green_kernel hc F hw R he hB τ hτ t A hA + exact h.trans (bind_kernel_add _ _ _ hR hG).symm + have htotal : P.bind μ=P.bind ν := by + simpa only [Measure.restrict_univ] using hevent univ MeasurableSet.univ + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hμmean (K : Set upperHalfPlane) (hK : IsCompact K) : P.bind μ K < ∞ := by + rw [hmean] + exact hK.measure_lt_top + have hνmean (K : Set upperHalfPlane) (hK : IsCompact K) : P.bind ν K < ∞ := by + rw [← htotal] + exact hμmean K hK + exact conditional_halfPlane_random_test (shiftedStoppingFiltration hτ t) + ((shiftedStoppingFiltration hτ).le t) P μ ν hμ hν + (ae_of_all _ (fun ω => (R.terminal_radon _).2)) + (ae_finite_on_compacts_of_compact_mean P ν hν hνmean) hμmean hevent f hf + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawReverseComposition.lean b/lean/OAI/Probability/SLE/Loewner/RawReverseComposition.lean index 142e9ffe6ca278587e56ffe402b2d39c7794091d..7447ad7674f2958408b961874027ee67e6c7ae2b 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawReverseComposition.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawReverseComposition.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Loewner.SmoothRawInverse -import OAI.Probability.SLE.Loewner.ReverseLongContraction - -namespace OAI - -/-! A tail of a reverse trajectory is the raw inverse at the earlier -capacity time. This identity needs no geometric trace. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma reverse_height_mono_nonneg (q : ℝ → ℂ) (u : ℝ → ℝ) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Ici 0) t) - (hp : ∀ t, 0 ≤ t → 0 < (q t).im) : - MonotoneOn (fun t => (q t).im) (Ici 0) := by - intro s hs t ht hst - exact reverse_height_monotone q u - (fun v hv => (hd v hv.1).mono Icc_subset_Ici_self) (fun v hv => hp v hv.1) - ⟨hs,hst⟩ ⟨ht,le_rfl⟩ hst - -namespace Target - -lemma raw_reverse_tail_inverse (b : Driver) (T S : ℝ≥0) (hST : S ≤ T) - (q : ℝ → ℂ) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt q - (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici 0) t) - (hp : ∀ t, 0 ≤ t → 0 < (q t).im) : - inverseFlow 1 b S (q ((T : ℝ)-S)) = q T := by - let a : ℝ := (T : ℝ)-S - have ha : 0 ≤ a := sub_nonneg.mpr hST - have hmono := reverse_height_mono_nonneg q (fun t => driverValue b ((T : ℝ)-t)) hd hp - let Q (s : ℝ) := q (a+s) - let η : ℝ := (q a).im/2 - have hη : 0 < η := half_pos (hp a ha) - have hheight (s : ℝ) (hs : 0 ≤ s) : η ≤ (Q s).im := - (half_le_self (hp a ha).le).trans - (hmono ha (add_nonneg ha hs) (le_add_of_nonneg_right hs)) - have hQ (s : ℝ) (hs : 0 ≤ s) : HasDerivWithinAt Q - (reverseField η (driverValue b ((S : ℝ)-1*s)) (Q s)) (Ici 0) s := by - have hi : HasDerivWithinAt (fun v : ℝ => a+v) 1 (Ici 0) s := by - simpa using ((hasDerivAt_id s).const_add a).hasDerivWithinAt - have hmaps : MapsTo (fun v : ℝ => a+v) (Ici 0) (Ici 0) := - fun v hv => add_nonneg ha hv - have hh := (hd (a+s) (add_nonneg ha hs)).scomp s hi hmaps - have he : (T : ℝ)-(a+s)=(S : ℝ)-1*s := by dsimp [a]; ring - rw [reverseField,reverseCenter_eq (hheight s hs)] - convert! hh using 1 - simp only [one_smul,he,Q,div_eq_mul_inv] - have hh := inverseFlow_eq_reverse_trajectory 1 (by norm_num) b S (q a) - η hη (half_le_self (hp a ha).le) Q (by simp [Q]) hQ - convert hh using 1 - simp only [div_one,Q,a,sub_add_cancel] - -end Target -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawInverse +import OAI.Probability.SLE.Loewner.ReverseLongContraction + +namespace OAI + +/-! A tail of a reverse trajectory is the raw inverse at the earlier +capacity time. This identity needs no geometric trace. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma reverse_height_mono_nonneg (q : ℝ → ℂ) (u : ℝ → ℝ) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Ici 0) t) + (hp : ∀ t, 0 ≤ t → 0 < (q t).im) : + MonotoneOn (fun t => (q t).im) (Ici 0) := by + intro s hs t ht hst + exact reverse_height_monotone q u + (fun v hv => (hd v hv.1).mono Icc_subset_Ici_self) (fun v hv => hp v hv.1) + ⟨hs,hst⟩ ⟨ht,le_rfl⟩ hst + +namespace Target + +lemma raw_reverse_tail_inverse (b : Driver) (T S : ℝ≥0) (hST : S ≤ T) + (q : ℝ → ℂ) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt q + (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici 0) t) + (hp : ∀ t, 0 ≤ t → 0 < (q t).im) : + inverseFlow 1 b S (q ((T : ℝ)-S)) = q T := by + let a : ℝ := (T : ℝ)-S + have ha : 0 ≤ a := sub_nonneg.mpr hST + have hmono := reverse_height_mono_nonneg q (fun t => driverValue b ((T : ℝ)-t)) hd hp + let Q (s : ℝ) := q (a+s) + let η : ℝ := (q a).im/2 + have hη : 0 < η := half_pos (hp a ha) + have hheight (s : ℝ) (hs : 0 ≤ s) : η ≤ (Q s).im := + (half_le_self (hp a ha).le).trans + (hmono ha (add_nonneg ha hs) (le_add_of_nonneg_right hs)) + have hQ (s : ℝ) (hs : 0 ≤ s) : HasDerivWithinAt Q + (reverseField η (driverValue b ((S : ℝ)-1*s)) (Q s)) (Ici 0) s := by + have hi : HasDerivWithinAt (fun v : ℝ => a+v) 1 (Ici 0) s := by + simpa using ((hasDerivAt_id s).const_add a).hasDerivWithinAt + have hmaps : MapsTo (fun v : ℝ => a+v) (Ici 0) (Ici 0) := + fun v hv => add_nonneg ha hv + have hh := (hd (a+s) (add_nonneg ha hs)).scomp s hi hmaps + have he : (T : ℝ)-(a+s)=(S : ℝ)-1*s := by dsimp [a]; ring + rw [reverseField,reverseCenter_eq (hheight s hs)] + convert! hh using 1 + simp only [one_smul,he,Q,div_eq_mul_inv] + have hh := inverseFlow_eq_reverse_trajectory 1 (by norm_num) b S (q a) + η hη (half_le_self (hp a ha).le) Q (by simp [Q]) hQ + convert hh using 1 + simp only [div_one,Q,a,sub_add_cancel] + +end Target +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawScaleTests.lean b/lean/OAI/Probability/SLE/Loewner/RawScaleTests.lean index 3c456970af376ec7f44638e260e9365e0a34767e..a6da22cc3e6d37cc80c2b985052bed9eafecca3d 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawScaleTests.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawScaleTests.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Mass.RandomDilationMass -import OAI.Probability.SLE.Mass.CanonicalMassScaling - -namespace OAI - -/-! Fixed-scale covariance of the same raw functional, expressed as -measurable scalar tests on continuous driver space. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -local instance rawScaleTestsMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawScaleTestsBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma RawCanonicalMass.scaled_parameter_measurable {Ω : Type*} [MeasurableSpace Ω] - (R : RawCanonicalMass) (c T : Ω → ℝ≥0) (hc : Measurable c) (hT : Measurable T) - (Y : Ω → C(ℝ≥0,ℝ)) (hY : Measurable Y) : - Measurable (fun ω => R.process ((c ω)^2*T ω) (scaleDriverPath (c ω) (Y ω))) := - _root_.OAI.MeasureTheory.IsProgressive.parameter_time_measurable R.progressive - (fun ω => scaleDriverPath (c ω) (Y ω)) - (measurable_scaleDriverPath.comp (hc.prodMk hY)) - (fun ω => (c ω)^2*T ω) ((hc.pow_const 2).mul hT) - -lemma CanonicalMass.raw_scaled_test_eq - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) - (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈E) - (hEp : ∀ p∈E, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (c : ℝ≥0) (hpos : 0 ∫⁻ z, f z ∂dilationMass κ c hpos (R.process t p)) := - (Measure.measurable_lintegral hf).comp - ((conformalMass_measurable _ _).comp (R.progressive.adapted.measurable (i:=t))) - have hright : Measurable (fun p => ∫⁻ z, f z ∂(R.process (c^2*t) (scaleDriverPath c p))) := - (Measure.measurable_lintegral hf).comp - ((R.progressive.adapted.measurable (i:=c^2*t)).comp - (measurable_scaleDriverPath.comp (measurable_const.prodMk measurable_id))) - apply (ae_map_iff (source_path_measurable _ hm).aemeasurable - (measurableSet_eq_fun hleft hright)).mpr - filter_upwards [he,C.scaled_raw_process_eq hB hc F hw R he E hE hEf hEp c hpos] with ω heω hs - rw [heω.1 t,hs t] - rfl - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.RandomDilationMass +import OAI.Probability.SLE.Mass.CanonicalMassScaling + +namespace OAI + +/-! Fixed-scale covariance of the same raw functional, expressed as +measurable scalar tests on continuous driver space. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +local instance rawScaleTestsMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawScaleTestsBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma RawCanonicalMass.scaled_parameter_measurable {Ω : Type*} [MeasurableSpace Ω] + (R : RawCanonicalMass) (c T : Ω → ℝ≥0) (hc : Measurable c) (hT : Measurable T) + (Y : Ω → C(ℝ≥0,ℝ)) (hY : Measurable Y) : + Measurable (fun ω => R.process ((c ω)^2*T ω) (scaleDriverPath (c ω) (Y ω))) := + _root_.OAI.MeasureTheory.IsProgressive.parameter_time_measurable R.progressive + (fun ω => scaleDriverPath (c ω) (Y ω)) + (measurable_scaleDriverPath.comp (hc.prodMk hY)) + (fun ω => (c ω)^2*T ω) ((hc.pow_const 2).mul hT) + +lemma CanonicalMass.raw_scaled_test_eq + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) + (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p∈E) + (hEp : ∀ p∈E, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (c : ℝ≥0) (hpos : 0 ∫⁻ z, f z ∂dilationMass κ c hpos (R.process t p)) := + (Measure.measurable_lintegral hf).comp + ((conformalMass_measurable _ _).comp (R.progressive.adapted.measurable (i:=t))) + have hright : Measurable (fun p => ∫⁻ z, f z ∂(R.process (c^2*t) (scaleDriverPath c p))) := + (Measure.measurable_lintegral hf).comp + ((R.progressive.adapted.measurable (i:=c^2*t)).comp + (measurable_scaleDriverPath.comp (measurable_const.prodMk measurable_id))) + apply (ae_map_iff (source_path_measurable _ hm).aemeasurable + (measurableSet_eq_fun hleft hright)).mpr + filter_upwards [he,C.scaled_raw_process_eq hB hc F hw R he E hE hEf hEp c hpos] with ω heω hs + rw [heω.1 t,hs t] + rfl + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawScaledGreenKernel.lean b/lean/OAI/Probability/SLE/Loewner/RawScaledGreenKernel.lean index 64b11978d3a9ba4c3dc06d4813c321bb39abeff9..a76ba5745a84fe6876f56e6ad02c8e3c0eaae76e 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawScaledGreenKernel.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawScaledGreenKernel.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Stochastic.DriverLawKernelTransfer -import OAI.Probability.SLE.Stochastic.ScaledDriverFiltration -import OAI.Probability.SLE.Loewner.RawRestartGreenKernel - -namespace OAI - -/-! The raw canonical mass kernel under Brownian scaling, conditioned on -the full original past at the corresponding physical time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -local instance rawScaledGreenMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawScaledGreenBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -lemma CanonicalMass.raw_scaled_green_kernel - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (c : ℝ≥0) (hcp : 0 < c) (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[(Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) : - let Y := sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) - (P.restrict E).bind (fun ω => R.terminal (Y ω))= - (P.restrict E).bind (fun ω => R.process (c^2*t) (Y ω)+ - canonicalGreenMass κ (c^2*t) (Y ω)) := by - dsimp only - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - apply source_driver_bind_of_same_law_kernel hB (scaledBrownianDriver_brownian c hcp hB) - hm (scaledBrownianDriver_measurable c B hm) hc (scaledBrownianDriver_continuous c B hc) - (scaledBrownianDriver_path_law c hcp B hB hm hc).symm (c^2*t) - R.terminal (fun p => R.process (c^2*t) p+canonicalGreenMass κ (c^2*t) p) - R.terminal_measurable - ((R.progressive.adapted (c^2*t)).add (canonicalGreenMass_measurable_past κ (c^2*t))) - (fun A hA => C.raw_green_kernel hc F hw R he (c^2*t) A hA) E - rwa [scaledBrownianDriver_filtration c hcp B hm t] - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Stochastic.DriverLawKernelTransfer +import OAI.Probability.SLE.Stochastic.ScaledDriverFiltration +import OAI.Probability.SLE.Loewner.RawRestartGreenKernel + +namespace OAI + +/-! The raw canonical mass kernel under Brownian scaling, conditioned on +the full original past at the corresponding physical time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +local instance rawScaledGreenMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawScaledGreenBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +lemma CanonicalMass.raw_scaled_green_kernel + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (c : ℝ≥0) (hcp : 0 < c) (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[(Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) t] E) : + let Y := sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) + (P.restrict E).bind (fun ω => R.terminal (Y ω))= + (P.restrict E).bind (fun ω => R.process (c^2*t) (Y ω)+ + canonicalGreenMass κ (c^2*t) (Y ω)) := by + dsimp only + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + apply source_driver_bind_of_same_law_kernel hB (scaledBrownianDriver_brownian c hcp hB) + hm (scaledBrownianDriver_measurable c B hm) hc (scaledBrownianDriver_continuous c B hc) + (scaledBrownianDriver_path_law c hcp B hB hm hc).symm (c^2*t) + R.terminal (fun p => R.process (c^2*t) p+canonicalGreenMass κ (c^2*t) p) + R.terminal_measurable + ((R.progressive.adapted (c^2*t)).add (canonicalGreenMass_measurable_past κ (c^2*t))) + (fun A hA => C.raw_green_kernel hc F hw R he (c^2*t) A hA) E + rwa [scaledBrownianDriver_filtration c hcp B hm t] + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawSurvivingAnalytic.lean b/lean/OAI/Probability/SLE/Loewner/RawSurvivingAnalytic.lean index 50870f5ba4c89a1b47ece417406e306c168e4a9f..04e45de2150796f2a6e411f18df5d3f04e9dc929 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawSurvivingAnalytic.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawSurvivingAnalytic.lean @@ -1,74 +1,74 @@ -import OAI.Probability.SLE.Loewner.RawInverseHolomorphic -import OAI.Probability.SLE.Loewner.TargetForwardAnalytic -import OAI.Probability.SLE.Loewner.SmoothRawTrace - -namespace OAI - -/-! The raw surviving set is an open conformal image of the half-plane. -For a smooth driver its radial tip lies on the boundary of this set. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma Target.surviving_eq_inverse_image (b : Driver) (T : ℝ≥0) : - {z | Surviving 1 b z T} = inverseFlow 1 b T '' upperHalfPlane := by - ext z - constructor - · intro hz - refine ⟨flow 1 b z T,surviving_flow_mem_upperHalfPlane 1 b z T T.coe_nonneg hz,?_⟩ - exact inverseFlow_eq_of_surviving 1 b T T.coe_nonneg z _ hz rfl - · rintro ⟨w,hw,rfl⟩ - exact inverseFlow_surviving 1 (by norm_num) b T w hw - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} - -lemma RegularSLEPresentation.raw_surviving_isOpen (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) : - IsOpen {z | Target.Surviving 1 b z T} := by - rw [Target.surviving_eq_inverse_image] - apply isOpen_iff_mem_nhds.mpr - rintro z ⟨w,hw,rfl⟩ - exact univalent_nhds_le_map isOpen_upperHalfPlane (a.raw_inverse_holomorphic hκ b T) - (Target.inverseFlow_injOn b T) hw (image_mem_map (isOpen_upperHalfPlane.mem_nhds hw)) - -lemma RegularSLEPresentation.raw_forward_holomorphic (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) : - DifferentiableOn ℂ (fun z => Target.flow 1 b z T) {z | Target.Surviving 1 b z T} := - Target.differentiableOn_flow_of_inverse 1 (by norm_num) b T - (a.raw_inverse_holomorphic hκ b T) (fun _ hz => a.raw_inverse_deriv_ne_zero hκ b T hz) - -theorem RegularSLEPresentation.smooth_raw_tip_not_surviving - (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) - (b : Target.Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (Target.driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : - ¬Target.Surviving 1 b (Target.trace 1 b T) T := by - intro hs - have hc := ((a.raw_forward_holomorphic hκ b T).differentiableAt - ((a.raw_surviving_isOpen hκ b T).mem_nhds hs)).continuousAt - have ht := hc.tendsto.comp (Target.tendsto_inverseFlow_smooth_trace b T b' hd hL hb) - have he : (fun y : ℝ => Target.flow 1 b - (Target.inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*Complex.I)) T) - =ᶠ[𝓝[>] (0 : ℝ)] (fun y => (b T : ℂ)+(y : ℂ)*Complex.I) := by - filter_upwards [self_mem_nhdsWithin] with y hy - exact Target.flow_inverseFlow 1 (by norm_num) b T _ - (by simpa [upperHalfPlane] using hy) - have hv : Tendsto (fun y : ℝ => (b T : ℂ)+(y : ℂ)*Complex.I) - (𝓝[>] 0) (𝓝 (b T : ℂ)) := by - have hc' : Continuous (fun y : ℝ => (b T : ℂ)+(y : ℂ)*Complex.I) := by fun_prop - simpa only [Complex.ofReal_zero,zero_mul,add_zero] using - (hc'.tendsto 0).mono_left nhdsWithin_le_nhds - have hval := tendsto_nhds_unique ht (hv.congr' he.symm) - have hp := Target.surviving_flow_mem_upperHalfPlane 1 b _ T T.coe_nonneg hs - rw [hval] at hp - simp [upperHalfPlane] at hp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawInverseHolomorphic +import OAI.Probability.SLE.Loewner.TargetForwardAnalytic +import OAI.Probability.SLE.Loewner.SmoothRawTrace + +namespace OAI + +/-! The raw surviving set is an open conformal image of the half-plane. +For a smooth driver its radial tip lies on the boundary of this set. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma Target.surviving_eq_inverse_image (b : Driver) (T : ℝ≥0) : + {z | Surviving 1 b z T} = inverseFlow 1 b T '' upperHalfPlane := by + ext z + constructor + · intro hz + refine ⟨flow 1 b z T,surviving_flow_mem_upperHalfPlane 1 b z T T.coe_nonneg hz,?_⟩ + exact inverseFlow_eq_of_surviving 1 b T T.coe_nonneg z _ hz rfl + · rintro ⟨w,hw,rfl⟩ + exact inverseFlow_surviving 1 (by norm_num) b T w hw + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} + +lemma RegularSLEPresentation.raw_surviving_isOpen (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) : + IsOpen {z | Target.Surviving 1 b z T} := by + rw [Target.surviving_eq_inverse_image] + apply isOpen_iff_mem_nhds.mpr + rintro z ⟨w,hw,rfl⟩ + exact univalent_nhds_le_map isOpen_upperHalfPlane (a.raw_inverse_holomorphic hκ b T) + (Target.inverseFlow_injOn b T) hw (image_mem_map (isOpen_upperHalfPlane.mem_nhds hw)) + +lemma RegularSLEPresentation.raw_forward_holomorphic (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (b : Target.Driver) (T : ℝ≥0) : + DifferentiableOn ℂ (fun z => Target.flow 1 b z T) {z | Target.Surviving 1 b z T} := + Target.differentiableOn_flow_of_inverse 1 (by norm_num) b T + (a.raw_inverse_holomorphic hκ b T) (fun _ hz => a.raw_inverse_deriv_ne_zero hκ b T hz) + +theorem RegularSLEPresentation.smooth_raw_tip_not_surviving + (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) + (b : Target.Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (Target.driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : + ¬Target.Surviving 1 b (Target.trace 1 b T) T := by + intro hs + have hc := ((a.raw_forward_holomorphic hκ b T).differentiableAt + ((a.raw_surviving_isOpen hκ b T).mem_nhds hs)).continuousAt + have ht := hc.tendsto.comp (Target.tendsto_inverseFlow_smooth_trace b T b' hd hL hb) + have he : (fun y : ℝ => Target.flow 1 b + (Target.inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*Complex.I)) T) + =ᶠ[𝓝[>] (0 : ℝ)] (fun y => (b T : ℂ)+(y : ℂ)*Complex.I) := by + filter_upwards [self_mem_nhdsWithin] with y hy + exact Target.flow_inverseFlow 1 (by norm_num) b T _ + (by simpa [upperHalfPlane] using hy) + have hv : Tendsto (fun y : ℝ => (b T : ℂ)+(y : ℂ)*Complex.I) + (𝓝[>] 0) (𝓝 (b T : ℂ)) := by + have hc' : Continuous (fun y : ℝ => (b T : ℂ)+(y : ℂ)*Complex.I) := by fun_prop + simpa only [Complex.ofReal_zero,zero_mul,add_zero] using + (hc'.tendsto 0).mono_left nhdsWithin_le_nhds + have hval := tendsto_nhds_unique ht (hv.congr' he.symm) + have hp := Target.surviving_flow_mem_upperHalfPlane 1 b _ T T.coe_nonneg hs + rw [hval] at hp + simp [upperHalfPlane] at hp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawSurvivingDerivative.lean b/lean/OAI/Probability/SLE/Loewner/RawSurvivingDerivative.lean index 3f25badbd5d796c8125c91d44707e12c6dd96641..015b99e6eabd9c213219f28022d8d38f356d2281 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawSurvivingDerivative.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawSurvivingDerivative.lean @@ -1,26 +1,26 @@ -import OAI.Probability.SLE.Angle.TargetFlowEvolution -import OAI.Probability.SLE.Loewner.TargetSurvivalExtension - -namespace OAI - -/-! The raw flow has its ordinary right evolution at every surviving -time, because the strict cutoff certificate persists a little longer. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -lemma flow_hasDerivWithinAt_of_survival (b : Driver) (z : ℂ) {T : ℝ} - (hT : 0 ≤ T) (hs : Surviving 1 b z T) : - HasDerivWithinAt (flow 1 b z) - (2/(flow 1 b z T-(b T.toNNReal:ℂ))) (Ici 0) T := by - obtain ⟨S,hTS,hS⟩ := exists_later_survival 1 b z hT hs - simpa only [a,div_one,Complex.ofReal_ofNat,driverValue] using - flow_hasDerivWithinAt_of_later_survival 1 b z T S hT hTS hS - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetFlowEvolution +import OAI.Probability.SLE.Loewner.TargetSurvivalExtension + +namespace OAI + +/-! The raw flow has its ordinary right evolution at every surviving +time, because the strict cutoff certificate persists a little longer. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +lemma flow_hasDerivWithinAt_of_survival (b : Driver) (z : ℂ) {T : ℝ} + (hT : 0 ≤ T) (hs : Surviving 1 b z T) : + HasDerivWithinAt (flow 1 b z) + (2/(flow 1 b z T-(b T.toNNReal:ℂ))) (Ici 0) T := by + obtain ⟨S,hTS,hS⟩ := exists_later_survival 1 b z hT hs + simpa only [a,div_one,Complex.ofReal_ofNat,driverValue] using + flow_hasDerivWithinAt_of_later_survival 1 b z T S hT hTS hS + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawSurvivingMonotone.lean b/lean/OAI/Probability/SLE/Loewner/RawSurvivingMonotone.lean index 72db37f299d960491de7b0c3985e9f7c360b5f31..7b052528eda73937ef80439f1fe3515a7fecde13 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawSurvivingMonotone.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawSurvivingMonotone.lean @@ -1,27 +1,27 @@ -import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory - -namespace OAI - -/-! Raw survival decreases with the time horizon. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge.Target - -lemma surviving_of_le_time (κ : ℝ) (b : Driver) (z : ℂ) {S T : ℝ} - (hS : 0 ≤ S) (hST : S ≤ T) (h : Surviving κ b z T) : Surviving κ b z S := by - obtain ⟨j,hj,hmin⟩ := h - refine ⟨j,hj,hmin.trans_le ?_⟩ - have hc : IsCompact ((fun t : ℝ => (cutoffFlow κ b j z t).im) '' Icc 0 T) := - isCompact_Icc.image_of_continuousOn (Complex.continuous_im.comp_continuousOn - ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self)) - apply le_csInf (Set.Nonempty.image _ (nonempty_Icc.mpr hS)) - rintro y ⟨s,hs,rfl⟩ - exact csInf_le hc.bddBelow ⟨s,⟨hs.1,hs.2.trans hST⟩,rfl⟩ - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory + +namespace OAI + +/-! Raw survival decreases with the time horizon. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge.Target + +lemma surviving_of_le_time (κ : ℝ) (b : Driver) (z : ℂ) {S T : ℝ} + (hS : 0 ≤ S) (hST : S ≤ T) (h : Surviving κ b z T) : Surviving κ b z S := by + obtain ⟨j,hj,hmin⟩ := h + refine ⟨j,hj,hmin.trans_le ?_⟩ + have hc : IsCompact ((fun t : ℝ => (cutoffFlow κ b j z t).im) '' Icc 0 T) := + isCompact_Icc.image_of_continuousOn (Complex.continuous_im.comp_continuousOn + ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self)) + apply le_csInf (Set.Nonempty.image _ (nonempty_Icc.mpr hS)) + rintro y ⟨s,hs,rfl⟩ + exact csInf_le hc.bddBelow ⟨s,⟨hs.1,hs.2.trans hST⟩,rfl⟩ + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawTerminalExhaustion.lean b/lean/OAI/Probability/SLE/Loewner/RawTerminalExhaustion.lean index 69a7db2ecbd786b27fac683b01c35cdc6b962e15..33c939b8d8e55d04f6347a7040fb017bd1bc4b60 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawTerminalExhaustion.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawTerminalExhaustion.lean @@ -1,153 +1,153 @@ -import OAI.Probability.SLE.Geometry.CanonicalRawTerminal -import OAI.Probability.SLE.Mass.RawMassPathLaw - -namespace OAI - -/-! A measurable terminal certificate for the raw finite-time mass. -Together with monotonicity it yields terminal domination and local -exhaustion at all real times, not just integer times. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge - -local instance rawExhaustionDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance rawExhaustionDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma ennreal_antitone_tendsto_zero_of_nat - (f : ℝ≥0 → ℝ≥0∞) (hf : Antitone f) - (hn : Tendsto (fun n : ℕ => f n) atTop (𝓝 0)) : - Tendsto f atTop (𝓝 0) := by - have hi : (⨅ t, f t)=0 := by - apply le_antisymm ?_ bot_le - exact le_of_tendsto_of_tendsto' tendsto_const_nhds hn (fun n => iInf_le f (n:ℝ≥0)) - simpa only [hi] using tendsto_atTop_iInf hf - -def RawTerminalGood (D : LocalTestFamily upperHalfPlane) - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) : Prop := - (∀ n : ℕ, ∀ j, (∫ z, (D.test j ⊔ 0) z ∂N n) ≤ ∫ z, (D.test j ⊔ 0) z ∂ν) ∧ - ∀ k, Tendsto (fun n : ℕ => ν (D.compactSet k)-N n (D.compactSet k)) atTop (𝓝 0) - -theorem measurableSet_rawTerminalGood - {Y : Type*} [MeasurableSpace Y] (D : LocalTestFamily upperHalfPlane) - (N : ℝ≥0 → Y → Measure upperHalfPlane) (ν : Y → Measure upperHalfPlane) - (hNm : ∀ n : ℕ, Measurable (N n)) (hνm : Measurable ν) - (hNr : ∀ n : ℕ, ∀ y, (N n y).Regular) (hνr : ∀ y, (ν y).Regular) : - MeasurableSet {y | RawTerminalGood D (fun t => N t y) (ν y)} := by - have ho : MeasurableSet {y | ∀ n : ℕ, ∀ j, - (∫ z, (D.test j ⊔ 0) z ∂N n y) ≤ ∫ z, (D.test j ⊔ 0) z ∂ν y} := by - simp only [Set.ofPred_forall] - exact MeasurableSet.iInter fun n => MeasurableSet.iInter fun j => - measurableSet_le - (measurable_halfPlane_compact_test_integral (N n) (hNm n) (hNr n) _) - (measurable_halfPlane_compact_test_integral ν hνm hνr _) - have ht : MeasurableSet {y | ∀ k, Tendsto - (fun n : ℕ => ν y (D.compactSet k)-N n y (D.compactSet k)) atTop (𝓝 0)} := by - simp only [Set.ofPred_forall] - apply MeasurableSet.iInter - intro k - exact measurableSet_tendsto (𝓝 (0:ℝ≥0∞)) (fun n => - ((Measure.measurable_coe (D.isCompact k).measurableSet).comp hνm).sub - ((Measure.measurable_coe (D.isCompact k).measurableSet).comp (hNm n))) - simpa only [RawTerminalGood,Set.ofPred_and] using ho.inter ht - -theorem RawTerminalGood.domination_exhaustion - (D : LocalTestFamily upperHalfPlane) - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) - (hNr : ∀ t, (N t).Regular) (hνr : ν.Regular) (hN : Monotone N) - (hg : RawTerminalGood D N ν) : - (∀ t, N t ≤ ν) ∧ ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0) := by - let := hνr - have hn (n : ℕ) : N n ≤ ν := by - let := hNr n - exact D.measure_le_of_positive_tests _ _ (hg.1 n) - have hdom (t : ℝ≥0) : N t ≤ ν := by - obtain ⟨n,hn'⟩ := exists_nat_ge (t:ℝ) - have ht : t ≤ (n:ℝ≥0) := by exact_mod_cast hn' - exact (hN ht).trans (hn n) - refine ⟨hdom,?_⟩ - intro K hK - obtain ⟨k,hKk⟩ := D.exhaustive K hK - have hanti : Antitone (fun t : ℝ≥0 => ν (D.compactSet k)-N t (D.compactSet k)) := by - intro s t hst - exact tsub_le_tsub_left (hN hst (D.compactSet k)) _ - have hk := ennreal_antitone_tendsto_zero_of_nat _ hanti (hg.2 k) - have he (t : ℝ≥0) : (ν-N t) (D.compactSet k) = - ν (D.compactSet k)-N t (D.compactSet k) := by - let := hNr t - exact sub_apply_of_finite_restriction (hdom t) (D.isCompact k).measurableSet - (D.isCompact k).measure_lt_top.ne - have hk' : Tendsto (fun t : ℝ≥0 => (ν-N t) (D.compactSet k)) atTop (𝓝 0) := by - simpa only [he] using hk - exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hk' - (fun _ => bot_le) (fun _ => measure_mono hKk) - -theorem rawTerminalGood_of_exhaustion - (D : LocalTestFamily upperHalfPlane) - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) - (hNr : ∀ t, (N t).Regular) (hνr : ν.Regular) - (hdom : ∀ t, N t ≤ ν) - (hlim : ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) : - RawTerminalGood D N ν := by - let := hνr - constructor - · intro n j - exact integral_mono_measure (hdom n) (ae_of_all _ (fun _ => le_max_right _ _)) - (D.test j ⊔ 0).integrable - · intro k - have he (n : ℕ) : (ν-N n) (D.compactSet k) = - ν (D.compactSet k)-N n (D.compactSet k) := by - let := hNr n - exact sub_apply_of_finite_restriction (hdom n) (D.isCompact k).measurableSet - (D.isCompact k).measure_lt_top.ne - simpa only [Function.comp_def,he] using (hlim _ (D.isCompact k)).comp (tendsto_natCast_atTop_atTop (R := ℝ≥0)) - -namespace Multipoint - -theorem CanonicalMass.raw_terminal_good_set - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hBc : ∀ ω, Continuous (fun t => B t ω)) - (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) - (hRp : IsProgressive driverPathFiltration R) - (hRr : ∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) - (he : ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω)=C.process t ω) : - let D := localTestFamily upperHalfPlane - let E : Set C(ℝ≥0,ℝ) := {p | RawTerminalGood D (fun t => R t p) (rawTerminalMass R p)} - MeasurableSet E ∧ (∀ᵐ p ∂P.map (sourceDriverPath B hBc), p ∈ E) ∧ - ∀ p ∈ E, Monotone (fun t => R t p) → - (∀ t, R t p ≤ rawTerminalMass R p) ∧ - ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (rawTerminalMass R p-R t p) K) atTop (𝓝 0) := by - dsimp only - have hE := measurableSet_rawTerminalGood (localTestFamily upperHalfPlane) R (rawTerminalMass R) - (fun n => hRp.adapted.measurable (i := (n:ℝ≥0))) - (rawTerminalMass_measurable R hRp hRr) (fun n p => (hRr n p).1) - (fun p => (rawTerminalMass_radon R p).1) - refine ⟨hE,?_,?_⟩ - · have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) - apply (ae_map_iff hp.aemeasurable hE).mpr - filter_upwards [he,C.rawTerminal_realizes hBc R he,C.paths,C.exhaustion] with ω hω hterm hpath hex - simp_rw [hω,hterm] - exact rawTerminalGood_of_exhaustion (localTestFamily upperHalfPlane) - (fun t => C.process t ω) (C.rational.terminal ω) - (fun t => (C.radon t ω).1) (C.rational.terminal_radon ω).1 hpath.dominated hex - · intro p hp hm - exact RawTerminalGood.domination_exhaustion (localTestFamily upperHalfPlane) - (fun t => R t p) (rawTerminalMass R p) (fun t => (hRr t p).1) - (rawTerminalMass_radon R p).1 hm hp - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalRawTerminal +import OAI.Probability.SLE.Mass.RawMassPathLaw + +namespace OAI + +/-! A measurable terminal certificate for the raw finite-time mass. +Together with monotonicity it yields terminal domination and local +exhaustion at all real times, not just integer times. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge + +local instance rawExhaustionDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance rawExhaustionDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma ennreal_antitone_tendsto_zero_of_nat + (f : ℝ≥0 → ℝ≥0∞) (hf : Antitone f) + (hn : Tendsto (fun n : ℕ => f n) atTop (𝓝 0)) : + Tendsto f atTop (𝓝 0) := by + have hi : (⨅ t, f t)=0 := by + apply le_antisymm ?_ bot_le + exact le_of_tendsto_of_tendsto' tendsto_const_nhds hn (fun n => iInf_le f (n:ℝ≥0)) + simpa only [hi] using tendsto_atTop_iInf hf + +def RawTerminalGood (D : LocalTestFamily upperHalfPlane) + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) : Prop := + (∀ n : ℕ, ∀ j, (∫ z, (D.test j ⊔ 0) z ∂N n) ≤ ∫ z, (D.test j ⊔ 0) z ∂ν) ∧ + ∀ k, Tendsto (fun n : ℕ => ν (D.compactSet k)-N n (D.compactSet k)) atTop (𝓝 0) + +theorem measurableSet_rawTerminalGood + {Y : Type*} [MeasurableSpace Y] (D : LocalTestFamily upperHalfPlane) + (N : ℝ≥0 → Y → Measure upperHalfPlane) (ν : Y → Measure upperHalfPlane) + (hNm : ∀ n : ℕ, Measurable (N n)) (hνm : Measurable ν) + (hNr : ∀ n : ℕ, ∀ y, (N n y).Regular) (hνr : ∀ y, (ν y).Regular) : + MeasurableSet {y | RawTerminalGood D (fun t => N t y) (ν y)} := by + have ho : MeasurableSet {y | ∀ n : ℕ, ∀ j, + (∫ z, (D.test j ⊔ 0) z ∂N n y) ≤ ∫ z, (D.test j ⊔ 0) z ∂ν y} := by + simp only [Set.ofPred_forall] + exact MeasurableSet.iInter fun n => MeasurableSet.iInter fun j => + measurableSet_le + (measurable_halfPlane_compact_test_integral (N n) (hNm n) (hNr n) _) + (measurable_halfPlane_compact_test_integral ν hνm hνr _) + have ht : MeasurableSet {y | ∀ k, Tendsto + (fun n : ℕ => ν y (D.compactSet k)-N n y (D.compactSet k)) atTop (𝓝 0)} := by + simp only [Set.ofPred_forall] + apply MeasurableSet.iInter + intro k + exact measurableSet_tendsto (𝓝 (0:ℝ≥0∞)) (fun n => + ((Measure.measurable_coe (D.isCompact k).measurableSet).comp hνm).sub + ((Measure.measurable_coe (D.isCompact k).measurableSet).comp (hNm n))) + simpa only [RawTerminalGood,Set.ofPred_and] using ho.inter ht + +theorem RawTerminalGood.domination_exhaustion + (D : LocalTestFamily upperHalfPlane) + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) + (hNr : ∀ t, (N t).Regular) (hνr : ν.Regular) (hN : Monotone N) + (hg : RawTerminalGood D N ν) : + (∀ t, N t ≤ ν) ∧ ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0) := by + let := hνr + have hn (n : ℕ) : N n ≤ ν := by + let := hNr n + exact D.measure_le_of_positive_tests _ _ (hg.1 n) + have hdom (t : ℝ≥0) : N t ≤ ν := by + obtain ⟨n,hn'⟩ := exists_nat_ge (t:ℝ) + have ht : t ≤ (n:ℝ≥0) := by exact_mod_cast hn' + exact (hN ht).trans (hn n) + refine ⟨hdom,?_⟩ + intro K hK + obtain ⟨k,hKk⟩ := D.exhaustive K hK + have hanti : Antitone (fun t : ℝ≥0 => ν (D.compactSet k)-N t (D.compactSet k)) := by + intro s t hst + exact tsub_le_tsub_left (hN hst (D.compactSet k)) _ + have hk := ennreal_antitone_tendsto_zero_of_nat _ hanti (hg.2 k) + have he (t : ℝ≥0) : (ν-N t) (D.compactSet k) = + ν (D.compactSet k)-N t (D.compactSet k) := by + let := hNr t + exact sub_apply_of_finite_restriction (hdom t) (D.isCompact k).measurableSet + (D.isCompact k).measure_lt_top.ne + have hk' : Tendsto (fun t : ℝ≥0 => (ν-N t) (D.compactSet k)) atTop (𝓝 0) := by + simpa only [he] using hk + exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hk' + (fun _ => bot_le) (fun _ => measure_mono hKk) + +theorem rawTerminalGood_of_exhaustion + (D : LocalTestFamily upperHalfPlane) + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) + (hNr : ∀ t, (N t).Regular) (hνr : ν.Regular) + (hdom : ∀ t, N t ≤ ν) + (hlim : ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) : + RawTerminalGood D N ν := by + let := hνr + constructor + · intro n j + exact integral_mono_measure (hdom n) (ae_of_all _ (fun _ => le_max_right _ _)) + (D.test j ⊔ 0).integrable + · intro k + have he (n : ℕ) : (ν-N n) (D.compactSet k) = + ν (D.compactSet k)-N n (D.compactSet k) := by + let := hNr n + exact sub_apply_of_finite_restriction (hdom n) (D.isCompact k).measurableSet + (D.isCompact k).measure_lt_top.ne + simpa only [Function.comp_def,he] using (hlim _ (D.isCompact k)).comp (tendsto_natCast_atTop_atTop (R := ℝ≥0)) + +namespace Multipoint + +theorem CanonicalMass.raw_terminal_good_set + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hBc : ∀ ω, Continuous (fun t => B t ω)) + (R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane) + (hRp : IsProgressive driverPathFiltration R) + (hRr : ∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) + (he : ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω)=C.process t ω) : + let D := localTestFamily upperHalfPlane + let E : Set C(ℝ≥0,ℝ) := {p | RawTerminalGood D (fun t => R t p) (rawTerminalMass R p)} + MeasurableSet E ∧ (∀ᵐ p ∂P.map (sourceDriverPath B hBc), p ∈ E) ∧ + ∀ p ∈ E, Monotone (fun t => R t p) → + (∀ t, R t p ≤ rawTerminalMass R p) ∧ + ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (rawTerminalMass R p-R t p) K) atTop (𝓝 0) := by + dsimp only + have hE := measurableSet_rawTerminalGood (localTestFamily upperHalfPlane) R (rawTerminalMass R) + (fun n => hRp.adapted.measurable (i := (n:ℝ≥0))) + (rawTerminalMass_measurable R hRp hRr) (fun n p => (hRr n p).1) + (fun p => (rawTerminalMass_radon R p).1) + refine ⟨hE,?_,?_⟩ + · have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) + apply (ae_map_iff hp.aemeasurable hE).mpr + filter_upwards [he,C.rawTerminal_realizes hBc R he,C.paths,C.exhaustion] with ω hω hterm hpath hex + simp_rw [hω,hterm] + exact rawTerminalGood_of_exhaustion (localTestFamily upperHalfPlane) + (fun t => C.process t ω) (C.rational.terminal ω) + (fun t => (C.radon t ω).1) (C.rational.terminal_radon ω).1 hpath.dominated hex + · intro p hp hm + exact RawTerminalGood.domination_exhaustion (localTestFamily upperHalfPlane) + (fun t => R t p) (rawTerminalMass R p) (fun t => (hRr t p).1) + (rawTerminalMass_radon R p).1 hm hp + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RawTipTimeContinuity.lean b/lean/OAI/Probability/SLE/Loewner/RawTipTimeContinuity.lean index 63c67578c2086b1e63d34059b4a3dbc8103b65d1..6e801ff83bdc42d819819c2b1eff72d130e9672e 100644 --- a/lean/OAI/Probability/SLE/Loewner/RawTipTimeContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/RawTipTimeContinuity.lean @@ -1,89 +1,89 @@ -import OAI.Probability.SLE.Loewner.SmoothRawInverse -import OAI.Probability.SLE.Loewner.ReverseFlowStability - -namespace OAI - -/-! Interior radial approximations to the raw tip vary continuously with -time for a Lipschitz driver. The bound is uniform on each finite interval. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma gronwallBound_scale_initial (δ K c t : ℝ) : - gronwallBound δ K (c*δ) t = δ*gronwallBound 1 K c t := by - unfold gronwallBound - split_ifs <;> ring - -namespace Target - -lemma raw_radial_time_distance (b : Driver) {L : ℝ≥0} - (hb : LipschitzWith L (driverValue b)) (R T S : ℝ≥0) - (hT : T ≤ R) (_hS : S ≤ R) {y : ℝ} (hy : 0 < y) : - dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) - (inverseFlow 1 b S ((b S : ℂ)+(y : ℂ)*I)) ≤ - ((L : ℝ)*gronwallBound 1 (4*y⁻¹^2) (2*y⁻¹^2) R+2*y⁻¹) * dist T S := by - obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T - ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) - obtain ⟨r,hr0,hry,hrd,her⟩ := exists_raw_reverse_trajectory b S - ((b S : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) - have hdQ (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt q - (reverseField y (driverValue b ((T : ℝ)-t)) (q t)) (Ici 0) t := by - rw [reverseField,reverseCenter_eq (by simpa using hqy t ht)] - exact hqd t ht - have hdR (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt r - (reverseField y (driverValue b ((S : ℝ)-t)) (r t)) (Ici 0) t := by - rw [reverseField,reverseCenter_eq (by simpa using hry t ht)] - exact hrd t ht - have hinit : dist (q 0) (r 0) ≤ (L : ℝ)*dist T S := by - rw [hq0,hr0,dist_add_right] - simpa only [driverValue,Real.toNNReal_coe,dist_eq_norm,← Complex.ofReal_sub, - Complex.norm_real,Real.norm_eq_abs,NNReal.dist_eq] using - hb.dist_le_mul (T : ℝ) (S : ℝ) - have hsame := reverse_flow_driver_distance hy - (fun t => driverValue b ((T : ℝ)-t)) (fun t => driverValue b ((S : ℝ)-t)) - q r (T : ℝ) (fun t ht => (hdQ t ht.1).mono Icc_subset_Ici_self) - (fun t ht => (hdR t ht.1).mono Icc_subset_Ici_self) - ((L : ℝ)*dist T S) hinit (fun t _ => by - simpa only [Real.dist_eq,sub_sub_sub_cancel_right,NNReal.dist_eq] using - hb.dist_le_mul ((T : ℝ)-t) ((S : ℝ)-t)) T ⟨T.coe_nonneg,le_rfl⟩ - have hmono := gronwallBound_mono (show 0 ≤ (L : ℝ)*dist T S by positivity) - (show 0 ≤ 2*y⁻¹^2*((L : ℝ)*dist T S) by positivity) - (show 0 ≤ 4*y⁻¹^2 by positivity) (show (T : ℝ) ≤ R from hT) - have htime := (reverse_flow_lipschitz hy _ r hdR).dist_le_mul - (T : ℝ) T.coe_nonneg (S : ℝ) S.coe_nonneg - change dist (r (T : ℝ)) (r (S : ℝ)) ≤ (2*y⁻¹)*dist (T : ℝ) (S : ℝ) at htime - rw [heq,her] - calc - dist (q T) (r S) ≤ dist (q T) (r T)+dist (r T) (r S) := dist_triangle _ _ _ - _ ≤ gronwallBound ((L : ℝ)*dist T S) (4*y⁻¹^2) - (2*y⁻¹^2*((L : ℝ)*dist T S)) R + (2*y⁻¹)*dist T S := by - exact add_le_add (hsame.trans hmono) (by simpa only [NNReal.coe_mk,NNReal.dist_eq,Real.dist_eq] using htime) - _ = _ := by rw [gronwallBound_scale_initial]; ring - -lemma continuous_raw_radial_time (b : Driver) {L : ℝ≥0} - (hb : LipschitzWith L (driverValue b)) {y : ℝ} (hy : 0 < y) : - Continuous (fun T : ℝ≥0 => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) := by - have hlocal (R : ℝ≥0) : ContinuousOn - (fun T : ℝ≥0 => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) (Iic R) := by - have hpos : 0 ≤ (L : ℝ)*gronwallBound 1 (4*y⁻¹^2) (2*y⁻¹^2) R+2*y⁻¹ := by - have hg := gronwallBound_mono (by norm_num : (0 : ℝ) ≤ 1) - (show 0 ≤ 2*y⁻¹^2 by positivity) (show 0 ≤ 4*y⁻¹^2 by positivity) - R.coe_nonneg - rw [gronwallBound_x0] at hg - positivity - apply LipschitzOnWith.continuousOn (K := ⟨_,hpos⟩) - apply LipschitzOnWith.of_dist_le_mul - intro T hT S hS - exact raw_radial_time_distance b hb R T S hT hS hy - apply continuous_iff_continuousAt.mpr - intro T - exact (hlocal (T+1)).continuousAt (Iic_mem_nhds (by simp)) - -end Target -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawInverse +import OAI.Probability.SLE.Loewner.ReverseFlowStability + +namespace OAI + +/-! Interior radial approximations to the raw tip vary continuously with +time for a Lipschitz driver. The bound is uniform on each finite interval. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma gronwallBound_scale_initial (δ K c t : ℝ) : + gronwallBound δ K (c*δ) t = δ*gronwallBound 1 K c t := by + unfold gronwallBound + split_ifs <;> ring + +namespace Target + +lemma raw_radial_time_distance (b : Driver) {L : ℝ≥0} + (hb : LipschitzWith L (driverValue b)) (R T S : ℝ≥0) + (hT : T ≤ R) (_hS : S ≤ R) {y : ℝ} (hy : 0 < y) : + dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) + (inverseFlow 1 b S ((b S : ℂ)+(y : ℂ)*I)) ≤ + ((L : ℝ)*gronwallBound 1 (4*y⁻¹^2) (2*y⁻¹^2) R+2*y⁻¹) * dist T S := by + obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T + ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) + obtain ⟨r,hr0,hry,hrd,her⟩ := exists_raw_reverse_trajectory b S + ((b S : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) + have hdQ (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt q + (reverseField y (driverValue b ((T : ℝ)-t)) (q t)) (Ici 0) t := by + rw [reverseField,reverseCenter_eq (by simpa using hqy t ht)] + exact hqd t ht + have hdR (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt r + (reverseField y (driverValue b ((S : ℝ)-t)) (r t)) (Ici 0) t := by + rw [reverseField,reverseCenter_eq (by simpa using hry t ht)] + exact hrd t ht + have hinit : dist (q 0) (r 0) ≤ (L : ℝ)*dist T S := by + rw [hq0,hr0,dist_add_right] + simpa only [driverValue,Real.toNNReal_coe,dist_eq_norm,← Complex.ofReal_sub, + Complex.norm_real,Real.norm_eq_abs,NNReal.dist_eq] using + hb.dist_le_mul (T : ℝ) (S : ℝ) + have hsame := reverse_flow_driver_distance hy + (fun t => driverValue b ((T : ℝ)-t)) (fun t => driverValue b ((S : ℝ)-t)) + q r (T : ℝ) (fun t ht => (hdQ t ht.1).mono Icc_subset_Ici_self) + (fun t ht => (hdR t ht.1).mono Icc_subset_Ici_self) + ((L : ℝ)*dist T S) hinit (fun t _ => by + simpa only [Real.dist_eq,sub_sub_sub_cancel_right,NNReal.dist_eq] using + hb.dist_le_mul ((T : ℝ)-t) ((S : ℝ)-t)) T ⟨T.coe_nonneg,le_rfl⟩ + have hmono := gronwallBound_mono (show 0 ≤ (L : ℝ)*dist T S by positivity) + (show 0 ≤ 2*y⁻¹^2*((L : ℝ)*dist T S) by positivity) + (show 0 ≤ 4*y⁻¹^2 by positivity) (show (T : ℝ) ≤ R from hT) + have htime := (reverse_flow_lipschitz hy _ r hdR).dist_le_mul + (T : ℝ) T.coe_nonneg (S : ℝ) S.coe_nonneg + change dist (r (T : ℝ)) (r (S : ℝ)) ≤ (2*y⁻¹)*dist (T : ℝ) (S : ℝ) at htime + rw [heq,her] + calc + dist (q T) (r S) ≤ dist (q T) (r T)+dist (r T) (r S) := dist_triangle _ _ _ + _ ≤ gronwallBound ((L : ℝ)*dist T S) (4*y⁻¹^2) + (2*y⁻¹^2*((L : ℝ)*dist T S)) R + (2*y⁻¹)*dist T S := by + exact add_le_add (hsame.trans hmono) (by simpa only [NNReal.coe_mk,NNReal.dist_eq,Real.dist_eq] using htime) + _ = _ := by rw [gronwallBound_scale_initial]; ring + +lemma continuous_raw_radial_time (b : Driver) {L : ℝ≥0} + (hb : LipschitzWith L (driverValue b)) {y : ℝ} (hy : 0 < y) : + Continuous (fun T : ℝ≥0 => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) := by + have hlocal (R : ℝ≥0) : ContinuousOn + (fun T : ℝ≥0 => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) (Iic R) := by + have hpos : 0 ≤ (L : ℝ)*gronwallBound 1 (4*y⁻¹^2) (2*y⁻¹^2) R+2*y⁻¹ := by + have hg := gronwallBound_mono (by norm_num : (0 : ℝ) ≤ 1) + (show 0 ≤ 2*y⁻¹^2 by positivity) (show 0 ≤ 4*y⁻¹^2 by positivity) + R.coe_nonneg + rw [gronwallBound_x0] at hg + positivity + apply LipschitzOnWith.continuousOn (K := ⟨_,hpos⟩) + apply LipschitzOnWith.of_dist_le_mul + intro T hT S hS + exact raw_radial_time_distance b hb R T S hT hS hy + apply continuous_iff_continuousAt.mpr + intro T + exact (hlocal (T+1)).continuousAt (Iic_mem_nhds (by simp)) + +end Target +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RealBoundaryDriverNeighborhood.lean b/lean/OAI/Probability/SLE/Loewner/RealBoundaryDriverNeighborhood.lean index 6ad28f47669e8b3261550ee97bb14de428621961..fdb209d1d90aa3527f914200ee6201705b28c995 100644 --- a/lean/OAI/Probability/SLE/Loewner/RealBoundaryDriverNeighborhood.lean +++ b/lean/OAI/Probability/SLE/Loewner/RealBoundaryDriverNeighborhood.lean @@ -1,115 +1,115 @@ -import OAI.Probability.SLE.Loewner.RightFieldDriverStability - -namespace OAI - -/-! A regular real Loewner trajectory has a whole surviving half-disk -which persists under small perturbations of the driving function. -/ -noncomputable section -open Set Filter Complex Metric MeasureTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem real_boundary_driver_neighborhood_control - (V : ℝ≥0 → ℝ) (T : ℝ≥0) (q : ℝ → ℝ) - {m : ℝ} (hm : 0 < m) - (hgap : ∀ t ∈ Icc 0 (T : ℝ), m ≤ q t-V t.toNNReal) - (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q - (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) : - ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ), Continuous U → - ∀ (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → - (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → - ∀ z ∈ ball ((q 0 : ℝ) : ℂ) δ ∩ upperHalfPlane, - z ∈ survivingDomain γ T ∧ ∀ t : ℝ≥0, t ≤ T → - dist (G t z) (q t : ℂ) < m/4 := by - let a : ℝ := m/4 - have ha : 0 < a := by dsimp [a]; positivity - let K : ℝ := 4*a⁻¹^2 - let C : ℝ := 2*a⁻¹^2 - have hcont : ContinuousAt (fun δ : ℝ => gronwallBound δ K (C*δ) T) 0 := by - unfold gronwallBound - split_ifs <;> fun_prop - obtain ⟨η,hη,hsmall⟩ := Metric.continuousAt_iff.mp hcont (m/4) (by positivity) - let δ := min (η/2) (m/4) - have hδ : 0 < δ := lt_min (by positivity) (by positivity) - have hδη : δ < η := (min_le_left _ _).trans_lt (half_lt_self hη) - have hδm : δ ≤ m/4 := min_le_right _ _ - have hb : gronwallBound δ K (C*δ) T < m/4 := by - have hh := hsmall (show dist δ 0 < η by - simpa only [Real.dist_eq,sub_zero,abs_of_pos hδ] using hδη) - simp only [mul_zero,gronwallBound_ε0_δ0,Real.dist_eq,sub_zero] at hh - exact (le_abs_self _).trans_lt hh - refine ⟨δ,hδ,?_⟩ - intro U hU γ G F hw hUV z hz - have hu : Continuous (fun t : ℝ => U t.toNNReal) := hU.comp continuous_real_toNNReal - obtain ⟨Y,_,hY0,hYd⟩ := exists_measurable_right_flow - (fun t (_ : Unit) => U t.toNNReal) (hu.measurable.comp measurable_fst) - (fun _ => hu) (fun _ => z) measurable_const ha - let Z : ℝ → ℂ := fun t => Y t () - have hZ0 : Z 0=z := hY0 () - have hZ (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt Z - (rightField a (U t.toNNReal) (Z t)) (Icc 0 (T : ℝ)) t := - (hYd () t ht.1).mono Icc_subset_Ici_self - have hQ (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt - (fun s => (q s : ℂ)) (rightField a (V t.toNNReal) (q t)) (Icc 0 (T : ℝ)) t := by - rw [rightField_eq (show a ≤ (q t : ℂ).re-V t.toNNReal by - simp only [Complex.ofReal_re]; dsimp [a]; linarith [hgap t ht])] - have hh := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t (hqd t ht) - simpa only [Function.comp_def,Complex.ofRealCLM_apply,Complex.ofReal_div, - Complex.ofReal_ofNat,Complex.ofReal_sub] using hh - have hclose (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : dist (Z t) (q t : ℂ) < m/4 := by - have he := right_flow_driver_distance ha (fun s => U s.toNNReal) - (fun s => V s.toNNReal) Z (fun s => (q s : ℂ)) T T.coe_nonneg hZ hQ δ - (by rw [hZ0]; exact (mem_ball.mp hz.1).le) - (fun s hs => hUV s.toNNReal ⟨zero_le,(Real.toNNReal_le_iff_le_coe).mpr hs.2⟩) t ht - have hmono := gronwallBound_mono hδ.le - (show 0 ≤ C*δ by dsimp [C]; positivity) (show 0 ≤ K by dsimp [K]; positivity) ht.2 - exact (he.trans hmono).trans_lt hb - have hclip (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : a ≤ (Z t).re-U t.toNNReal := by - have hr := Complex.abs_re_le_norm (Z t-(q t : ℂ)) - have he := hclose t ht - rw [dist_eq_norm] at he - have hreal : |(Z t).re-q t| < m/4 := by - simpa only [Complex.sub_re,Complex.ofReal_re] using hr.trans_lt he - have hd := hUV t.toNNReal ⟨zero_le,(Real.toNNReal_le_iff_le_coe).mpr ht.2⟩ - dsimp [a] - linarith [(abs_lt.mp hreal).1,(abs_le.mp hd).2,hgap t ht] - have hpos (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : 0 < (Z t).im := by - apply right_flow_im_pos hu ha ht.1 - · intro s hs - exact (hZ s ⟨hs.1,hs.2.trans ht.2⟩).mono - (Icc_subset_Icc le_rfl ht.2) - · simpa only [hZ0,upperHalfPlane,mem_ofPred_eq] using hz.2 - have hode (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt Z - (2/(Z t-(U t.toNNReal : ℂ))) (Icc 0 (T : ℝ)) t := by - rw [← rightField_eq (hclip t ht)] - exact hZ t ht - have hs := (hw.forward_flow_survives T Z hpos hode).1 - refine ⟨by simpa only [hZ0] using hs,?_⟩ - intro t ht - have hsub : Icc 0 (t : ℝ) ⊆ Icc 0 (T : ℝ) := - Icc_subset_Icc le_rfl (by exact_mod_cast ht) - have he := (hw.forward_flow_survives t Z - (fun s hs => hpos s (hsub hs)) - (fun s hs => (hode s (hsub hs)).mono hsub)).2 - rw [hZ0] at he - rw [he] - exact hclose t ⟨t.coe_nonneg,by exact_mod_cast ht⟩ - -theorem real_boundary_driver_neighborhood - (V : ℝ≥0 → ℝ) (T : ℝ≥0) (q : ℝ → ℝ) {m : ℝ} (hm : 0 < m) - (hgap : ∀ t ∈ Icc 0 (T : ℝ), m ≤ q t-V t.toNNReal) - (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q - (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) : - ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ), Continuous U → - ∀ (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → - (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → - ball ((q 0 : ℝ) : ℂ) δ ∩ upperHalfPlane ⊆ survivingDomain γ T := by - obtain ⟨δ,hδ,hh⟩ := real_boundary_driver_neighborhood_control V T q hm hgap hqd - exact ⟨δ,hδ,fun U hU γ G F hw hUV z hz => (hh U hU γ G F hw hUV z hz).1⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RightFieldDriverStability + +namespace OAI + +/-! A regular real Loewner trajectory has a whole surviving half-disk +which persists under small perturbations of the driving function. -/ +noncomputable section +open Set Filter Complex Metric MeasureTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem real_boundary_driver_neighborhood_control + (V : ℝ≥0 → ℝ) (T : ℝ≥0) (q : ℝ → ℝ) + {m : ℝ} (hm : 0 < m) + (hgap : ∀ t ∈ Icc 0 (T : ℝ), m ≤ q t-V t.toNNReal) + (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q + (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) : + ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ), Continuous U → + ∀ (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → + (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → + ∀ z ∈ ball ((q 0 : ℝ) : ℂ) δ ∩ upperHalfPlane, + z ∈ survivingDomain γ T ∧ ∀ t : ℝ≥0, t ≤ T → + dist (G t z) (q t : ℂ) < m/4 := by + let a : ℝ := m/4 + have ha : 0 < a := by dsimp [a]; positivity + let K : ℝ := 4*a⁻¹^2 + let C : ℝ := 2*a⁻¹^2 + have hcont : ContinuousAt (fun δ : ℝ => gronwallBound δ K (C*δ) T) 0 := by + unfold gronwallBound + split_ifs <;> fun_prop + obtain ⟨η,hη,hsmall⟩ := Metric.continuousAt_iff.mp hcont (m/4) (by positivity) + let δ := min (η/2) (m/4) + have hδ : 0 < δ := lt_min (by positivity) (by positivity) + have hδη : δ < η := (min_le_left _ _).trans_lt (half_lt_self hη) + have hδm : δ ≤ m/4 := min_le_right _ _ + have hb : gronwallBound δ K (C*δ) T < m/4 := by + have hh := hsmall (show dist δ 0 < η by + simpa only [Real.dist_eq,sub_zero,abs_of_pos hδ] using hδη) + simp only [mul_zero,gronwallBound_ε0_δ0,Real.dist_eq,sub_zero] at hh + exact (le_abs_self _).trans_lt hh + refine ⟨δ,hδ,?_⟩ + intro U hU γ G F hw hUV z hz + have hu : Continuous (fun t : ℝ => U t.toNNReal) := hU.comp continuous_real_toNNReal + obtain ⟨Y,_,hY0,hYd⟩ := exists_measurable_right_flow + (fun t (_ : Unit) => U t.toNNReal) (hu.measurable.comp measurable_fst) + (fun _ => hu) (fun _ => z) measurable_const ha + let Z : ℝ → ℂ := fun t => Y t () + have hZ0 : Z 0=z := hY0 () + have hZ (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt Z + (rightField a (U t.toNNReal) (Z t)) (Icc 0 (T : ℝ)) t := + (hYd () t ht.1).mono Icc_subset_Ici_self + have hQ (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt + (fun s => (q s : ℂ)) (rightField a (V t.toNNReal) (q t)) (Icc 0 (T : ℝ)) t := by + rw [rightField_eq (show a ≤ (q t : ℂ).re-V t.toNNReal by + simp only [Complex.ofReal_re]; dsimp [a]; linarith [hgap t ht])] + have hh := Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt t (hqd t ht) + simpa only [Function.comp_def,Complex.ofRealCLM_apply,Complex.ofReal_div, + Complex.ofReal_ofNat,Complex.ofReal_sub] using hh + have hclose (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : dist (Z t) (q t : ℂ) < m/4 := by + have he := right_flow_driver_distance ha (fun s => U s.toNNReal) + (fun s => V s.toNNReal) Z (fun s => (q s : ℂ)) T T.coe_nonneg hZ hQ δ + (by rw [hZ0]; exact (mem_ball.mp hz.1).le) + (fun s hs => hUV s.toNNReal ⟨zero_le,(Real.toNNReal_le_iff_le_coe).mpr hs.2⟩) t ht + have hmono := gronwallBound_mono hδ.le + (show 0 ≤ C*δ by dsimp [C]; positivity) (show 0 ≤ K by dsimp [K]; positivity) ht.2 + exact (he.trans hmono).trans_lt hb + have hclip (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : a ≤ (Z t).re-U t.toNNReal := by + have hr := Complex.abs_re_le_norm (Z t-(q t : ℂ)) + have he := hclose t ht + rw [dist_eq_norm] at he + have hreal : |(Z t).re-q t| < m/4 := by + simpa only [Complex.sub_re,Complex.ofReal_re] using hr.trans_lt he + have hd := hUV t.toNNReal ⟨zero_le,(Real.toNNReal_le_iff_le_coe).mpr ht.2⟩ + dsimp [a] + linarith [(abs_lt.mp hreal).1,(abs_le.mp hd).2,hgap t ht] + have hpos (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : 0 < (Z t).im := by + apply right_flow_im_pos hu ha ht.1 + · intro s hs + exact (hZ s ⟨hs.1,hs.2.trans ht.2⟩).mono + (Icc_subset_Icc le_rfl ht.2) + · simpa only [hZ0,upperHalfPlane,mem_ofPred_eq] using hz.2 + have hode (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt Z + (2/(Z t-(U t.toNNReal : ℂ))) (Icc 0 (T : ℝ)) t := by + rw [← rightField_eq (hclip t ht)] + exact hZ t ht + have hs := (hw.forward_flow_survives T Z hpos hode).1 + refine ⟨by simpa only [hZ0] using hs,?_⟩ + intro t ht + have hsub : Icc 0 (t : ℝ) ⊆ Icc 0 (T : ℝ) := + Icc_subset_Icc le_rfl (by exact_mod_cast ht) + have he := (hw.forward_flow_survives t Z + (fun s hs => hpos s (hsub hs)) + (fun s hs => (hode s (hsub hs)).mono hsub)).2 + rw [hZ0] at he + rw [he] + exact hclose t ⟨t.coe_nonneg,by exact_mod_cast ht⟩ + +theorem real_boundary_driver_neighborhood + (V : ℝ≥0 → ℝ) (T : ℝ≥0) (q : ℝ → ℝ) {m : ℝ} (hm : 0 < m) + (hgap : ∀ t ∈ Icc 0 (T : ℝ), m ≤ q t-V t.toNNReal) + (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q + (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) : + ∃ δ : ℝ, 0 < δ ∧ ∀ (U : ℝ≥0 → ℝ), Continuous U → + ∀ (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → + (∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) → + ball ((q 0 : ℝ) : ℂ) δ ∩ upperHalfPlane ⊆ survivingDomain γ T := by + obtain ⟨δ,hδ,hh⟩ := real_boundary_driver_neighborhood_control V T q hm hgap hqd + exact ⟨δ,hδ,fun U hU γ G F hw hUV z hz => (hh U hU γ G F hw hUV z hz).1⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RealBoundaryTraceAvoidance.lean b/lean/OAI/Probability/SLE/Loewner/RealBoundaryTraceAvoidance.lean index b746e4050ba3b8939709c9019cfbe32935f7fd1c..c4f2d66023fd0d4159f076f6512f61bf396adbab 100644 --- a/lean/OAI/Probability/SLE/Loewner/RealBoundaryTraceAvoidance.lean +++ b/lean/OAI/Probability/SLE/Loewner/RealBoundaryTraceAvoidance.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Loewner.RealBoundaryDriverNeighborhood - -namespace OAI - -/-! A regular real characteristic excludes an entire neighborhood from -the perturbed trace, including its real boundary points. -/ -noncomputable section -open Set Filter Complex Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem real_boundary_driver_trace_avoidance - (V : ℝ≥0 → ℝ) (T : ℝ≥0) (q : ℝ → ℝ) {m : ℝ} (hm : 0 < m) - (hgap : ∀ t ∈ Icc 0 (T : ℝ), m ≤ q t-V t.toNNReal) - (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q - (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) : - ∃ ε : ℝ, 0 < ε ∧ ∀ (U : ℝ≥0 → ℝ), Continuous U → - ∀ (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → - (∀ t ∈ Icc 0 T, |U t-V t| ≤ ε) → - ∀ t ∈ Icc 0 T, γ t ∉ ball ((q 0 : ℝ) : ℂ) ε := by - obtain ⟨δ,hδ,hcontrol⟩ := real_boundary_driver_neighborhood_control V T q hm hgap hqd - let ε := min δ (m/4) - have hε : 0 < ε := lt_min hδ (by positivity) - refine ⟨ε,hε,?_⟩ - intro U hU γ G F hw hUV t ht hball - have hUVδ : ∀ s ∈ Icc 0 T, |U s-V s| ≤ δ := - fun s hs => (hUV s hs).trans (min_le_left _ _) - have hballδ : γ t ∈ ball ((q 0 : ℝ) : ℂ) δ := - (ball_subset_ball (min_le_left _ _)) hball - have hnear : ∀ᶠ y : ℝ in 𝓝[>] 0, - F t ((U t : ℂ)+(y : ℂ)*I) ∈ ball ((q 0 : ℝ) : ℂ) δ := - (hw.2.2.2.2.2.2.2.2.2.2.2.2 t).eventually (isOpen_ball.mem_nhds hballδ) - have he : ∀ᶠ y : ℝ in 𝓝[>] 0, - dist ((U t : ℂ)+(y : ℂ)*I) (q t : ℂ) ≤ m/4 := by - filter_upwards [hnear,self_mem_nhdsWithin] with y hy hy0 - have hwH : (U t : ℂ)+(y : ℂ)*I ∈ upperHalfPlane := by - simpa [upperHalfPlane] using hy0 - have hzH := (hw.2.2.2.2.2.2.1 t hwH).1.1 - have hh := (hcontrol U hU γ G F hw hUVδ _ ⟨hy,hzH⟩).2 t ht.2 - rw [hw.2.2.2.2.2.2.2.2.1 t _ hwH] at hh - exact hh.le - have hd : Tendsto (fun y : ℝ => dist ((U t : ℂ)+(y : ℂ)*I) (q t : ℂ)) - (𝓝[>] 0) (𝓝 (dist (U t : ℂ) (q t : ℂ))) := by - have hc : Continuous (fun y : ℝ => dist ((U t : ℂ)+(y : ℂ)*I) (q t : ℂ)) := by fun_prop - simpa only [Complex.ofReal_zero,zero_mul,add_zero] using - (hc.tendsto 0).mono_left nhdsWithin_le_nhds - have hb := le_of_tendsto hd he - have hr : |U t-q t| ≤ m/4 := by - simpa only [dist_eq_norm,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs] using hb - have hg : m ≤ q t-V t := by - simpa only [Real.toNNReal_coe] using hgap t ⟨t.coe_nonneg,by exact_mod_cast ht.2⟩ - have hu : |U t-V t| ≤ m/4 := (hUV t ht).trans (min_le_right _ _) - linarith [(abs_le.mp hr).1,(abs_le.mp hu).2] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RealBoundaryDriverNeighborhood + +namespace OAI + +/-! A regular real characteristic excludes an entire neighborhood from +the perturbed trace, including its real boundary points. -/ +noncomputable section +open Set Filter Complex Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem real_boundary_driver_trace_avoidance + (V : ℝ≥0 → ℝ) (T : ℝ≥0) (q : ℝ → ℝ) {m : ℝ} (hm : 0 < m) + (hgap : ∀ t ∈ Icc 0 (T : ℝ), m ≤ q t-V t.toNNReal) + (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q + (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) : + ∃ ε : ℝ, 0 < ε ∧ ∀ (U : ℝ≥0 → ℝ), Continuous U → + ∀ (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → + (∀ t ∈ Icc 0 T, |U t-V t| ≤ ε) → + ∀ t ∈ Icc 0 T, γ t ∉ ball ((q 0 : ℝ) : ℂ) ε := by + obtain ⟨δ,hδ,hcontrol⟩ := real_boundary_driver_neighborhood_control V T q hm hgap hqd + let ε := min δ (m/4) + have hε : 0 < ε := lt_min hδ (by positivity) + refine ⟨ε,hε,?_⟩ + intro U hU γ G F hw hUV t ht hball + have hUVδ : ∀ s ∈ Icc 0 T, |U s-V s| ≤ δ := + fun s hs => (hUV s hs).trans (min_le_left _ _) + have hballδ : γ t ∈ ball ((q 0 : ℝ) : ℂ) δ := + (ball_subset_ball (min_le_left _ _)) hball + have hnear : ∀ᶠ y : ℝ in 𝓝[>] 0, + F t ((U t : ℂ)+(y : ℂ)*I) ∈ ball ((q 0 : ℝ) : ℂ) δ := + (hw.2.2.2.2.2.2.2.2.2.2.2.2 t).eventually (isOpen_ball.mem_nhds hballδ) + have he : ∀ᶠ y : ℝ in 𝓝[>] 0, + dist ((U t : ℂ)+(y : ℂ)*I) (q t : ℂ) ≤ m/4 := by + filter_upwards [hnear,self_mem_nhdsWithin] with y hy hy0 + have hwH : (U t : ℂ)+(y : ℂ)*I ∈ upperHalfPlane := by + simpa [upperHalfPlane] using hy0 + have hzH := (hw.2.2.2.2.2.2.1 t hwH).1.1 + have hh := (hcontrol U hU γ G F hw hUVδ _ ⟨hy,hzH⟩).2 t ht.2 + rw [hw.2.2.2.2.2.2.2.2.1 t _ hwH] at hh + exact hh.le + have hd : Tendsto (fun y : ℝ => dist ((U t : ℂ)+(y : ℂ)*I) (q t : ℂ)) + (𝓝[>] 0) (𝓝 (dist (U t : ℂ) (q t : ℂ))) := by + have hc : Continuous (fun y : ℝ => dist ((U t : ℂ)+(y : ℂ)*I) (q t : ℂ)) := by fun_prop + simpa only [Complex.ofReal_zero,zero_mul,add_zero] using + (hc.tendsto 0).mono_left nhdsWithin_le_nhds + have hb := le_of_tendsto hd he + have hr : |U t-q t| ≤ m/4 := by + simpa only [dist_eq_norm,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs] using hb + have hg : m ≤ q t-V t := by + simpa only [Real.toNNReal_coe] using hgap t ⟨t.coe_nonneg,by exact_mod_cast ht.2⟩ + have hu : |U t-V t| ≤ m/4 := (hUV t ht).trans (min_le_right _ _) + linarith [(abs_le.mp hr).1,(abs_le.mp hu).2] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RealCharacteristicAvoidance.lean b/lean/OAI/Probability/SLE/Loewner/RealCharacteristicAvoidance.lean index 9385328fb87dff448181aa7059fb9f108cd61a79..428049d3aad4f7fadbb5b707ea59d85b0bf847fc 100644 --- a/lean/OAI/Probability/SLE/Loewner/RealCharacteristicAvoidance.lean +++ b/lean/OAI/Probability/SLE/Loewner/RealCharacteristicAvoidance.lean @@ -1,85 +1,85 @@ -import OAI.Probability.SLE.Geometry.StableTraceAvoidance - -namespace OAI - -/-! Both signs of a noncolliding real characteristic give stable trace -avoidance. A continuous real characteristic cannot change sides of the -driver without colliding with it. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma StableTraceAvoidance.of_mirror - {V : ℝ≥0 → ℝ} {T : ℝ≥0} {z : ℂ} - (h : StableTraceAvoidance (fun t => -V t) T (loewnerMirror z)) : - StableTraceAvoidance V T z := by - obtain ⟨r,δ,hr,hδ,hh⟩ := h - refine ⟨r,δ,hr,hδ,?_⟩ - intro U hU γ G F hw hUV t ht hb - have hneg : ∀ s ∈ Icc 0 T, |-U s-(-V s)| ≤ δ := by - intro s hs - simpa only [neg_sub_neg,abs_sub_comm] using hUV s hs - apply hh (fun t => -U t) hU.neg (loewnerMirror ∘ γ) - (fun t z => loewnerMirror (G t (loewnerMirror z))) - (fun t z => loewnerMirror (F t (loewnerMirror z))) hw.mirror - hneg t ht - simpa only [Function.comp_apply,mem_ball,loewnerMirror_dist] using hb - -lemma real_characteristic_one_side - (V : ℝ≥0 → ℝ) (hV : Continuous V) (T : ℝ≥0) (q : ℝ → ℝ) - (hq : ContinuousOn q (Icc 0 (T : ℝ))) - (hn : ∀ t ∈ Icc 0 (T : ℝ), q t-V t.toNNReal ≠ 0) - (h0 : 0 < q 0-V 0) : - ∀ t ∈ Icc 0 (T : ℝ), 0 < q t-V t.toNNReal := by - intro t ht - by_contra hh - have hc : ContinuousOn (fun s => -(q s-V s.toNNReal)) (Icc 0 (T : ℝ)) := - (hq.sub (hV.comp continuous_real_toNNReal).continuousOn).neg - obtain ⟨s,hs,he⟩ := intermediate_value_Icc ht.1 - (hc.mono (Icc_subset_Icc le_rfl ht.2)) - (show (0 : ℝ) ∈ Icc (-(q 0-V (Real.toNNReal 0))) (-(q t-V t.toNNReal)) by - simp only [Real.toNNReal_zero] - exact ⟨by linarith,by linarith⟩) - exact hn s ⟨hs.1,hs.2.trans ht.2⟩ (neg_eq_zero.mp he) - -theorem stableTraceAvoidance_of_real_characteristic - (V : ℝ≥0 → ℝ) (hV : Continuous V) (T : ℝ≥0) (q : ℝ → ℝ) - (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q - (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) - (hn : ∀ t ∈ Icc 0 (T : ℝ), q t-V t.toNNReal ≠ 0) : - StableTraceAvoidance V T (q 0 : ℂ) := by - have hq : ContinuousOn q (Icc 0 (T : ℝ)) := - fun t ht => (hqd t ht).continuousWithinAt - have h0 : q 0-V 0 ≠ 0 := by simpa only [Real.toNNReal_zero] using hn 0 ⟨le_rfl,T.coe_nonneg⟩ - rcases lt_or_gt_of_ne h0 with hneg | hpos - · have hnd (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt - (fun s => -q s) (2/(-q t-(-V t.toNNReal))) (Icc 0 (T : ℝ)) t := by - rw [show -q t-(-V t.toNNReal)=-(q t-V t.toNNReal) by ring,div_neg] - exact (hqd t ht).neg - have hnn (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : -q t-(-V t.toNNReal) ≠ 0 := by - rw [show -q t-(-V t.toNNReal)=-(q t-V t.toNNReal) by ring] - exact neg_ne_zero.mpr (hn t ht) - have hside := real_characteristic_one_side (fun t => -V t) hV.neg T - (fun s => -q s) hq.neg hnn (by linarith) - have hc : ContinuousOn (fun s => -q s-(-V s.toNNReal)) (Icc 0 (T : ℝ)) := - hq.neg.sub ((hV.comp continuous_real_toNNReal).neg.continuousOn) - obtain ⟨s,hs,hmin⟩ := isCompact_Icc.exists_isMinOn - (nonempty_Icc.mpr T.coe_nonneg) hc - have hp := stableTraceAvoidance_of_real_trajectory (fun t => -V t) T (fun s => -q s) - (hside s hs) hmin hnd - apply StableTraceAvoidance.of_mirror - simpa only [loewnerMirror_apply,Complex.conj_ofReal,Complex.ofReal_neg] using hp - · have hside := real_characteristic_one_side V hV T q hq hn hpos - have hc : ContinuousOn (fun s => q s-V s.toNNReal) (Icc 0 (T : ℝ)) := - hq.sub (hV.comp continuous_real_toNNReal).continuousOn - obtain ⟨s,hs,hmin⟩ := isCompact_Icc.exists_isMinOn - (nonempty_Icc.mpr T.coe_nonneg) hc - exact stableTraceAvoidance_of_real_trajectory V T q (hside s hs) hmin hqd - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.StableTraceAvoidance + +namespace OAI + +/-! Both signs of a noncolliding real characteristic give stable trace +avoidance. A continuous real characteristic cannot change sides of the +driver without colliding with it. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma StableTraceAvoidance.of_mirror + {V : ℝ≥0 → ℝ} {T : ℝ≥0} {z : ℂ} + (h : StableTraceAvoidance (fun t => -V t) T (loewnerMirror z)) : + StableTraceAvoidance V T z := by + obtain ⟨r,δ,hr,hδ,hh⟩ := h + refine ⟨r,δ,hr,hδ,?_⟩ + intro U hU γ G F hw hUV t ht hb + have hneg : ∀ s ∈ Icc 0 T, |-U s-(-V s)| ≤ δ := by + intro s hs + simpa only [neg_sub_neg,abs_sub_comm] using hUV s hs + apply hh (fun t => -U t) hU.neg (loewnerMirror ∘ γ) + (fun t z => loewnerMirror (G t (loewnerMirror z))) + (fun t z => loewnerMirror (F t (loewnerMirror z))) hw.mirror + hneg t ht + simpa only [Function.comp_apply,mem_ball,loewnerMirror_dist] using hb + +lemma real_characteristic_one_side + (V : ℝ≥0 → ℝ) (hV : Continuous V) (T : ℝ≥0) (q : ℝ → ℝ) + (hq : ContinuousOn q (Icc 0 (T : ℝ))) + (hn : ∀ t ∈ Icc 0 (T : ℝ), q t-V t.toNNReal ≠ 0) + (h0 : 0 < q 0-V 0) : + ∀ t ∈ Icc 0 (T : ℝ), 0 < q t-V t.toNNReal := by + intro t ht + by_contra hh + have hc : ContinuousOn (fun s => -(q s-V s.toNNReal)) (Icc 0 (T : ℝ)) := + (hq.sub (hV.comp continuous_real_toNNReal).continuousOn).neg + obtain ⟨s,hs,he⟩ := intermediate_value_Icc ht.1 + (hc.mono (Icc_subset_Icc le_rfl ht.2)) + (show (0 : ℝ) ∈ Icc (-(q 0-V (Real.toNNReal 0))) (-(q t-V t.toNNReal)) by + simp only [Real.toNNReal_zero] + exact ⟨by linarith,by linarith⟩) + exact hn s ⟨hs.1,hs.2.trans ht.2⟩ (neg_eq_zero.mp he) + +theorem stableTraceAvoidance_of_real_characteristic + (V : ℝ≥0 → ℝ) (hV : Continuous V) (T : ℝ≥0) (q : ℝ → ℝ) + (hqd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q + (2/(q t-V t.toNNReal)) (Icc 0 (T : ℝ)) t) + (hn : ∀ t ∈ Icc 0 (T : ℝ), q t-V t.toNNReal ≠ 0) : + StableTraceAvoidance V T (q 0 : ℂ) := by + have hq : ContinuousOn q (Icc 0 (T : ℝ)) := + fun t ht => (hqd t ht).continuousWithinAt + have h0 : q 0-V 0 ≠ 0 := by simpa only [Real.toNNReal_zero] using hn 0 ⟨le_rfl,T.coe_nonneg⟩ + rcases lt_or_gt_of_ne h0 with hneg | hpos + · have hnd (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt + (fun s => -q s) (2/(-q t-(-V t.toNNReal))) (Icc 0 (T : ℝ)) t := by + rw [show -q t-(-V t.toNNReal)=-(q t-V t.toNNReal) by ring,div_neg] + exact (hqd t ht).neg + have hnn (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : -q t-(-V t.toNNReal) ≠ 0 := by + rw [show -q t-(-V t.toNNReal)=-(q t-V t.toNNReal) by ring] + exact neg_ne_zero.mpr (hn t ht) + have hside := real_characteristic_one_side (fun t => -V t) hV.neg T + (fun s => -q s) hq.neg hnn (by linarith) + have hc : ContinuousOn (fun s => -q s-(-V s.toNNReal)) (Icc 0 (T : ℝ)) := + hq.neg.sub ((hV.comp continuous_real_toNNReal).neg.continuousOn) + obtain ⟨s,hs,hmin⟩ := isCompact_Icc.exists_isMinOn + (nonempty_Icc.mpr T.coe_nonneg) hc + have hp := stableTraceAvoidance_of_real_trajectory (fun t => -V t) T (fun s => -q s) + (hside s hs) hmin hnd + apply StableTraceAvoidance.of_mirror + simpa only [loewnerMirror_apply,Complex.conj_ofReal,Complex.ofReal_neg] using hp + · have hside := real_characteristic_one_side V hV T q hq hn hpos + have hc : ContinuousOn (fun s => q s-V s.toNNReal) (Icc 0 (T : ℝ)) := + hq.sub (hV.comp continuous_real_toNNReal).continuousOn + obtain ⟨s,hs,hmin⟩ := isCompact_Icc.exists_isMinOn + (nonempty_Icc.mpr T.coe_nonneg) hc + exact stableTraceAvoidance_of_real_trajectory V T q (hside s hs) hmin hqd + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReflectedCapacityChart.lean b/lean/OAI/Probability/SLE/Loewner/ReflectedCapacityChart.lean index ad0fd925a6d90e8878b7652d5b0c6eeb5ae00703..408598876809c46ca09ae522c5686776312b9f6d 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReflectedCapacityChart.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReflectedCapacityChart.lean @@ -1,94 +1,94 @@ -import OAI.Probability.SLE.Loewner.SlitChartUniqueness - -namespace OAI - -/-! Retain the reflected Riemann-map data of a normalized slit chart, so -that its analytic continuation outside the reflected slit is available -for the local evolution estimates. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -structure ReflectedCapacityChart (p : C(unitInterval,ℂ)) where - chart : SimpleSlitChart p - diskMap : ℂ → ℂ - slope : ℝ - slope_pos : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) := by - have hh := slitExteriorMap_laurent_limit (hf.analyticAt hD) hf0 ha.ne' hfa - have he := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD - (dslope_reflection hD (slitInfinityFactor_reflection hD hs a))) - rw [he] at hh - exact hh - have hc0 : 0≤c := by - apply hydrodynamic_coefficient_nonneg hc - rw [eventually_inf_principal] - filter_upwards [(isCompact_range p.continuous).compl_mem_cocompact] with z hz - exact fun hzH => hheight z ⟨hzH,hz⟩ - have hcpos : 0 by - change 0≤(F z-z).im - simpa only [sub_im] using sub_nonneg.mpr - (normalized_halfPlane_height hdF (hmF.mono_right sdiff_subset) hn hz)) hfcoef - have hp1 : p 1∈upperHalfPlane := hH 1 (by norm_num) - have hFeq : F (p 1)=p 1 := sub_eq_zero.mp (hh (p 1) hp1) - have hmem := hmF hp1 - rw [hFeq] at hmem - exact hmem.2 (mem_range_self 1) - let Q : SimpleSlitChart p := ⟨G,F,c,hcpos,hG,hdG,hdF,hmF,hl,hr,hc,hn,hheight⟩ - exact ⟨⟨Q,f,a,ha,hb,hf,hf0,hfa,hs,rfl⟩⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartUniqueness + +namespace OAI + +/-! Retain the reflected Riemann-map data of a normalized slit chart, so +that its analytic continuation outside the reflected slit is available +for the local evolution estimates. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +structure ReflectedCapacityChart (p : C(unitInterval,ℂ)) where + chart : SimpleSlitChart p + diskMap : ℂ → ℂ + slope : ℝ + slope_pos : 0 z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) := by + have hh := slitExteriorMap_laurent_limit (hf.analyticAt hD) hf0 ha.ne' hfa + have he := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD + (dslope_reflection hD (slitInfinityFactor_reflection hD hs a))) + rw [he] at hh + exact hh + have hc0 : 0≤c := by + apply hydrodynamic_coefficient_nonneg hc + rw [eventually_inf_principal] + filter_upwards [(isCompact_range p.continuous).compl_mem_cocompact] with z hz + exact fun hzH => hheight z ⟨hzH,hz⟩ + have hcpos : 0 by + change 0≤(F z-z).im + simpa only [sub_im] using sub_nonneg.mpr + (normalized_halfPlane_height hdF (hmF.mono_right sdiff_subset) hn hz)) hfcoef + have hp1 : p 1∈upperHalfPlane := hH 1 (by norm_num) + have hFeq : F (p 1)=p 1 := sub_eq_zero.mp (hh (p 1) hp1) + have hmem := hmF hp1 + rw [hFeq] at hmem + exact hmem.2 (mem_range_self 1) + let Q : SimpleSlitChart p := ⟨G,F,c,hcpos,hG,hdG,hdF,hmF,hl,hr,hc,hn,hheight⟩ + exact ⟨⟨Q,f,a,ha,hb,hf,hf0,hfa,hs,rfl⟩⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReflectedChartLocalBound.lean b/lean/OAI/Probability/SLE/Loewner/ReflectedChartLocalBound.lean index cfbc22d5de17d03162fae32b627c7c95e250cf3c..bb2e523797178f7854f6e3f5746637ff34638547 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReflectedChartLocalBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReflectedChartLocalBound.lean @@ -1,117 +1,117 @@ -import OAI.Probability.SLE.Loewner.ReflectedExteriorAnalytic -import OAI.Probability.SLE.Loewner.UnivalentLocalBound -import OAI.Probability.SLE.Loewner.SlitCapacityRemainder - -namespace OAI - -/-! Coarse uniform control of the reflected chart near a small slit. -Only a fixed numerical constant is needed for driver continuity. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma reflected_disk_ball_subset {p : C(unitInterval,ℂ)} {L : ℝ} (hL : 0 hz (Or.inl ⟨0,h0.trans he.symm⟩) - have hf0 : Q.diskMap z⁻¹≠0 := by - intro he - exact inv_ne_zero hz0 (Q.disk_bijOn.injOn (Q.inverse_mem hz) - (zero_mem_invertedExterior _) (he.trans Q.disk_zero.symm)) - have hfn : 0<‖Q.diskMap z⁻¹‖ := norm_pos_iff.mpr hf0 - change Q.slope*‖(Q.diskMap z⁻¹)⁻¹‖≤(128*Real.pi)*L - rw [norm_inv,←div_eq_mul_inv] - apply (div_le_iff₀ hfn).mpr - have hmul := mul_le_mul_of_nonneg_right ((div_le_iff₀ hC).mp hb) hL.le - have he : (1/L)*Q.slope*L=Q.slope := by field_simp - rw [he] at hmul - nlinarith - -lemma ReflectedCapacityChart.joukowski_scaled_bound {p : C(unitInterval,ℂ)} - (Q : ReflectedCapacityChart p) (h0 : p 0=0) {L : ℝ} (hL : 0 hz (Or.inl ⟨0,h0.trans he.symm⟩) + have hf0 : Q.diskMap z⁻¹≠0 := by + intro he + exact inv_ne_zero hz0 (Q.disk_bijOn.injOn (Q.inverse_mem hz) + (zero_mem_invertedExterior _) (he.trans Q.disk_zero.symm)) + have hfn : 0<‖Q.diskMap z⁻¹‖ := norm_pos_iff.mpr hf0 + change Q.slope*‖(Q.diskMap z⁻¹)⁻¹‖≤(128*Real.pi)*L + rw [norm_inv,←div_eq_mul_inv] + apply (div_le_iff₀ hfn).mpr + have hmul := mul_le_mul_of_nonneg_right ((div_le_iff₀ hC).mp hb) hL.le + have he : (1/L)*Q.slope*L=Q.slope := by field_simp + rw [he] at hmul + nlinarith + +lemma ReflectedCapacityChart.joukowski_scaled_bound {p : C(unitInterval,ℂ)} + (Q : ReflectedCapacityChart p) (h0 : p 0=0) {L : ℝ} (hL : 0 (R.slope:ℂ)* - ((R.diskMap y⁻¹)⁻¹+R.diskMap y⁻¹)-deriv (slitInfinityFactor R.diskMap R.slope) 0) z at hh - rw [R.forward_eq] - have he : slitHydrodynamicMap R.diskMap R.slope = fun y : ℂ => (R.slope:ℂ)* - ((R.diskMap y⁻¹)⁻¹+R.diskMap y⁻¹)-deriv (slitInfinityFactor R.diskMap R.slope) 0 := by - funext y - simp only [slitHydrodynamicMap,exteriorJoukowski,slitExteriorMap,inv_inv] - rw [he] - exact hh.differentiableWithinAt - -theorem ReflectedCapacityChart.forward_injOn {p : C(unitInterval,ℂ)} - (R : ReflectedCapacityChart p) (h0 : p 0=0) : - InjOn R.chart.forward (reflectedArcSet p)ᶜ := by - intro z hz w hw he - rw [R.forward_eq] at he - have hj : exteriorJoukowski (slitExteriorMap R.diskMap z)= - exteriorJoukowski (slitExteriorMap R.diskMap w) := by - apply mul_left_cancel₀ (ofReal_ne_zero.mpr R.slope_pos.ne') - exact sub_left_inj.mp he - have hW := exteriorJoukowski_injOn (R.exterior_norm h0 hz) (R.exterior_norm h0 hw) hj - have hf := inv_injective hW - exact inv_injective (R.disk_bijOn.injOn (R.inverse_mem hz) (R.inverse_mem hw) hf) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReflectedCapacityChart + +namespace OAI + +/-! The retained chart is univalent and holomorphic on the full +complement of the reflected slit, not only on its upper side. -/ +noncomputable section +open Set Function Filter Complex Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +lemma ReflectedCapacityChart.inverse_mem {p : C(unitInterval,ℂ)} (_R : ReflectedCapacityChart p) + {z : ℂ} (hz : z∉reflectedArcSet p) : z⁻¹∈invertedExterior (reflectedArcSet p) := by + exact Or.inr (by simpa only [inv_inv] using hz) + +lemma ReflectedCapacityChart.exterior_norm {p : C(unitInterval,ℂ)} (R : ReflectedCapacityChart p) + (h0 : p 0=0) {z : ℂ} (hz : z∉reflectedArcSet p) : + 1<‖slitExteriorMap R.diskMap z‖ := by + have hz0 : z≠0 := by + intro he + subst z + exact hz (Or.inl ⟨0,h0⟩) + have hm := R.inverse_mem hz + have hn : ‖R.diskMap z⁻¹‖<1 := by + simpa only [mem_ball_zero_iff] using R.disk_bijOn.mapsTo hm + have hf0 : R.diskMap z⁻¹≠0 := by + intro he + have hh := R.disk_bijOn.injOn hm (zero_mem_invertedExterior _) (he.trans R.disk_zero.symm) + exact inv_ne_zero hz0 hh + change 1<‖(R.diskMap z⁻¹)⁻¹‖ + rw [norm_inv] + exact (one_lt_inv₀ (norm_pos_iff.mpr hf0)).mpr hn + +theorem ReflectedCapacityChart.forward_differentiable {p : C(unitInterval,ℂ)} + (R : ReflectedCapacityChart p) (h0 : p 0=0) : + DifferentiableOn ℂ R.chart.forward (reflectedArcSet p)ᶜ := by + intro z hz + have hz0 : z≠0 := by + intro he + subst z + exact hz (Or.inl ⟨0,h0⟩) + have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj) + have hm := R.inverse_mem hz + have hf := (R.disk_differentiable.differentiableAt + ((invertedExterior_isOpen hK).mem_nhds hm)).comp z (differentiableAt_id.inv hz0) + have hf0 : R.diskMap z⁻¹≠0 := by + intro he + exact inv_ne_zero hz0 (R.disk_bijOn.injOn hm (zero_mem_invertedExterior _) + (he.trans R.disk_zero.symm)) + have hh := (((hf.inv hf0).add hf).const_mul (R.slope:ℂ)).sub_const + (deriv (slitInfinityFactor R.diskMap R.slope) 0) + change DifferentiableAt ℂ (fun y : ℂ => (R.slope:ℂ)* + ((R.diskMap y⁻¹)⁻¹+R.diskMap y⁻¹)-deriv (slitInfinityFactor R.diskMap R.slope) 0) z at hh + rw [R.forward_eq] + have he : slitHydrodynamicMap R.diskMap R.slope = fun y : ℂ => (R.slope:ℂ)* + ((R.diskMap y⁻¹)⁻¹+R.diskMap y⁻¹)-deriv (slitInfinityFactor R.diskMap R.slope) 0 := by + funext y + simp only [slitHydrodynamicMap,exteriorJoukowski,slitExteriorMap,inv_inv] + rw [he] + exact hh.differentiableWithinAt + +theorem ReflectedCapacityChart.forward_injOn {p : C(unitInterval,ℂ)} + (R : ReflectedCapacityChart p) (h0 : p 0=0) : + InjOn R.chart.forward (reflectedArcSet p)ᶜ := by + intro z hz w hw he + rw [R.forward_eq] at he + have hj : exteriorJoukowski (slitExteriorMap R.diskMap z)= + exteriorJoukowski (slitExteriorMap R.diskMap w) := by + apply mul_left_cancel₀ (ofReal_ne_zero.mpr R.slope_pos.ne') + exact sub_left_inj.mp he + have hW := exteriorJoukowski_injOn (R.exterior_norm h0 hz) (R.exterior_norm h0 hw) hj + have hf := inv_injective hW + exact inv_injective (R.disk_bijOn.injOn (R.inverse_mem hz) (R.inverse_mem hw) hf) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReflectedExteriorHeight.lean b/lean/OAI/Probability/SLE/Loewner/ReflectedExteriorHeight.lean index 998aba1404d91a459fb9eb5baa0465656a129b73..6ac5440f61919a414ea58be60fa1cabcae96aa10 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReflectedExteriorHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReflectedExteriorHeight.lean @@ -1,54 +1,54 @@ -import OAI.Probability.SLE.Loewner.ReflectedExteriorAnalytic - -namespace OAI - -/-! Reflection supplies the lower-half-plane version of the normalized -height inequality. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -lemma ReflectedCapacityChart.forward_reflection {p : C(unitInterval,ℂ)} - (R : ReflectedCapacityChart p) {z : ℂ} (hz : z∉reflectedArcSet p) : - R.chart.forward (conj z)=conj (R.chart.forward z) := by - have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union - ((isCompact_range p.continuous).image Complex.continuous_conj) - have hD := (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) - have hd := derivative_real_of_reflection hD - (slitInfinityFactor_reflection hD R.disk_reflection R.slope) - have hf : R.diskMap ((conj z)⁻¹)=conj (R.diskMap z⁻¹) := by - simpa only [map_inv₀] using R.disk_reflection z⁻¹ (R.inverse_mem hz) - rw [R.forward_eq] - simp only [slitHydrodynamicMap,exteriorJoukowski,slitExteriorMap, - map_inv₀,inv_inv,map_sub,map_mul,map_add,conj_ofReal,hd,hf] - -lemma ReflectedCapacityChart.height_displacement_sign {p : C(unitInterval,ℂ)} - (R : ReflectedCapacityChart p) {z : ℂ} (hz : z∉reflectedArcSet p) : - z.im*(R.chart.forward z-z).im≤0 := by - rcases lt_trichotomy z.im 0 with hneg|heq|hpos - · have hcz : conj z∉reflectedArcSet p := by - intro hh - have hh' := reflectedArcSet_conj p hh - exact hz (by simpa only [conj_conj] using hh') - have hH : conj z∈upperHalfPlane := by - change 0<(conj z).im - simpa only [conj_im] using neg_pos.mpr hneg - have hn : conj z∉range p := fun hh => hcz (Or.inl hh) - have hh := R.chart.height_le (conj z) ⟨hH,hn⟩ - rw [R.forward_reflection hz] at hh - simp only [conj_im] at hh - apply mul_nonpos_of_nonpos_of_nonneg hneg.le - simp only [sub_im] - linarith - · simp only [heq,zero_mul,le_refl] - · have hn : z∉range p := fun hh => hz (Or.inl hh) - exact mul_nonpos_of_nonneg_of_nonpos hpos.le - (sub_nonpos.mpr (R.chart.height_le z ⟨hpos,hn⟩)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReflectedExteriorAnalytic + +namespace OAI + +/-! Reflection supplies the lower-half-plane version of the normalized +height inequality. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +lemma ReflectedCapacityChart.forward_reflection {p : C(unitInterval,ℂ)} + (R : ReflectedCapacityChart p) {z : ℂ} (hz : z∉reflectedArcSet p) : + R.chart.forward (conj z)=conj (R.chart.forward z) := by + have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj) + have hD := (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) + have hd := derivative_real_of_reflection hD + (slitInfinityFactor_reflection hD R.disk_reflection R.slope) + have hf : R.diskMap ((conj z)⁻¹)=conj (R.diskMap z⁻¹) := by + simpa only [map_inv₀] using R.disk_reflection z⁻¹ (R.inverse_mem hz) + rw [R.forward_eq] + simp only [slitHydrodynamicMap,exteriorJoukowski,slitExteriorMap, + map_inv₀,inv_inv,map_sub,map_mul,map_add,conj_ofReal,hd,hf] + +lemma ReflectedCapacityChart.height_displacement_sign {p : C(unitInterval,ℂ)} + (R : ReflectedCapacityChart p) {z : ℂ} (hz : z∉reflectedArcSet p) : + z.im*(R.chart.forward z-z).im≤0 := by + rcases lt_trichotomy z.im 0 with hneg|heq|hpos + · have hcz : conj z∉reflectedArcSet p := by + intro hh + have hh' := reflectedArcSet_conj p hh + exact hz (by simpa only [conj_conj] using hh') + have hH : conj z∈upperHalfPlane := by + change 0<(conj z).im + simpa only [conj_im] using neg_pos.mpr hneg + have hn : conj z∉range p := fun hh => hcz (Or.inl hh) + have hh := R.chart.height_le (conj z) ⟨hH,hn⟩ + rw [R.forward_reflection hz] at hh + simp only [conj_im] at hh + apply mul_nonpos_of_nonpos_of_nonneg hneg.le + simp only [sub_im] + linarith + · simp only [heq,zero_mul,le_refl] + · have hn : z∉range p := fun hh => hz (Or.inl hh) + exact mul_nonpos_of_nonneg_of_nonpos hpos.le + (sub_nonpos.mpr (R.chart.height_le z ⟨hpos,hn⟩)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReflectedForwardContinuity.lean b/lean/OAI/Probability/SLE/Loewner/ReflectedForwardContinuity.lean index 2c366f3ce3b675097af3c4f9298a3c0b60aaae8f..8b1c7359ac46a157a4b4696eeb1b5230b646d436 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReflectedForwardContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReflectedForwardContinuity.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Loewner.InitialReflectedForward -import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryExtension -import TauCeti.Analysis.Complex.Conformal.NormalFamilies - -namespace OAI - -/-! Uniform analytic bounds carry time continuity of the initial slit -maps from the open half-plane to the surviving real boundary. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma initialReflectedForward_equicontinuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) : - EquicontinuousOn (initialReflectedForward p hi h0 hH) (reflectedArcSet p)ᶜ := by - have hb : TauCeti.IsLocallyBoundedOn (initialReflectedForward p hi h0 hH) - (reflectedArcSet p)ᶜ := by - intro K hKD hK - obtain ⟨A,hA⟩ := hK.isBounded.exists_norm_le - obtain ⟨B,hB⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le - let R := |A|+|B|+1 - have hR : 0 initialReflectedForward p hi h0 hH s (x:ℂ)) := by - have hD : IsOpen (reflectedArcSet p)ᶜ := - ((isCompact_range p.continuous).union - ((isCompact_range p.continuous).image Complex.continuous_conj)).isClosed.isOpen_compl - have hxD : (x:ℂ)∈(reflectedArcSet p)ᶜ := real_not_reflected_arc p h0 hH hx - apply continuous_iff_continuousAt.mpr - intro s - apply (initialReflectedForward_equicontinuousAt_real p hi h0 hH hx).tendsto_of_mem_closure - (((initialReflectedForward_differentiable p hi h0 hH s).differentiableAt - (hD.mem_nhds hxD)).continuousAt.tendsto.mono_left nhdsWithin_le_nhds) - (s := upperHalfPlane ∩ (reflectedArcSet p)ᶜ) - · intro z hz - have hzD (t : unitInterval) : z∈upperHalfPlane \ range (initialSubarc p t) := - ⟨hz.1,fun he => hz.2 (reflected_initial_subset p t (Or.inl he))⟩ - have hc : Continuous (fun t => initialForwardChart p hi h0 hH t z) := by - apply SeqContinuous.continuous - intro t r ht - exact initialForwardChart_tendsto p hi h0 hH ht tendsto_const_nhds (hzD r) (fun n => hzD (t n)) - have he : (fun t => initialReflectedForward p hi h0 hH t z)= - (fun t => initialForwardChart p hi h0 hH t z) := - funext (fun t => initialReflectedForward_eq p hi h0 hH t (hzD t)) - simpa only [he,initialReflectedForward_eq p hi h0 hH s (hzD s)] using hc.continuousAt.tendsto - · exact hD.closure_inter ⟨closedHalfPlane_subset_closure_upperHalfPlane (by simp),hxD⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialReflectedForward +import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryExtension +import TauCeti.Analysis.Complex.Conformal.NormalFamilies + +namespace OAI + +/-! Uniform analytic bounds carry time continuity of the initial slit +maps from the open half-plane to the surviving real boundary. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +lemma initialReflectedForward_equicontinuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) : + EquicontinuousOn (initialReflectedForward p hi h0 hH) (reflectedArcSet p)ᶜ := by + have hb : TauCeti.IsLocallyBoundedOn (initialReflectedForward p hi h0 hH) + (reflectedArcSet p)ᶜ := by + intro K hKD hK + obtain ⟨A,hA⟩ := hK.isBounded.exists_norm_le + obtain ⟨B,hB⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le + let R := |A|+|B|+1 + have hR : 0 initialReflectedForward p hi h0 hH s (x:ℂ)) := by + have hD : IsOpen (reflectedArcSet p)ᶜ := + ((isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj)).isClosed.isOpen_compl + have hxD : (x:ℂ)∈(reflectedArcSet p)ᶜ := real_not_reflected_arc p h0 hH hx + apply continuous_iff_continuousAt.mpr + intro s + apply (initialReflectedForward_equicontinuousAt_real p hi h0 hH hx).tendsto_of_mem_closure + (((initialReflectedForward_differentiable p hi h0 hH s).differentiableAt + (hD.mem_nhds hxD)).continuousAt.tendsto.mono_left nhdsWithin_le_nhds) + (s := upperHalfPlane ∩ (reflectedArcSet p)ᶜ) + · intro z hz + have hzD (t : unitInterval) : z∈upperHalfPlane \ range (initialSubarc p t) := + ⟨hz.1,fun he => hz.2 (reflected_initial_subset p t (Or.inl he))⟩ + have hc : Continuous (fun t => initialForwardChart p hi h0 hH t z) := by + apply SeqContinuous.continuous + intro t r ht + exact initialForwardChart_tendsto p hi h0 hH ht tendsto_const_nhds (hzD r) (fun n => hzD (t n)) + have he : (fun t => initialReflectedForward p hi h0 hH t z)= + (fun t => initialForwardChart p hi h0 hH t z) := + funext (fun t => initialReflectedForward_eq p hi h0 hH t (hzD t)) + simpa only [he,initialReflectedForward_eq p hi h0 hH s (hzD s)] using hc.continuousAt.tendsto + · exact hD.closure_inter ⟨closedHalfPlane_subset_closure_upperHalfPlane (by simp),hxD⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReflectedSlitRiemann.lean b/lean/OAI/Probability/SLE/Loewner/ReflectedSlitRiemann.lean index 0dc2bc2929fdef211d4e6b6b2f0e19322b0145cb..09a989e897bc2a914813e23cdfc0f8cb2a2f56aa 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReflectedSlitRiemann.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReflectedSlitRiemann.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Geometry.InvertedArcComponents -import OAI.Probability.SLE.Geometry.SimpleSlitRiemann - -namespace OAI - -/-! A Riemann chart including infinity for the complement of a reflected -simple slit. Zero in this domain represents infinity before inversion. -/ -noncomputable section -open Set Complex Metric Function -open scoped ComplexConjugate - -namespace SLEExactGauge - -theorem reflected_slit_inverted_simplyConnected - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - IsSimplyConnected (invertedExterior (reflectedArcSet p)) := by - let q : C(unitInterval,ℂ) := ⟨fun t => conj (p t),Complex.continuous_conj.comp p.continuous⟩ - have hq0 : q 0=0 := by simp [q,h0] - have hp : ∀ t : unitInterval, 0 < t → p t ≠ 0 := by - intro t ht he - have hh := hH t ht.ne' - simp only [he,upperHalfPlane,mem_ofPred_eq,zero_im,lt_self_iff_false] at hh - have hq : ∀ t : unitInterval, 0 < t → q t ≠ 0 := by - intro t ht he - apply hp t ht - have hh := congrArg conj he - simpa [q] using hh - have he : reflectedArcSet p=range p ∪ range q := by - unfold reflectedArcSet - congr 1 - exact (range_comp conj p).symm - have hK : IsCompact (reflectedArcSet p) := by - rw [he] - exact (isCompact_range p.continuous).union (isCompact_range q.continuous) - have hK0 : (0:ℂ) ∈ reflectedArcSet p := Or.inl ⟨0,h0⟩ - apply TauCeti.isSimplyConnected_of_filledHull_subset (invertedExterior_isOpen hK) - (invertedExterior_connected hK.isBounded hK0 - (reflectedArc_complement_connected p hi h0 hH)) - intro z hz - by_contra hzD - have hb : Bornology.IsBounded - (connectedComponentIn (invertedExterior (reflectedArcSet p))ᶜ z) := hz - rw [he] at hzD hb - exact invertedExterior_two_arcs_components_unbounded p q h0 hq0 hp hq hzD hb - -theorem reflected_slit_exists_inverted_disk_chart - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ∃ f : ℂ → ℂ, BijOn f (invertedExterior (reflectedArcSet p)) (ball 0 1) ∧ - DifferentiableOn ℂ f (invertedExterior (reflectedArcSet p)) ∧ - DifferentiableOn ℂ (invFunOn f (invertedExterior (reflectedArcSet p))) (ball 0 1) ∧ - LeftInvOn (invFunOn f (invertedExterior (reflectedArcSet p))) f - (invertedExterior (reflectedArcSet p)) ∧ - RightInvOn (invFunOn f (invertedExterior (reflectedArcSet p))) f (ball 0 1) := by - have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union - ((isCompact_range p.continuous).image Complex.continuous_conj) - apply TauCeti.exists_bijOn_ball_differentiableOn_invFunOn (invertedExterior_isOpen hK) - (reflected_slit_inverted_simplyConnected p hi h0 hH) - intro he - have hp1 : p 1 ≠ 0 := by - intro hp1 - have ht := hi (hp1.trans h0.symm) - norm_num at ht - have hm : (p 1)⁻¹ ∈ invertedExterior (reflectedArcSet p) := by rw [he]; trivial - rcases hm with hz|hz - · exact inv_ne_zero hp1 hz - · apply hz - rw [inv_inv] - exact Or.inl (mem_range_self 1) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.InvertedArcComponents +import OAI.Probability.SLE.Geometry.SimpleSlitRiemann + +namespace OAI + +/-! A Riemann chart including infinity for the complement of a reflected +simple slit. Zero in this domain represents infinity before inversion. -/ +noncomputable section +open Set Complex Metric Function +open scoped ComplexConjugate + +namespace SLEExactGauge + +theorem reflected_slit_inverted_simplyConnected + (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + IsSimplyConnected (invertedExterior (reflectedArcSet p)) := by + let q : C(unitInterval,ℂ) := ⟨fun t => conj (p t),Complex.continuous_conj.comp p.continuous⟩ + have hq0 : q 0=0 := by simp [q,h0] + have hp : ∀ t : unitInterval, 0 < t → p t ≠ 0 := by + intro t ht he + have hh := hH t ht.ne' + simp only [he,upperHalfPlane,mem_ofPred_eq,zero_im,lt_self_iff_false] at hh + have hq : ∀ t : unitInterval, 0 < t → q t ≠ 0 := by + intro t ht he + apply hp t ht + have hh := congrArg conj he + simpa [q] using hh + have he : reflectedArcSet p=range p ∪ range q := by + unfold reflectedArcSet + congr 1 + exact (range_comp conj p).symm + have hK : IsCompact (reflectedArcSet p) := by + rw [he] + exact (isCompact_range p.continuous).union (isCompact_range q.continuous) + have hK0 : (0:ℂ) ∈ reflectedArcSet p := Or.inl ⟨0,h0⟩ + apply TauCeti.isSimplyConnected_of_filledHull_subset (invertedExterior_isOpen hK) + (invertedExterior_connected hK.isBounded hK0 + (reflectedArc_complement_connected p hi h0 hH)) + intro z hz + by_contra hzD + have hb : Bornology.IsBounded + (connectedComponentIn (invertedExterior (reflectedArcSet p))ᶜ z) := hz + rw [he] at hzD hb + exact invertedExterior_two_arcs_components_unbounded p q h0 hq0 hp hq hzD hb + +theorem reflected_slit_exists_inverted_disk_chart + (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + ∃ f : ℂ → ℂ, BijOn f (invertedExterior (reflectedArcSet p)) (ball 0 1) ∧ + DifferentiableOn ℂ f (invertedExterior (reflectedArcSet p)) ∧ + DifferentiableOn ℂ (invFunOn f (invertedExterior (reflectedArcSet p))) (ball 0 1) ∧ + LeftInvOn (invFunOn f (invertedExterior (reflectedArcSet p))) f + (invertedExterior (reflectedArcSet p)) ∧ + RightInvOn (invFunOn f (invertedExterior (reflectedArcSet p))) f (ball 0 1) := by + have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj) + apply TauCeti.exists_bijOn_ball_differentiableOn_invFunOn (invertedExterior_isOpen hK) + (reflected_slit_inverted_simplyConnected p hi h0 hH) + intro he + have hp1 : p 1 ≠ 0 := by + intro hp1 + have ht := hi (hp1.trans h0.symm) + norm_num at ht + have hm : (p 1)⁻¹ ∈ invertedExterior (reflectedArcSet p) := by rw [he]; trivial + rcases hm with hz|hz + · exact inv_ne_zero hp1 hz + · apply hz + rw [inv_inv] + exact Or.inl (mem_range_self 1) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RestartDomainEquality.lean b/lean/OAI/Probability/SLE/Loewner/RestartDomainEquality.lean index 3edfe581dd65d64deebf7bc17170ea0cfb50995f..0d567a737a5824d74c584808a1d2af6537db976b 100644 --- a/lean/OAI/Probability/SLE/Loewner/RestartDomainEquality.lean +++ b/lean/OAI/Probability/SLE/Loewner/RestartDomainEquality.lean @@ -1,103 +1,103 @@ -import OAI.Probability.SLE.Geometry.RestartGreenKernel -import OAI.Probability.SLE.Gauge.DenseRadiusGap - -namespace OAI - -/-! Surjectivity of the actual restarted surviving domain. Inverse-map -composition identifies every original future live point with a live fresh -point, so reaching a deeper radius is equivalent in the two charts. -/ - -noncomputable section -open Set Filter -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma IsCapacityTwoWitness.restart_live_preimage - {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) - (hb : IsCapacityTwoWitness (fun t => U (T + t) - U T) β G' F') - (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ (T + t)) : - G T z - (U T : ℂ) ∈ survivingDomain β t := by - let v := G (T + t) z - (U T : ℂ) - have hv : v ∈ upperHalfPlane := by - simpa only [v, upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, sub_zero] - using hw.2.2.2.2.2.1 (T + t) hz - have hvadd : v + (U T : ℂ) = G (T + t) z := sub_add_cancel _ _ - have hcomp := hw.restart_inverse_composition T hb t v hv - rw [hvadd, hw.2.2.2.2.2.2.2.1 (T + t) z hz] at hcomp - have hFv : F' t v + (U T : ℂ) ∈ upperHalfPlane := by - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im, add_zero] - using (hb.2.2.2.2.2.2.1 t hv).1.1 - have he : F' t v + (U T : ℂ) = G T z := by - rw [← hw.2.2.2.2.2.2.2.2.1 T _ hFv, ← hcomp] - have he' : F' t v = G T z - (U T : ℂ) := by - rw [← he, add_sub_cancel_right] - rw [← he'] - exact hb.2.2.2.2.2.2.1 t hv - -lemma IsCapacityTwoWitness.restart_domain_iff - {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) - (hb : IsCapacityTwoWitness (fun t => U (T + t) - U T) β G' F') - (t : ℝ≥0) (w : ℂ) (hwH : w ∈ upperHalfPlane) : - F T (w + (U T : ℂ)) ∈ survivingDomain γ (T + t) ↔ w ∈ survivingDomain β t := by - constructor - · intro hz - have hwadd : w + (U T : ℂ) ∈ upperHalfPlane := by - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im, add_zero] - using hwH - have h := hw.restart_live_preimage T hb t _ hz - rwa [hw.2.2.2.2.2.2.2.2.1 T _ hwadd, add_sub_cancel_right] at h - · exact fun ht => (hw.restart_forward_composition T hb t w ht).1 - -theorem IsCapacityTwoWitness.restart_radius_finite_iff - {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T : ℝ≥0) - (hb : IsCapacityTwoWitness (fun t => U (T + t) - U T) β G' F') - (w : ℂ) (hwH : w ∈ upperHalfPlane) (u v : ℝ) (hu : 0 ≤ u) (hv : 0 < v) - (hbase : fullRadius G T (F T (w + (U T : ℂ))) = - (2 * (F T (w + (U T : ℂ))).im) * Real.exp (-u)) : - radiusLevelTime γ G (F T (w + (U T : ℂ))) (u + v) < ∞ ↔ - radiusLevelTime β G' w v < ∞ := by - have hUp : Continuous (fun t => U (T + t) - U T) := - (hU.comp (continuous_const.add continuous_id)).sub continuous_const - have hwadd : w + (U T : ℂ) ∈ upperHalfPlane := by - simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im, add_zero] using hwH - constructor - · intro hfinite - obtain ⟨S, hSlive, hSR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G _ (u + v) hfinite - have hS := radiusLevelTime_eq_of_live_level hU hw _ (u + v) S hSlive hSR - have hbefore := hw.before_deeper_radius_level hU _ u v hu hv T - (hw.2.2.2.2.2.2.1 T hwadd) hbase - rw [hS] at hbefore - have hTS : T ≤ S := by exact_mod_cast hbefore.le - let t := S - T - have htime : T + t = S := add_tsub_cancel_of_le hTS - have htlive : w ∈ survivingDomain β t := - (hw.restart_domain_iff T hb t w hwH).mp (htime ▸ hSlive) - have hR := hw.restart_radius_composition hU T hb t w htlive - rw [htime, hSR] at hR - have hR0 := hw.restart_initial_radius hU T hb w hwH - rw [hbase] at hR0 - have hD := hw.restart_inverse_derivative_norm_pos hU T hb w hwH - have hsourceR : fullRadius G' t w = (2 * w.im) * Real.exp (-v) := by - apply mul_left_cancel₀ hD.ne' - calc - _ = (2 * (F T (w + (U T : ℂ))).im) * Real.exp (-(u + v)) := hR.symm - _ = ((2 * (F T (w + (U T : ℂ))).im) * Real.exp (-u)) * Real.exp (-v) := by - rw [neg_add, Real.exp_add] - ring - _ = _ := by rw [hR0]; ring - rw [radiusLevelTime_eq_of_live_level hUp hb w v t htlive hsourceR] - exact ENNReal.coe_lt_top - · intro hfinite - obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top β G' w v hfinite - rw [(hw.restart_level_of_live_source hU T hb w u v hbase t ht hR).2] - exact ENNReal.coe_lt_top - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.RestartGreenKernel +import OAI.Probability.SLE.Gauge.DenseRadiusGap + +namespace OAI + +/-! Surjectivity of the actual restarted surviving domain. Inverse-map +composition identifies every original future live point with a live fresh +point, so reaching a deeper radius is equivalent in the two charts. -/ + +noncomputable section +open Set Filter +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma IsCapacityTwoWitness.restart_live_preimage + {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) + (hb : IsCapacityTwoWitness (fun t => U (T + t) - U T) β G' F') + (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ (T + t)) : + G T z - (U T : ℂ) ∈ survivingDomain β t := by + let v := G (T + t) z - (U T : ℂ) + have hv : v ∈ upperHalfPlane := by + simpa only [v, upperHalfPlane, mem_ofPred_eq, Complex.sub_im, Complex.ofReal_im, sub_zero] + using hw.2.2.2.2.2.1 (T + t) hz + have hvadd : v + (U T : ℂ) = G (T + t) z := sub_add_cancel _ _ + have hcomp := hw.restart_inverse_composition T hb t v hv + rw [hvadd, hw.2.2.2.2.2.2.2.1 (T + t) z hz] at hcomp + have hFv : F' t v + (U T : ℂ) ∈ upperHalfPlane := by + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im, add_zero] + using (hb.2.2.2.2.2.2.1 t hv).1.1 + have he : F' t v + (U T : ℂ) = G T z := by + rw [← hw.2.2.2.2.2.2.2.2.1 T _ hFv, ← hcomp] + have he' : F' t v = G T z - (U T : ℂ) := by + rw [← he, add_sub_cancel_right] + rw [← he'] + exact hb.2.2.2.2.2.2.1 t hv + +lemma IsCapacityTwoWitness.restart_domain_iff + {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) + (hb : IsCapacityTwoWitness (fun t => U (T + t) - U T) β G' F') + (t : ℝ≥0) (w : ℂ) (hwH : w ∈ upperHalfPlane) : + F T (w + (U T : ℂ)) ∈ survivingDomain γ (T + t) ↔ w ∈ survivingDomain β t := by + constructor + · intro hz + have hwadd : w + (U T : ℂ) ∈ upperHalfPlane := by + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im, add_zero] + using hwH + have h := hw.restart_live_preimage T hb t _ hz + rwa [hw.2.2.2.2.2.2.2.2.1 T _ hwadd, add_sub_cancel_right] at h + · exact fun ht => (hw.restart_forward_composition T hb t w ht).1 + +theorem IsCapacityTwoWitness.restart_radius_finite_iff + {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F G' F' : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hU : Continuous U) (T : ℝ≥0) + (hb : IsCapacityTwoWitness (fun t => U (T + t) - U T) β G' F') + (w : ℂ) (hwH : w ∈ upperHalfPlane) (u v : ℝ) (hu : 0 ≤ u) (hv : 0 < v) + (hbase : fullRadius G T (F T (w + (U T : ℂ))) = + (2 * (F T (w + (U T : ℂ))).im) * Real.exp (-u)) : + radiusLevelTime γ G (F T (w + (U T : ℂ))) (u + v) < ∞ ↔ + radiusLevelTime β G' w v < ∞ := by + have hUp : Continuous (fun t => U (T + t) - U T) := + (hU.comp (continuous_const.add continuous_id)).sub continuous_const + have hwadd : w + (U T : ℂ) ∈ upperHalfPlane := by + simpa only [upperHalfPlane, mem_ofPred_eq, Complex.add_im, Complex.ofReal_im, add_zero] using hwH + constructor + · intro hfinite + obtain ⟨S, hSlive, hSR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G _ (u + v) hfinite + have hS := radiusLevelTime_eq_of_live_level hU hw _ (u + v) S hSlive hSR + have hbefore := hw.before_deeper_radius_level hU _ u v hu hv T + (hw.2.2.2.2.2.2.1 T hwadd) hbase + rw [hS] at hbefore + have hTS : T ≤ S := by exact_mod_cast hbefore.le + let t := S - T + have htime : T + t = S := add_tsub_cancel_of_le hTS + have htlive : w ∈ survivingDomain β t := + (hw.restart_domain_iff T hb t w hwH).mp (htime ▸ hSlive) + have hR := hw.restart_radius_composition hU T hb t w htlive + rw [htime, hSR] at hR + have hR0 := hw.restart_initial_radius hU T hb w hwH + rw [hbase] at hR0 + have hD := hw.restart_inverse_derivative_norm_pos hU T hb w hwH + have hsourceR : fullRadius G' t w = (2 * w.im) * Real.exp (-v) := by + apply mul_left_cancel₀ hD.ne' + calc + _ = (2 * (F T (w + (U T : ℂ))).im) * Real.exp (-(u + v)) := hR.symm + _ = ((2 * (F T (w + (U T : ℂ))).im) * Real.exp (-u)) * Real.exp (-v) := by + rw [neg_add, Real.exp_add] + ring + _ = _ := by rw [hR0]; ring + rw [radiusLevelTime_eq_of_live_level hUp hb w v t htlive hsourceR] + exact ENNReal.coe_lt_top + · intro hfinite + obtain ⟨t, ht, hR⟩ := exists_live_level_of_radiusLevelTime_lt_top β G' w v hfinite + rw [(hw.restart_level_of_live_source hU T hb w u v hbase t ht hR).2] + exact ENNReal.coe_lt_top + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RestartDriverScaling.lean b/lean/OAI/Probability/SLE/Loewner/RestartDriverScaling.lean index 45a8aeef19cc843fa3c5b0aceb5965255876e510..cb078e09df4ba3bb7463b392182b1cbb9a8e3120 100644 --- a/lean/OAI/Probability/SLE/Loewner/RestartDriverScaling.lean +++ b/lean/OAI/Probability/SLE/Loewner/RestartDriverScaling.lean @@ -1,131 +1,131 @@ -import OAI.Probability.SLE.Loewner.ScaleTraceLaw -import OAI.Probability.SLE.Geometry.RandomRestartEvents - -namespace OAI - -/-! Scaling the fresh Brownian driver by a scale chosen from the stopped -past preserves its exact restricted path law. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -local instance restartScaleDriverRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance restartScaleDriverRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ - -def scaleDriverPath (c : ℝ≥0) (p : C(ℝ≥0, ℝ)) : C(ℝ≥0, ℝ) := - ⟨fun t => (c : ℝ) * p (sourceScaleTime c t), - continuous_const.mul (p.continuous.comp (continuous_id.div_const (c ^ 2)))⟩ - -lemma measurable_scaleDriverPath : - Measurable (fun x : ℝ≥0 × C(ℝ≥0, ℝ) => scaleDriverPath x.1 x.2) := by - apply source_path_measurable - intro t - change Measurable (fun x : ℝ≥0 × C(ℝ≥0, ℝ) => - (x.1 : ℝ) * x.2 (t / x.1 ^ 2)) - exact NNReal.continuous_coe.measurable.comp measurable_fst |>.mul - (continuous_eval.measurable.comp - (measurable_snd.prodMk (measurable_const.div (measurable_fst.pow_const 2)))) - -lemma brownian_scaled_driver_path_law - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (c : ℝ≥0) (hcpos : 0 < c) : - (P.map (sourceDriverPath B hc)).map (scaleDriverPath c) = - P.map (sourceDriverPath B hc) := by - let Bc : ℝ≥0 → Ω → ℝ := fun t ω => (c : ℝ) * B (sourceScaleTime c t) ω - have hBc : IsBrownianReal Bc P := by - have hh := hB.smul (c := (c ^ 2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hcpos.ne')) - simpa only [Bc, sourceScaleTime, NNReal.coe_inv, NNReal.coe_pow, Real.sqrt_inv, - Real.sqrt_sq c.coe_nonneg, inv_inv, div_eq_mul_inv, mul_comm] using hh - have hmc : ∀ t, Measurable (Bc t) := fun t => measurable_const.mul (hm _) - have hcc : ∀ ω, Continuous (fun t => Bc t ω) := fun ω => - continuous_const.mul ((hc ω).comp (continuous_id.div_const (c ^ 2))) - have hs : Measurable (scaleDriverPath c) := - measurable_scaleDriverPath.comp measurable_prodMk_left - rw [Measure.map_map hs (source_path_measurable _ hm)] - exact (ScaleTraceLaw.brownian_continuous_path_law B Bc hB hBc hm hmc hc hcc).symm - -/-- A measurable transformation of the fresh path may depend on the stopped -past if each of its sections preserves the fresh path law. -/ -theorem restart_random_transform_law - {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] - (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν] - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - (Y : Ω → X) (hY : Measurable[mΩ] Y) - (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → - @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) - (H : Ω × X → X) (hH : @Measurable (Ω × X) X (m.prod mX) mX H) - (hsections : ∀ ω, ν.map (fun x => H (ω, x)) = ν) - (A : Set Ω) (hA : MeasurableSet[m] A) : - @Measure.map Ω X mΩ mX (fun ω => H (ω, Y ω)) (P.restrict A) = P A • ν := by - let : MeasurableSpace Ω := mΩ - have hj : @Measurable Ω (Ω × X) mΩ (m.prod mX) (fun ω => (ω, Y ω)) := - (measurable_id'' hm).prodMk hY - apply Measure.ext - intro D hD - let E : Set (Ω × X) := H ⁻¹' D - have hE : @MeasurableSet (Ω × X) (m.prod mX) E := hH hD - let f : Ω × X → ℝ≥0∞ := (A ×ˢ univ).indicator (E.indicator (fun _ => 1)) - have hf : @Measurable (Ω × X) ℝ≥0∞ (m.prod mX) _ f := - (measurable_const.indicator hE).indicator (hA.prod MeasurableSet.univ) - have he := @restart_lintegral_random_test Ω X mΩ mX P inferInstance ν inferInstance - m hm Y hY hlaw f hf - have hsec (ω : Ω) : (∫⁻ x, f (ω, x) ∂ν) = A.indicator (fun _ => ν D) ω := by - by_cases hω : ω ∈ A - · have hHx : Measurable (fun x => H (ω, x)) := hH.comp measurable_prodMk_left - have hfun : (fun x => f (ω, x)) = - ((fun x => H (ω, x)) ⁻¹' D).indicator (fun _ => 1) := by - funext x - by_cases hd : H (ω, x) ∈ D <;> simp [f, E, hω, hd] - rw [hfun, lintegral_indicator (hHx hD), setLIntegral_const, one_mul, - ← Measure.map_apply hHx hD, hsections ω, indicator_of_mem hω] - · simp [f, hω] - have hfun : (fun ω => f (ω, Y ω)) = - A.indicator ((fun ω => H (ω, Y ω)) ⁻¹' D |>.indicator (fun _ => 1)) := by - funext ω - by_cases hω : ω ∈ A <;> by_cases hd : H (ω, Y ω) ∈ D <;> simp [f, E, hω, hd] - rw [hfun] at he - simp_rw [hsec] at he - rw [lintegral_indicator (hm A hA), - lintegral_indicator (hm A hA), setLIntegral_const] at he - have hscaled : Measurable (fun ω => H (ω, Y ω)) := hH.comp hj - rw [Measure.map_apply hscaled hD, Measure.restrict_apply (hscaled hD), - Measure.smul_apply, smul_eq_mul] - rw [setLIntegral_indicator (hscaled hD), setLIntegral_const, one_mul] at he - simpa only [mul_comm] using he - -/-- The normalized fresh driver has the original continuous Brownian law -on every event in the stopped past, even with a random past-measurable scale. -/ -theorem source_restart_random_scale_driver_law - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - (c : Ω → ℝ≥0) (hcm : Measurable[hτ.measurableSpace] c) (hcpos : ∀ ω, 0 < c ω) - (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) : - (P.restrict A).map (fun ω => scaleDriverPath (c ω) (sourceRestartDriverPath B hc τ ω)) = - P A • P.map (sourceDriverPath B hc) := by - have hrm : Measurable (sourceRestartDriverPath B hc τ) := - source_path_measurable _ (fun t => (measurable_pi_apply t).comp - (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) - let ν := P.map (sourceDriverPath B hc) - have : IsProbabilityMeasure ν := - (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance - apply restart_random_transform_law P ν hτ.measurableSpace hτ.measurableSpace_le - (sourceRestartDriverPath B hc τ) hrm - (fun A hA => source_restart_continuous_driver_law hB hm hc τ hτ hA) - (fun p => scaleDriverPath (c p.1) p.2) - (measurable_scaleDriverPath.comp ((hcm.comp measurable_fst).prodMk measurable_snd)) - (fun ω => brownian_scaled_driver_path_law P B hB hm hc (c ω) (hcpos ω)) A hA - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ScaleTraceLaw +import OAI.Probability.SLE.Geometry.RandomRestartEvents + +namespace OAI + +/-! Scaling the fresh Brownian driver by a scale chosen from the stopped +past preserves its exact restricted path law. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +local instance restartScaleDriverRealMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance restartScaleDriverRealBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ + +def scaleDriverPath (c : ℝ≥0) (p : C(ℝ≥0, ℝ)) : C(ℝ≥0, ℝ) := + ⟨fun t => (c : ℝ) * p (sourceScaleTime c t), + continuous_const.mul (p.continuous.comp (continuous_id.div_const (c ^ 2)))⟩ + +lemma measurable_scaleDriverPath : + Measurable (fun x : ℝ≥0 × C(ℝ≥0, ℝ) => scaleDriverPath x.1 x.2) := by + apply source_path_measurable + intro t + change Measurable (fun x : ℝ≥0 × C(ℝ≥0, ℝ) => + (x.1 : ℝ) * x.2 (t / x.1 ^ 2)) + exact NNReal.continuous_coe.measurable.comp measurable_fst |>.mul + (continuous_eval.measurable.comp + (measurable_snd.prodMk (measurable_const.div (measurable_fst.pow_const 2)))) + +lemma brownian_scaled_driver_path_law + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (c : ℝ≥0) (hcpos : 0 < c) : + (P.map (sourceDriverPath B hc)).map (scaleDriverPath c) = + P.map (sourceDriverPath B hc) := by + let Bc : ℝ≥0 → Ω → ℝ := fun t ω => (c : ℝ) * B (sourceScaleTime c t) ω + have hBc : IsBrownianReal Bc P := by + have hh := hB.smul (c := (c ^ 2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hcpos.ne')) + simpa only [Bc, sourceScaleTime, NNReal.coe_inv, NNReal.coe_pow, Real.sqrt_inv, + Real.sqrt_sq c.coe_nonneg, inv_inv, div_eq_mul_inv, mul_comm] using hh + have hmc : ∀ t, Measurable (Bc t) := fun t => measurable_const.mul (hm _) + have hcc : ∀ ω, Continuous (fun t => Bc t ω) := fun ω => + continuous_const.mul ((hc ω).comp (continuous_id.div_const (c ^ 2))) + have hs : Measurable (scaleDriverPath c) := + measurable_scaleDriverPath.comp measurable_prodMk_left + rw [Measure.map_map hs (source_path_measurable _ hm)] + exact (ScaleTraceLaw.brownian_continuous_path_law B Bc hB hBc hm hmc hc hcc).symm + +/-- A measurable transformation of the fresh path may depend on the stopped +past if each of its sections preserves the fresh path law. -/ +theorem restart_random_transform_law + {Ω X : Type*} [mΩ : MeasurableSpace Ω] [mX : MeasurableSpace X] + (P : Measure Ω) [IsFiniteMeasure P] (ν : Measure X) [IsFiniteMeasure ν] + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + (Y : Ω → X) (hY : Measurable[mΩ] Y) + (hlaw : ∀ A : Set Ω, MeasurableSet[m] A → + @Measure.map Ω X mΩ mX Y (P.restrict A) = P A • ν) + (H : Ω × X → X) (hH : @Measurable (Ω × X) X (m.prod mX) mX H) + (hsections : ∀ ω, ν.map (fun x => H (ω, x)) = ν) + (A : Set Ω) (hA : MeasurableSet[m] A) : + @Measure.map Ω X mΩ mX (fun ω => H (ω, Y ω)) (P.restrict A) = P A • ν := by + let : MeasurableSpace Ω := mΩ + have hj : @Measurable Ω (Ω × X) mΩ (m.prod mX) (fun ω => (ω, Y ω)) := + (measurable_id'' hm).prodMk hY + apply Measure.ext + intro D hD + let E : Set (Ω × X) := H ⁻¹' D + have hE : @MeasurableSet (Ω × X) (m.prod mX) E := hH hD + let f : Ω × X → ℝ≥0∞ := (A ×ˢ univ).indicator (E.indicator (fun _ => 1)) + have hf : @Measurable (Ω × X) ℝ≥0∞ (m.prod mX) _ f := + (measurable_const.indicator hE).indicator (hA.prod MeasurableSet.univ) + have he := @restart_lintegral_random_test Ω X mΩ mX P inferInstance ν inferInstance + m hm Y hY hlaw f hf + have hsec (ω : Ω) : (∫⁻ x, f (ω, x) ∂ν) = A.indicator (fun _ => ν D) ω := by + by_cases hω : ω ∈ A + · have hHx : Measurable (fun x => H (ω, x)) := hH.comp measurable_prodMk_left + have hfun : (fun x => f (ω, x)) = + ((fun x => H (ω, x)) ⁻¹' D).indicator (fun _ => 1) := by + funext x + by_cases hd : H (ω, x) ∈ D <;> simp [f, E, hω, hd] + rw [hfun, lintegral_indicator (hHx hD), setLIntegral_const, one_mul, + ← Measure.map_apply hHx hD, hsections ω, indicator_of_mem hω] + · simp [f, hω] + have hfun : (fun ω => f (ω, Y ω)) = + A.indicator ((fun ω => H (ω, Y ω)) ⁻¹' D |>.indicator (fun _ => 1)) := by + funext ω + by_cases hω : ω ∈ A <;> by_cases hd : H (ω, Y ω) ∈ D <;> simp [f, E, hω, hd] + rw [hfun] at he + simp_rw [hsec] at he + rw [lintegral_indicator (hm A hA), + lintegral_indicator (hm A hA), setLIntegral_const] at he + have hscaled : Measurable (fun ω => H (ω, Y ω)) := hH.comp hj + rw [Measure.map_apply hscaled hD, Measure.restrict_apply (hscaled hD), + Measure.smul_apply, smul_eq_mul] + rw [setLIntegral_indicator (hscaled hD), setLIntegral_const, one_mul] at he + simpa only [mul_comm] using he + +/-- The normalized fresh driver has the original continuous Brownian law +on every event in the stopped past, even with a random past-measurable scale. -/ +theorem source_restart_random_scale_driver_law + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + (c : Ω → ℝ≥0) (hcm : Measurable[hτ.measurableSpace] c) (hcpos : ∀ ω, 0 < c ω) + (A : Set Ω) (hA : MeasurableSet[hτ.measurableSpace] A) : + (P.restrict A).map (fun ω => scaleDriverPath (c ω) (sourceRestartDriverPath B hc τ ω)) = + P A • P.map (sourceDriverPath B hc) := by + have hrm : Measurable (sourceRestartDriverPath B hc τ) := + source_path_measurable _ (fun t => (measurable_pi_apply t).comp + (brownianRestart_measurable hm hc (stopping_time_measurable_nnreal hτ))) + let ν := P.map (sourceDriverPath B hc) + have : IsProbabilityMeasure ν := + (Measure.isProbabilityMeasure_map_iff (source_path_measurable _ hm).aemeasurable).mpr inferInstance + apply restart_random_transform_law P ν hτ.measurableSpace hτ.measurableSpace_le + (sourceRestartDriverPath B hc τ) hrm + (fun A hA => source_restart_continuous_driver_law hB hm hc τ hτ hA) + (fun p => scaleDriverPath (c p.1) p.2) + (measurable_scaleDriverPath.comp ((hcm.comp measurable_fst).prodMk measurable_snd)) + (fun ω => brownian_scaled_driver_path_law P B hB hm hc (c ω) (hcpos ω)) A hA + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseBoundaryCone.lean b/lean/OAI/Probability/SLE/Loewner/ReverseBoundaryCone.lean index 1ff42f5cbafd5fc2e394948edfbebad3e6101490..47e56fed5961d6497b1d14e711759b51c99d1ce5 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseBoundaryCone.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseBoundaryCone.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Loewner.ReverseBoundarySquare - -namespace OAI - -/-! A reverse trajectory starting at the driver enters a vertical cone. -The squared coordinate removes the singularity at the initial point. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -theorem reverse_boundary_cone_of_small_center - {T L M : ℝ} (hT : 0 ≤ T) (hL : 0 ≤ L) (hsmall : 2*L*M ≤ 1) - (q : ℝ → ℂ) (u u' : ℝ → ℝ) - (hc : ContinuousOn q (Icc 0 T)) (huc : ContinuousOn u (Icc 0 T)) - (hq0 : q 0=(u 0 : ℂ)) - (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Ici t) t) - (hu : ∀ t ∈ Ioc 0 T, HasDerivWithinAt u (u' t) (Ici t) t) - (hp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) - (hb : ∀ t ∈ Ioc 0 T, |u' t| ≤ L) - (hM : ∀ t ∈ Ioc 0 T, ‖q t-(u t : ℂ)‖ ≤ M) : - ∀ t ∈ Icc 0 T, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2 := by - have he := reverse_boundary_square_estimate hT q u u' hc huc hq0 hq hu hp hb hM hL - intro t ht - let z : ℂ := q t-(u t : ℂ) - have hnorm : ‖z^2+4*(t : ℂ)‖ ≤ t := - (he t ht).trans (by nlinarith [mul_le_mul_of_nonneg_right hsmall ht.1]) - have hre : (z^2+4*(t : ℂ)).re ≤ t := (Complex.re_le_norm _).trans hnorm - have hz : ‖z^2‖ ≤ 5*t := by - calc - ‖z^2‖ = ‖(z^2+4*(t : ℂ))-4*(t : ℂ)‖ := by congr 1; ring - _ ≤ ‖z^2+4*(t : ℂ)‖+‖4*(t : ℂ)‖ := norm_sub_le _ _ - _ ≤ t+4*t := by - rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] - norm_num only [norm_ofNat] - exact add_le_add hnorm le_rfl - _ = 5*t := by ring - have hnormsq : ‖z^2‖ = z.re^2+z.im^2 := by - rw [norm_pow,← Complex.normSq_eq_norm_sq,Complex.normSq_apply] - ring - have here : (z^2+4*(t : ℂ)).re=z.re^2-z.im^2+4*t := by - simp [pow_two,Complex.mul_re] - rw [hnormsq] at hz - rw [here] at hre - have hx : z.re=(q t).re-u t := by simp [z] - have hy : z.im=(q t).im := by simp [z] - rw [hx,hy] at hre hz - constructor <;> nlinarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReverseBoundarySquare + +namespace OAI + +/-! A reverse trajectory starting at the driver enters a vertical cone. +The squared coordinate removes the singularity at the initial point. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +theorem reverse_boundary_cone_of_small_center + {T L M : ℝ} (hT : 0 ≤ T) (hL : 0 ≤ L) (hsmall : 2*L*M ≤ 1) + (q : ℝ → ℂ) (u u' : ℝ → ℝ) + (hc : ContinuousOn q (Icc 0 T)) (huc : ContinuousOn u (Icc 0 T)) + (hq0 : q 0=(u 0 : ℂ)) + (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Ici t) t) + (hu : ∀ t ∈ Ioc 0 T, HasDerivWithinAt u (u' t) (Ici t) t) + (hp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) + (hb : ∀ t ∈ Ioc 0 T, |u' t| ≤ L) + (hM : ∀ t ∈ Ioc 0 T, ‖q t-(u t : ℂ)‖ ≤ M) : + ∀ t ∈ Icc 0 T, ((q t).re-u t)^2 ≤ t ∧ 3*t ≤ (q t).im^2 := by + have he := reverse_boundary_square_estimate hT q u u' hc huc hq0 hq hu hp hb hM hL + intro t ht + let z : ℂ := q t-(u t : ℂ) + have hnorm : ‖z^2+4*(t : ℂ)‖ ≤ t := + (he t ht).trans (by nlinarith [mul_le_mul_of_nonneg_right hsmall ht.1]) + have hre : (z^2+4*(t : ℂ)).re ≤ t := (Complex.re_le_norm _).trans hnorm + have hz : ‖z^2‖ ≤ 5*t := by + calc + ‖z^2‖ = ‖(z^2+4*(t : ℂ))-4*(t : ℂ)‖ := by congr 1; ring + _ ≤ ‖z^2+4*(t : ℂ)‖+‖4*(t : ℂ)‖ := norm_sub_le _ _ + _ ≤ t+4*t := by + rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] + norm_num only [norm_ofNat] + exact add_le_add hnorm le_rfl + _ = 5*t := by ring + have hnormsq : ‖z^2‖ = z.re^2+z.im^2 := by + rw [norm_pow,← Complex.normSq_eq_norm_sq,Complex.normSq_apply] + ring + have here : (z^2+4*(t : ℂ)).re=z.re^2-z.im^2+4*t := by + simp [pow_two,Complex.mul_re] + rw [hnormsq] at hz + rw [here] at hre + have hx : z.re=(q t).re-u t := by simp [z] + have hy : z.im=(q t).im := by simp [z] + rw [hx,hy] at hre hz + constructor <;> nlinarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseBoundarySquare.lean b/lean/OAI/Probability/SLE/Loewner/ReverseBoundarySquare.lean index 7a4f6853929e3458e77e53c2261954da81e0179a..69d23b1e8244dbb26f01bbe03747850785d9f4ac 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseBoundarySquare.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseBoundarySquare.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.Geometry.EndpointDerivativeBound - -namespace OAI - -/-! Squared centered reverse coordinates have a regular derivative -even when the initial point is the boundary tip. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -lemma reverse_centered_square_derivative {S : Set ℝ} {t : ℝ} - (q : ℝ → ℂ) (u : ℝ → ℝ) (u' : ℝ) - (hq : HasDerivWithinAt q (-2/(q t-(u t : ℂ))) S t) - (hu : HasDerivWithinAt u u' S t) (hp : 0 < (q t).im) : - HasDerivWithinAt (fun s => (q s-(u s : ℂ))^2) - (-4-2*(u' : ℂ)*(q t-(u t : ℂ))) S t := by - have hn : q t-(u t : ℂ) ≠ 0 := by - intro he - have hi := congrArg Complex.im he - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi - exact hp.ne' hi - have huc : HasDerivWithinAt (fun s => (u s : ℂ)) (u' : ℂ) S t := hu.ofReal_comp - have h := (hq.sub huc).pow 2 - simp only [Pi.sub_apply,Nat.cast_ofNat,Nat.reduceSub,pow_one] at h - convert! h using 1 - field_simp [hn] - ring - -lemma reverse_boundary_square_estimate {T L M : ℝ} (hT : 0 ≤ T) - (q : ℝ → ℂ) (u u' : ℝ → ℝ) - (hc : ContinuousOn q (Icc 0 T)) (huc : ContinuousOn u (Icc 0 T)) - (hq0 : q 0=(u 0 : ℂ)) - (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Ici t) t) - (hu : ∀ t ∈ Ioc 0 T, HasDerivWithinAt u (u' t) (Ici t) t) - (hp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) - (hb : ∀ t ∈ Ioc 0 T, |u' t| ≤ L) - (hM : ∀ t ∈ Ioc 0 T, ‖q t-(u t : ℂ)‖ ≤ M) (hL : 0 ≤ L) : - ∀ t ∈ Icc 0 T, ‖(q t-(u t : ℂ))^2+4*(t : ℂ)‖ ≤ (2*L*M)*t := by - let f (t : ℝ) : ℂ := (q t-(u t : ℂ))^2+4*(t : ℂ) - have hfc : ContinuousOn f (Icc 0 T) := - ((hc.sub (Complex.continuous_ofReal.comp_continuousOn huc)).pow 2).add - (continuous_const.mul Complex.continuous_ofReal).continuousOn - have hf0 : f 0=0 := by simp [f,hq0] - have hfd (t : ℝ) (ht : t ∈ Ioc 0 T) : HasDerivWithinAt f - (-2*(u' t : ℂ)*(q t-(u t : ℂ))) (Ici t) t := by - have hid : HasDerivAt (fun s : ℝ => (s : ℂ)) 1 t := by - convert! (hasDerivAt_id t).ofReal_comp using 1 - have h := (reverse_centered_square_derivative q u (u' t) (hq t ht) (hu t ht) (hp t ht)).add - ((hid.const_mul 4).hasDerivWithinAt) - convert! h using 1 - ring - have hbd (t : ℝ) (ht : t ∈ Ioc 0 T) : - ‖-2*(u' t : ℂ)*(q t-(u t : ℂ))‖ ≤ 2*L*M := by - rw [norm_mul,norm_mul,norm_neg,norm_ofNat,Complex.norm_real,Real.norm_eq_abs] - exact mul_le_mul (mul_le_mul_of_nonneg_left (hb t ht) (by norm_num)) (hM t ht) - (norm_nonneg _) (mul_nonneg (by norm_num) hL) - simpa only [hf0,sub_zero,f] using norm_sub_initial_le_of_deriv_on_Ioc hT f _ hfc hfd hbd - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.EndpointDerivativeBound + +namespace OAI + +/-! Squared centered reverse coordinates have a regular derivative +even when the initial point is the boundary tip. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +lemma reverse_centered_square_derivative {S : Set ℝ} {t : ℝ} + (q : ℝ → ℂ) (u : ℝ → ℝ) (u' : ℝ) + (hq : HasDerivWithinAt q (-2/(q t-(u t : ℂ))) S t) + (hu : HasDerivWithinAt u u' S t) (hp : 0 < (q t).im) : + HasDerivWithinAt (fun s => (q s-(u s : ℂ))^2) + (-4-2*(u' : ℂ)*(q t-(u t : ℂ))) S t := by + have hn : q t-(u t : ℂ) ≠ 0 := by + intro he + have hi := congrArg Complex.im he + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi + exact hp.ne' hi + have huc : HasDerivWithinAt (fun s => (u s : ℂ)) (u' : ℂ) S t := hu.ofReal_comp + have h := (hq.sub huc).pow 2 + simp only [Pi.sub_apply,Nat.cast_ofNat,Nat.reduceSub,pow_one] at h + convert! h using 1 + field_simp [hn] + ring + +lemma reverse_boundary_square_estimate {T L M : ℝ} (hT : 0 ≤ T) + (q : ℝ → ℂ) (u u' : ℝ → ℝ) + (hc : ContinuousOn q (Icc 0 T)) (huc : ContinuousOn u (Icc 0 T)) + (hq0 : q 0=(u 0 : ℂ)) + (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Ici t) t) + (hu : ∀ t ∈ Ioc 0 T, HasDerivWithinAt u (u' t) (Ici t) t) + (hp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) + (hb : ∀ t ∈ Ioc 0 T, |u' t| ≤ L) + (hM : ∀ t ∈ Ioc 0 T, ‖q t-(u t : ℂ)‖ ≤ M) (hL : 0 ≤ L) : + ∀ t ∈ Icc 0 T, ‖(q t-(u t : ℂ))^2+4*(t : ℂ)‖ ≤ (2*L*M)*t := by + let f (t : ℝ) : ℂ := (q t-(u t : ℂ))^2+4*(t : ℂ) + have hfc : ContinuousOn f (Icc 0 T) := + ((hc.sub (Complex.continuous_ofReal.comp_continuousOn huc)).pow 2).add + (continuous_const.mul Complex.continuous_ofReal).continuousOn + have hf0 : f 0=0 := by simp [f,hq0] + have hfd (t : ℝ) (ht : t ∈ Ioc 0 T) : HasDerivWithinAt f + (-2*(u' t : ℂ)*(q t-(u t : ℂ))) (Ici t) t := by + have hid : HasDerivAt (fun s : ℝ => (s : ℂ)) 1 t := by + convert! (hasDerivAt_id t).ofReal_comp using 1 + have h := (reverse_centered_square_derivative q u (u' t) (hq t ht) (hu t ht) (hp t ht)).add + ((hid.const_mul 4).hasDerivWithinAt) + convert! h using 1 + ring + have hbd (t : ℝ) (ht : t ∈ Ioc 0 T) : + ‖-2*(u' t : ℂ)*(q t-(u t : ℂ))‖ ≤ 2*L*M := by + rw [norm_mul,norm_mul,norm_neg,norm_ofNat,Complex.norm_real,Real.norm_eq_abs] + exact mul_le_mul (mul_le_mul_of_nonneg_left (hb t ht) (by norm_num)) (hM t ht) + (norm_nonneg _) (mul_nonneg (by norm_num) hL) + simpa only [hf0,sub_zero,f] using norm_sub_initial_le_of_deriv_on_Ioc hT f _ hfc hfd hbd + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseConeContraction.lean b/lean/OAI/Probability/SLE/Loewner/ReverseConeContraction.lean index dd96f63e903759bf5393c2740db54cabe005f5e3..45982e3cc8bdfa6a74761a83f7bae48c9f05b624 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseConeContraction.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseConeContraction.lean @@ -1,73 +1,73 @@ -import OAI.Probability.SLE.Geometry.ComplexDissipation -import OAI.Probability.SLE.Loewner.LocalReverseCone - -namespace OAI - -/-! Reverse trajectories in a vertical cone have a dissipative difference -equation. The contraction includes a boundary initial point. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -lemma reverse_coefficient_re_nonpos {q r : ℂ} {u t : ℝ} - (ht : 0 ≤ t) (hqp : 0 ≤ q.im) (hrp : 0 ≤ r.im) - (hqx : (q.re-u)^2 ≤ t) (hrx : (r.re-u)^2 ≤ t) - (hqy : t ≤ q.im^2) (hry : t ≤ r.im^2) : - (2/((q-(u : ℂ))*(r-(u : ℂ)))).re ≤ 0 := by - have hxx : (q.re-u)*(r.re-u) ≤ t := by - nlinarith [sq_nonneg ((q.re-u)-(r.re-u))] - have hqroot : Real.sqrt t ≤ q.im := by - nlinarith [Real.sq_sqrt ht,Real.sqrt_nonneg t] - have hrroot : Real.sqrt t ≤ r.im := by - nlinarith [Real.sq_sqrt ht,Real.sqrt_nonneg t] - have hyy : t ≤ q.im*r.im := by - have hh := mul_le_mul hqroot hrroot (Real.sqrt_nonneg t) hqp - nlinarith [Real.sq_sqrt ht] - have hreal : ((q-(u : ℂ))*(r-(u : ℂ))).re ≤ 0 := by - simp only [Complex.mul_re,Complex.sub_re,Complex.ofReal_re, - Complex.sub_im,Complex.ofReal_im,sub_zero] - linarith - simp only [Complex.div_re,Complex.re_ofNat,Complex.im_ofNat,zero_mul,zero_div,add_zero] - exact div_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonneg_of_nonpos (by norm_num) hreal) - (Complex.normSq_nonneg _) - -theorem reverse_trajectories_contract_in_cones {T : ℝ} (hT : 0 ≤ T) - (q r : ℝ → ℂ) (u : ℝ → ℝ) - (hqc : ContinuousOn q (Icc 0 T)) (hrc : ContinuousOn r (Icc 0 T)) - (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Ici t) t) - (hr : ∀ t ∈ Ioc 0 T, HasDerivWithinAt r - (-2/(r t-(u t : ℂ))) (Ici t) t) - (hqp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) - (hrp : ∀ t ∈ Ioc 0 T, 0 < (r t).im) - (hqx : ∀ t ∈ Ioc 0 T, ((q t).re-u t)^2 ≤ t) - (hrx : ∀ t ∈ Ioc 0 T, ((r t).re-u t)^2 ≤ t) - (hqy : ∀ t ∈ Ioc 0 T, t ≤ (q t).im^2) - (hry : ∀ t ∈ Ioc 0 T, t ≤ (r t).im^2) : - ∀ t ∈ Icc 0 T, dist (q t) (r t) ≤ dist (q 0) (r 0) := by - let c (t : ℝ) : ℂ := 2/((q t-(u t : ℂ))*(r t-(u t : ℂ))) - have hn (v : ℝ → ℂ) (hp : ∀ t ∈ Ioc 0 T, 0 < (v t).im) - (t : ℝ) (ht : t ∈ Ioc 0 T) : v t-(u t : ℂ) ≠ 0 := by - intro he - have hi := congrArg Complex.im he - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi - exact (hp t ht).ne' hi - have hd (t : ℝ) (ht : t ∈ Ioc 0 T) : - HasDerivWithinAt (fun s => q s-r s) (c t*(q t-r t)) (Ici t) t := by - convert! (hq t ht).sub (hr t ht) using 1 - dsimp [c] - field_simp [hn q hqp t ht,hn r hrp t ht] - ring - have hc (t : ℝ) (ht : t ∈ Ioc 0 T) : (c t).re ≤ 0 := - reverse_coefficient_re_nonpos ht.1.le (hqp t ht).le (hrp t ht).le - (hqx t ht) (hrx t ht) (hqy t ht) (hry t ht) - simpa only [dist_eq_norm] using - complex_linear_endpoint_contraction hT (fun s => q s-r s) c (hqc.sub hrc) hd hc - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ComplexDissipation +import OAI.Probability.SLE.Loewner.LocalReverseCone + +namespace OAI + +/-! Reverse trajectories in a vertical cone have a dissipative difference +equation. The contraction includes a boundary initial point. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +lemma reverse_coefficient_re_nonpos {q r : ℂ} {u t : ℝ} + (ht : 0 ≤ t) (hqp : 0 ≤ q.im) (hrp : 0 ≤ r.im) + (hqx : (q.re-u)^2 ≤ t) (hrx : (r.re-u)^2 ≤ t) + (hqy : t ≤ q.im^2) (hry : t ≤ r.im^2) : + (2/((q-(u : ℂ))*(r-(u : ℂ)))).re ≤ 0 := by + have hxx : (q.re-u)*(r.re-u) ≤ t := by + nlinarith [sq_nonneg ((q.re-u)-(r.re-u))] + have hqroot : Real.sqrt t ≤ q.im := by + nlinarith [Real.sq_sqrt ht,Real.sqrt_nonneg t] + have hrroot : Real.sqrt t ≤ r.im := by + nlinarith [Real.sq_sqrt ht,Real.sqrt_nonneg t] + have hyy : t ≤ q.im*r.im := by + have hh := mul_le_mul hqroot hrroot (Real.sqrt_nonneg t) hqp + nlinarith [Real.sq_sqrt ht] + have hreal : ((q-(u : ℂ))*(r-(u : ℂ))).re ≤ 0 := by + simp only [Complex.mul_re,Complex.sub_re,Complex.ofReal_re, + Complex.sub_im,Complex.ofReal_im,sub_zero] + linarith + simp only [Complex.div_re,Complex.re_ofNat,Complex.im_ofNat,zero_mul,zero_div,add_zero] + exact div_nonpos_of_nonpos_of_nonneg (mul_nonpos_of_nonneg_of_nonpos (by norm_num) hreal) + (Complex.normSq_nonneg _) + +theorem reverse_trajectories_contract_in_cones {T : ℝ} (hT : 0 ≤ T) + (q r : ℝ → ℂ) (u : ℝ → ℝ) + (hqc : ContinuousOn q (Icc 0 T)) (hrc : ContinuousOn r (Icc 0 T)) + (hq : ∀ t ∈ Ioc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Ici t) t) + (hr : ∀ t ∈ Ioc 0 T, HasDerivWithinAt r + (-2/(r t-(u t : ℂ))) (Ici t) t) + (hqp : ∀ t ∈ Ioc 0 T, 0 < (q t).im) + (hrp : ∀ t ∈ Ioc 0 T, 0 < (r t).im) + (hqx : ∀ t ∈ Ioc 0 T, ((q t).re-u t)^2 ≤ t) + (hrx : ∀ t ∈ Ioc 0 T, ((r t).re-u t)^2 ≤ t) + (hqy : ∀ t ∈ Ioc 0 T, t ≤ (q t).im^2) + (hry : ∀ t ∈ Ioc 0 T, t ≤ (r t).im^2) : + ∀ t ∈ Icc 0 T, dist (q t) (r t) ≤ dist (q 0) (r 0) := by + let c (t : ℝ) : ℂ := 2/((q t-(u t : ℂ))*(r t-(u t : ℂ))) + have hn (v : ℝ → ℂ) (hp : ∀ t ∈ Ioc 0 T, 0 < (v t).im) + (t : ℝ) (ht : t ∈ Ioc 0 T) : v t-(u t : ℂ) ≠ 0 := by + intro he + have hi := congrArg Complex.im he + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi + exact (hp t ht).ne' hi + have hd (t : ℝ) (ht : t ∈ Ioc 0 T) : + HasDerivWithinAt (fun s => q s-r s) (c t*(q t-r t)) (Ici t) t := by + convert! (hq t ht).sub (hr t ht) using 1 + dsimp [c] + field_simp [hn q hqp t ht,hn r hrp t ht] + ring + have hc (t : ℝ) (ht : t ∈ Ioc 0 T) : (c t).re ≤ 0 := + reverse_coefficient_re_nonpos ht.1.le (hqp t ht).le (hrp t ht).le + (hqx t ht) (hrx t ht) (hqy t ht) (hry t ht) + simpa only [dist_eq_norm] using + complex_linear_endpoint_contraction hT (fun s => q s-r s) c (hqc.sub hrc) hd hc + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseFlowStability.lean b/lean/OAI/Probability/SLE/Loewner/ReverseFlowStability.lean index 4d66eb7a30039053a8dbad21bec1f58888aee303..841697e2271163ccbd223b93ce4aef4e91581948 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseFlowStability.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseFlowStability.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Loewner.LoewnerInverseStability - -namespace OAI - -/-! Stability of raw reverse trajectories, including dependence on their -initial point. This is used before a geometric trace has been constructed. -/ -noncomputable section -open Set Filter Complex Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma reverse_flow_driver_distance {a : ℝ} (ha : 0 < a) - (U V : ℝ → ℝ) (Z Q : ℝ → ℂ) (T : ℝ) - (hZ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z - (reverseField a (U t) (Z t)) (Icc 0 T) t) - (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q - (reverseField a (V t) (Q t)) (Icc 0 T) t) - (δ : ℝ) (hinit : dist (Z 0) (Q 0) ≤ δ) - (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) : - ∀ t ∈ Icc 0 T, dist (Z t) (Q t) ≤ - gronwallBound δ (4*a⁻¹^2) (2*a⁻¹^2*δ) t := by - intro t ht - have he := dist_le_of_approx_trajectories_ODE - (K := (⟨4*a⁻¹^2,by positivity⟩ : ℝ≥0)) - (v := fun s => reverseField a (U s)) (fun _ => reverseField_lipschitz ha _) - (f := Z) (fun s hs => (hZ s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hZ s ⟨hs.1,hs.2.le⟩)) - (εf := 0) (fun _ _ => by simp) - (g := Q) (fun s hs => (hQ s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQ s ⟨hs.1,hs.2.le⟩)) - (εg := 2*a⁻¹^2*δ) (fun s hs => ?_) hinit t ht - · change dist (Z t) (Q t) ≤ gronwallBound δ (4*a⁻¹^2) (0+2*a⁻¹^2*δ) (t-0) at he - simpa only [zero_add,sub_zero] using he - · rw [dist_comm] - exact (reverseField_driver_dist ha _ _ _).trans - (mul_le_mul_of_nonneg_left (hUV s (Ico_subset_Icc_self hs)) (by positivity)) - -lemma reverse_flow_lipschitz {a : ℝ} (ha : 0 < a) - (U : ℝ → ℝ) (Q : ℝ → ℂ) - (hQ : ∀ t ∈ Ici (0 : ℝ), HasDerivWithinAt Q - (reverseField a (U t) (Q t)) (Ici 0) t) : - LipschitzOnWith ⟨2*a⁻¹,by positivity⟩ Q (Ici 0) := by - apply (convex_Ici (0 : ℝ)).lipschitzOnWith_of_nnnorm_hasDerivWithin_le hQ - intro t ht - exact_mod_cast reverseField_bound ha (U t) (Q t) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerInverseStability + +namespace OAI + +/-! Stability of raw reverse trajectories, including dependence on their +initial point. This is used before a geometric trace has been constructed. -/ +noncomputable section +open Set Filter Complex Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma reverse_flow_driver_distance {a : ℝ} (ha : 0 < a) + (U V : ℝ → ℝ) (Z Q : ℝ → ℂ) (T : ℝ) + (hZ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z + (reverseField a (U t) (Z t)) (Icc 0 T) t) + (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q + (reverseField a (V t) (Q t)) (Icc 0 T) t) + (δ : ℝ) (hinit : dist (Z 0) (Q 0) ≤ δ) + (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) : + ∀ t ∈ Icc 0 T, dist (Z t) (Q t) ≤ + gronwallBound δ (4*a⁻¹^2) (2*a⁻¹^2*δ) t := by + intro t ht + have he := dist_le_of_approx_trajectories_ODE + (K := (⟨4*a⁻¹^2,by positivity⟩ : ℝ≥0)) + (v := fun s => reverseField a (U s)) (fun _ => reverseField_lipschitz ha _) + (f := Z) (fun s hs => (hZ s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hZ s ⟨hs.1,hs.2.le⟩)) + (εf := 0) (fun _ _ => by simp) + (g := Q) (fun s hs => (hQ s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQ s ⟨hs.1,hs.2.le⟩)) + (εg := 2*a⁻¹^2*δ) (fun s hs => ?_) hinit t ht + · change dist (Z t) (Q t) ≤ gronwallBound δ (4*a⁻¹^2) (0+2*a⁻¹^2*δ) (t-0) at he + simpa only [zero_add,sub_zero] using he + · rw [dist_comm] + exact (reverseField_driver_dist ha _ _ _).trans + (mul_le_mul_of_nonneg_left (hUV s (Ico_subset_Icc_self hs)) (by positivity)) + +lemma reverse_flow_lipschitz {a : ℝ} (ha : 0 < a) + (U : ℝ → ℝ) (Q : ℝ → ℂ) + (hQ : ∀ t ∈ Ici (0 : ℝ), HasDerivWithinAt Q + (reverseField a (U t) (Q t)) (Ici 0) t) : + LipschitzOnWith ⟨2*a⁻¹,by positivity⟩ Q (Ici 0) := by + apply (convex_Ici (0 : ℝ)).lipschitzOnWith_of_nnnorm_hasDerivWithin_le hQ + intro t ht + exact_mod_cast reverseField_bound ha (U t) (Q t) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseHeightUpper.lean b/lean/OAI/Probability/SLE/Loewner/ReverseHeightUpper.lean index 339b9ef3019d369fdaad40ac5dd5fa55598c21a0..14c88e96dc3e97ef9fb0a78b6a647202b064cd74 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseHeightUpper.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseHeightUpper.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Loewner.SmoothRawInverse - -namespace OAI - -/-! The elementary upper height bound for reverse trajectories applies -before a trace or a geometric hull is available. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma reverse_height_square_le {T : ℝ} (hT : 0 ≤ T) - (q : ℝ → ℂ) (u : ℝ → ℝ) - (hd : ∀ s ∈ Icc 0 T, HasDerivWithinAt q (-2/(q s-(u s : ℂ))) (Icc 0 T) s) - (hp : ∀ s ∈ Icc 0 T, 0 < (q s).im) : - (q T).im^2 ≤ (q 0).im^2+4*T := by - have hsq (s : ℝ) (hs : s ∈ Icc 0 T) : - HasDerivWithinAt (fun t => (q t).im^2) - (4*(q s).im^2/normSq (q s-(u s : ℂ))) (Icc 0 T) s := by - have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt s (hd s hs) - change HasDerivWithinAt (fun t => (q t).im) - ((-2/(q s-(u s : ℂ))).im) _ _ at hi - convert! hi.pow 2 using 1 - simp only [Complex.div_im,Complex.neg_re,Complex.re_ofNat,Complex.neg_im, - Complex.im_ofNat,neg_zero,zero_mul,Complex.sub_im,Complex.ofReal_im,sub_zero] - ring - apply image_le_of_deriv_right_le_deriv_boundary - (fun s hs => (hsq s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hsq s ⟨hs.1,hs.2.le⟩)) - (B := fun s => (q 0).im^2+4*s) (B' := fun _ => 4) - (by simp) (by fun_prop) - (fun s _ => by - convert (((hasDerivAt_id s).const_mul 4).const_add ((q 0).im^2)).hasDerivWithinAt using 1 <;> simp) - (fun s hs => ?_) ⟨hT,le_rfl⟩ - apply (div_le_iff₀ ?_).mpr - · simp only [Complex.normSq_apply,Complex.sub_re,Complex.ofReal_re, - Complex.sub_im,Complex.ofReal_im,sub_zero] - nlinarith [sq_nonneg ((q s).re-u s)] - · apply Complex.normSq_pos.mpr - intro he - have hi := congrArg Complex.im he - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi - exact (hp s (Ico_subset_Icc_self hs)).ne' hi - -namespace Target - -lemma raw_inverse_height_square_le (b : Driver) (T : ℝ≥0) (w : ℂ) - (hw : w ∈ upperHalfPlane) : (inverseFlow 1 b T w).im^2 ≤ w.im^2+4*T := by - obtain ⟨q,hq0,hheight,hd,he⟩ := exists_raw_reverse_trajectory b T w hw - rw [he,← hq0] - exact reverse_height_square_le T.coe_nonneg q (fun s => driverValue b ((T : ℝ)-s)) - (fun s hs => (hd s hs.1).mono Icc_subset_Ici_self) - (fun s hs => hw.trans_le (hheight s hs.1)) - -end Target -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawInverse + +namespace OAI + +/-! The elementary upper height bound for reverse trajectories applies +before a trace or a geometric hull is available. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma reverse_height_square_le {T : ℝ} (hT : 0 ≤ T) + (q : ℝ → ℂ) (u : ℝ → ℝ) + (hd : ∀ s ∈ Icc 0 T, HasDerivWithinAt q (-2/(q s-(u s : ℂ))) (Icc 0 T) s) + (hp : ∀ s ∈ Icc 0 T, 0 < (q s).im) : + (q T).im^2 ≤ (q 0).im^2+4*T := by + have hsq (s : ℝ) (hs : s ∈ Icc 0 T) : + HasDerivWithinAt (fun t => (q t).im^2) + (4*(q s).im^2/normSq (q s-(u s : ℂ))) (Icc 0 T) s := by + have hi := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt s (hd s hs) + change HasDerivWithinAt (fun t => (q t).im) + ((-2/(q s-(u s : ℂ))).im) _ _ at hi + convert! hi.pow 2 using 1 + simp only [Complex.div_im,Complex.neg_re,Complex.re_ofNat,Complex.neg_im, + Complex.im_ofNat,neg_zero,zero_mul,Complex.sub_im,Complex.ofReal_im,sub_zero] + ring + apply image_le_of_deriv_right_le_deriv_boundary + (fun s hs => (hsq s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hsq s ⟨hs.1,hs.2.le⟩)) + (B := fun s => (q 0).im^2+4*s) (B' := fun _ => 4) + (by simp) (by fun_prop) + (fun s _ => by + convert (((hasDerivAt_id s).const_mul 4).const_add ((q 0).im^2)).hasDerivWithinAt using 1 <;> simp) + (fun s hs => ?_) ⟨hT,le_rfl⟩ + apply (div_le_iff₀ ?_).mpr + · simp only [Complex.normSq_apply,Complex.sub_re,Complex.ofReal_re, + Complex.sub_im,Complex.ofReal_im,sub_zero] + nlinarith [sq_nonneg ((q s).re-u s)] + · apply Complex.normSq_pos.mpr + intro he + have hi := congrArg Complex.im he + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hi + exact (hp s (Ico_subset_Icc_self hs)).ne' hi + +namespace Target + +lemma raw_inverse_height_square_le (b : Driver) (T : ℝ≥0) (w : ℂ) + (hw : w ∈ upperHalfPlane) : (inverseFlow 1 b T w).im^2 ≤ w.im^2+4*T := by + obtain ⟨q,hq0,hheight,hd,he⟩ := exists_raw_reverse_trajectory b T w hw + rw [he,← hq0] + exact reverse_height_square_le T.coe_nonneg q (fun s => driverValue b ((T : ℝ)-s)) + (fun s hs => (hd s hs.1).mono Icc_subset_Ici_self) + (fun s hs => hw.trans_le (hheight s hs.1)) + +end Target +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseLongContraction.lean b/lean/OAI/Probability/SLE/Loewner/ReverseLongContraction.lean index bfa59f6c923be71cc3536ca6e049ce7a215d3a86..46e097b473fe717c0e5eaaad7e47078838b3b32d 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseLongContraction.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseLongContraction.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Loewner.SmoothReverseContraction - -namespace OAI - -/-! A short initial contraction, followed by positive-height Grönwall -control, bounds reverse trajectories on any finite time interval. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge - -lemma reverse_height_monotone {T : ℝ} (q : ℝ → ℂ) (u : ℝ → ℝ) - (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Icc 0 T) t) - (hp : ∀ t ∈ Icc 0 T, 0 < (q t).im) : - MonotoneOn (fun t => (q t).im) (Icc 0 T) := by - have hd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt (fun s => (q s).im) - (2*(q t).im/Complex.normSq (q t-(u t : ℂ))) (Icc 0 T) t := by - have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hq t ht) - change HasDerivWithinAt (fun s => (q s).im) - ((-2/(q t-(u t : ℂ))).im) (Icc 0 T) t at hh - have he : (-2/(q t-(u t : ℂ))).im = - 2*(q t).im/Complex.normSq (q t-(u t : ℂ)) := by - simp only [Complex.div_im,Complex.neg_re,Complex.re_ofNat,Complex.neg_im, - Complex.im_ofNat,neg_zero,zero_mul,Complex.sub_im,Complex.ofReal_im,sub_zero] - ring - rwa [he] at hh - apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _) - (fun t ht => (hd t ht).continuousWithinAt) - (fun t ht => (hd t (interior_subset ht)).mono interior_subset) - intro t ht - exact div_nonneg (mul_nonneg (by norm_num) (hp t (interior_subset ht)).le) - (Complex.normSq_nonneg _) - -theorem smooth_reverse_long_distance {T a L : ℝ} (ha : 0 < a) (haT : a ≤ T) (hL : 0 ≤ L) - (q r : ℝ → ℂ) (u u' : ℝ → ℝ) - (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Icc 0 T) t) - (hr : ∀ t ∈ Icc 0 T, HasDerivWithinAt r (-2/(r t-(u t : ℂ))) (Icc 0 T) t) - (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) - (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) - (hqp : ∀ t ∈ Icc 0 T, 0 < (q t).im) - (hrp : ∀ t ∈ Icc 0 T, 0 < (r t).im) - (hq0 : (q 0).re=u 0) (hr0 : (r 0).re=u 0) - (hsmall : L^2*a ≤ 1) : - dist (q T) (r T) ≤ dist (q 0) (r 0)*Real.exp ((4*(Real.sqrt a)⁻¹^2)*(T-a)) := by - have hsub : Icc 0 a ⊆ Icc 0 T := Icc_subset_Icc_right haT - have hqd := fun t ht => (hq t (hsub ht)).mono hsub - have hrd := fun t ht => (hr t (hsub ht)).mono hsub - have hud := fun t ht => (hu t (hsub ht)).mono hsub - have hb' := fun t ht => hb t (hsub ht) - have hqp' := fun t ht => hqp t (hsub ht) - have hrp' := fun t ht => hrp t (hsub ht) - have hinit := smooth_reverse_trajectories_contract ha.le hL q r u u' - hqd hrd hud hb' hqp' hrp' hq0 hr0 hsmall - have hqa := reverse_loewner_height_lower hL q u u' hqd hud hb' hqp' hq0 hsmall a ⟨ha.le,le_rfl⟩ - have hra := reverse_loewner_height_lower hL r u u' hrd hud hb' hrp' hr0 hsmall a ⟨ha.le,le_rfl⟩ - have hη : 0 < Real.sqrt a := Real.sqrt_pos.mpr ha - have hroot : (Real.sqrt a)^2=a := Real.sq_sqrt ha.le - have hqa' : Real.sqrt a ≤ (q a).im := by nlinarith [hqp' a ⟨ha.le,le_rfl⟩] - have hra' : Real.sqrt a ≤ (r a).im := by nlinarith [hrp' a ⟨ha.le,le_rfl⟩] - have hmQ := reverse_height_monotone q u hq hqp - have hmR := reverse_height_monotone r u hr hrp - have hpost : Icc a T ⊆ Icc 0 T := Icc_subset_Icc_left ha.le - have hQ (t : ℝ) (ht : t ∈ Icc a T) : Real.sqrt a ≤ (q t).im := - hqa'.trans (hmQ ⟨ha.le,haT⟩ (hpost ht) ht.1) - have hR (t : ℝ) (ht : t ∈ Icc a T) : Real.sqrt a ≤ (r t).im := - hra'.trans (hmR ⟨ha.le,haT⟩ (hpost ht) ht.1) - have hdQ (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt q - (reverseField (Real.sqrt a) (u t) (q t)) (Ici t) t := by - rw [reverseField,reverseCenter_eq (hQ t (Ico_subset_Icc_self ht))] - simpa only [div_eq_mul_inv] using - hasDerivWithinAt_Ici_of_Icc ht ((hq t (hpost (Ico_subset_Icc_self ht))).mono hpost) - have hdR (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt r - (reverseField (Real.sqrt a) (u t) (r t)) (Ici t) t := by - rw [reverseField,reverseCenter_eq (hR t (Ico_subset_Icc_self ht))] - simpa only [div_eq_mul_inv] using - hasDerivWithinAt_Ici_of_Icc ht ((hr t (hpost (Ico_subset_Icc_self ht))).mono hpost) - exact dist_le_of_trajectories_ODE (fun t => reverseField_lipschitz hη (u t)) - (fun t ht => (hq t (hpost ht)).continuousWithinAt.mono hpost) hdQ - (fun t ht => (hr t (hpost ht)).continuousWithinAt.mono hpost) hdR hinit T ⟨haT,le_rfl⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothReverseContraction + +namespace OAI + +/-! A short initial contraction, followed by positive-height Grönwall +control, bounds reverse trajectories on any finite time interval. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge + +lemma reverse_height_monotone {T : ℝ} (q : ℝ → ℂ) (u : ℝ → ℝ) + (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Icc 0 T) t) + (hp : ∀ t ∈ Icc 0 T, 0 < (q t).im) : + MonotoneOn (fun t => (q t).im) (Icc 0 T) := by + have hd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt (fun s => (q s).im) + (2*(q t).im/Complex.normSq (q t-(u t : ℂ))) (Icc 0 T) t := by + have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hq t ht) + change HasDerivWithinAt (fun s => (q s).im) + ((-2/(q t-(u t : ℂ))).im) (Icc 0 T) t at hh + have he : (-2/(q t-(u t : ℂ))).im = + 2*(q t).im/Complex.normSq (q t-(u t : ℂ)) := by + simp only [Complex.div_im,Complex.neg_re,Complex.re_ofNat,Complex.neg_im, + Complex.im_ofNat,neg_zero,zero_mul,Complex.sub_im,Complex.ofReal_im,sub_zero] + ring + rwa [he] at hh + apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Icc _ _) + (fun t ht => (hd t ht).continuousWithinAt) + (fun t ht => (hd t (interior_subset ht)).mono interior_subset) + intro t ht + exact div_nonneg (mul_nonneg (by norm_num) (hp t (interior_subset ht)).le) + (Complex.normSq_nonneg _) + +theorem smooth_reverse_long_distance {T a L : ℝ} (ha : 0 < a) (haT : a ≤ T) (hL : 0 ≤ L) + (q r : ℝ → ℂ) (u u' : ℝ → ℝ) + (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Icc 0 T) t) + (hr : ∀ t ∈ Icc 0 T, HasDerivWithinAt r (-2/(r t-(u t : ℂ))) (Icc 0 T) t) + (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) + (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) + (hqp : ∀ t ∈ Icc 0 T, 0 < (q t).im) + (hrp : ∀ t ∈ Icc 0 T, 0 < (r t).im) + (hq0 : (q 0).re=u 0) (hr0 : (r 0).re=u 0) + (hsmall : L^2*a ≤ 1) : + dist (q T) (r T) ≤ dist (q 0) (r 0)*Real.exp ((4*(Real.sqrt a)⁻¹^2)*(T-a)) := by + have hsub : Icc 0 a ⊆ Icc 0 T := Icc_subset_Icc_right haT + have hqd := fun t ht => (hq t (hsub ht)).mono hsub + have hrd := fun t ht => (hr t (hsub ht)).mono hsub + have hud := fun t ht => (hu t (hsub ht)).mono hsub + have hb' := fun t ht => hb t (hsub ht) + have hqp' := fun t ht => hqp t (hsub ht) + have hrp' := fun t ht => hrp t (hsub ht) + have hinit := smooth_reverse_trajectories_contract ha.le hL q r u u' + hqd hrd hud hb' hqp' hrp' hq0 hr0 hsmall + have hqa := reverse_loewner_height_lower hL q u u' hqd hud hb' hqp' hq0 hsmall a ⟨ha.le,le_rfl⟩ + have hra := reverse_loewner_height_lower hL r u u' hrd hud hb' hrp' hr0 hsmall a ⟨ha.le,le_rfl⟩ + have hη : 0 < Real.sqrt a := Real.sqrt_pos.mpr ha + have hroot : (Real.sqrt a)^2=a := Real.sq_sqrt ha.le + have hqa' : Real.sqrt a ≤ (q a).im := by nlinarith [hqp' a ⟨ha.le,le_rfl⟩] + have hra' : Real.sqrt a ≤ (r a).im := by nlinarith [hrp' a ⟨ha.le,le_rfl⟩] + have hmQ := reverse_height_monotone q u hq hqp + have hmR := reverse_height_monotone r u hr hrp + have hpost : Icc a T ⊆ Icc 0 T := Icc_subset_Icc_left ha.le + have hQ (t : ℝ) (ht : t ∈ Icc a T) : Real.sqrt a ≤ (q t).im := + hqa'.trans (hmQ ⟨ha.le,haT⟩ (hpost ht) ht.1) + have hR (t : ℝ) (ht : t ∈ Icc a T) : Real.sqrt a ≤ (r t).im := + hra'.trans (hmR ⟨ha.le,haT⟩ (hpost ht) ht.1) + have hdQ (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt q + (reverseField (Real.sqrt a) (u t) (q t)) (Ici t) t := by + rw [reverseField,reverseCenter_eq (hQ t (Ico_subset_Icc_self ht))] + simpa only [div_eq_mul_inv] using + hasDerivWithinAt_Ici_of_Icc ht ((hq t (hpost (Ico_subset_Icc_self ht))).mono hpost) + have hdR (t : ℝ) (ht : t ∈ Ico a T) : HasDerivWithinAt r + (reverseField (Real.sqrt a) (u t) (r t)) (Ici t) t := by + rw [reverseField,reverseCenter_eq (hR t (Ico_subset_Icc_self ht))] + simpa only [div_eq_mul_inv] using + hasDerivWithinAt_Ici_of_Icc ht ((hr t (hpost (Ico_subset_Icc_self ht))).mono hpost) + exact dist_le_of_trajectories_ODE (fun t => reverseField_lipschitz hη (u t)) + (fun t ht => (hq t (hpost ht)).continuousWithinAt.mono hpost) hdQ + (fun t ht => (hr t (hpost ht)).continuousWithinAt.mono hpost) hdR hinit T ⟨haT,le_rfl⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseSmoothBounds.lean b/lean/OAI/Probability/SLE/Loewner/ReverseSmoothBounds.lean index 7f3c85536e271a85c6e802da6d604aabfc4bc559..16c88a199d27c5110f100b2d6c569f8954bff41b 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseSmoothBounds.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseSmoothBounds.lean @@ -1,93 +1,93 @@ -import OAI.Probability.SLE.Loewner.LoewnerDriverBounds - -namespace OAI - -/-! Elementary barriers for the reverse Loewner equation with a bounded -derivative of the driving function. The estimates are uniform in the -positive starting height. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma damped_real_barrier {T L : ℝ} (hL : 0 ≤ L) - (x b D : ℝ → ℝ) (hx0 : x 0=0) - (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt x - (-2*x t/D t+b t) (Icc 0 T) t) - (hD : ∀ t ∈ Icc 0 T, 0 < D t) - (hb : ∀ t ∈ Icc 0 T, b t ≤ L) : - ∀ t ∈ Icc 0 T, x t ≤ L*t := by - intro t ht - apply le_of_forall_pos_le_add - intro ε hε - apply image_le_of_deriv_right_lt_deriv_boundary - (fun s hs => (hd s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hd s ⟨hs.1,hs.2.le⟩)) - (B := fun s => L*s+ε) (B' := fun _ => L) - (by simp only [hx0,mul_zero,zero_add]; exact hε.le) - (fun s => by simpa using ((hasDerivAt_id s).const_mul L).add_const ε) - (fun s hs he => ?_) ht - have hxs : 0 < x s := by - rw [he] - exact add_pos_of_nonneg_of_pos (mul_nonneg hL hs.1) hε - have hn : -2*x s/D s < 0 := div_neg_of_neg_of_pos (by linarith) (hD s ⟨hs.1,hs.2.le⟩) - linarith [hb s ⟨hs.1,hs.2.le⟩] - -lemma damped_real_abs_barrier {T L : ℝ} (hL : 0 ≤ L) - (x b D : ℝ → ℝ) (hx0 : x 0=0) - (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt x - (-2*x t/D t+b t) (Icc 0 T) t) - (hD : ∀ t ∈ Icc 0 T, 0 < D t) - (hb : ∀ t ∈ Icc 0 T, |b t| ≤ L) : - ∀ t ∈ Icc 0 T, |x t| ≤ L*t := by - intro t ht - refine abs_le.mpr ⟨?_,damped_real_barrier hL x b D hx0 hd hD - (fun s hs => (le_abs_self _).trans (hb s hs)) t ht⟩ - have hn := damped_real_barrier hL (fun s => -x s) (fun s => -b s) D - (by simp [hx0]) (fun s hs => ?_) hD - (fun s hs => (neg_le_abs _).trans (hb s hs)) t ht - · linarith - · convert (hd s hs).neg using 1; ring - -lemma reverse_height_square_barrier {T L : ℝ} - (x y : ℝ → ℝ) - (hy : ∀ t ∈ Icc 0 T, 0 < y t) - (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y - (2*y t/((x t)^2+(y t)^2)) (Icc 0 T) t) - (hx : ∀ t ∈ Icc 0 T, |x t| ≤ L*t) - (hsmall : ∀ t ∈ Icc 0 T, L^2*t ≤ 1) : - ∀ t ∈ Icc 0 T, 2*t ≤ (y t)^2 := by - intro t ht - have hh := image_le_of_deriv_right_lt_deriv_boundary - (f := fun s => -(y s)^2) (f' := fun s => -(4*(y s)^2/((x s)^2+(y s)^2))) - (fun s hs => ((hd s hs).pow 2).neg.continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs ?_) - (B := fun s => -2*s) (B' := fun _ => -2) - (by nlinarith [sq_nonneg (y 0)]) - (fun s => by simpa using (hasDerivAt_id s).const_mul (-2)) - (fun s hs he => ?_) ht - · linarith - · convert ((hd s ⟨hs.1,hs.2.le⟩).pow 2).neg using 1; dsimp; ring - · have hs' : s ∈ Icc 0 T := ⟨hs.1,hs.2.le⟩ - have hy2 : (y s)^2=2*s := by linarith - have hspos : 0 < s := by nlinarith [sq_pos_of_pos (hy s hs')] - have hxx : (x s)^2 ≤ L^2*s^2 := by - have ha := hx s hs' - have hb : |x s|^2 ≤ (L*s)^2 := pow_le_pow_left₀ (abs_nonneg _) ha 2 - simpa only [sq_abs,mul_pow] using hb - have hyp := hy s hs' - have hden : 0 < (x s)^2+(y s)^2 := by positivity - have hxs : (x s)^2 ≤ s := by - have hh := mul_le_mul_of_nonneg_right (hsmall s hs') hs.1 - nlinarith - have hp : 2 < 4*(y s)^2/((x s)^2+(y s)^2) := by - apply (lt_div_iff₀ hden).mpr - nlinarith - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerDriverBounds + +namespace OAI + +/-! Elementary barriers for the reverse Loewner equation with a bounded +derivative of the driving function. The estimates are uniform in the +positive starting height. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma damped_real_barrier {T L : ℝ} (hL : 0 ≤ L) + (x b D : ℝ → ℝ) (hx0 : x 0=0) + (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt x + (-2*x t/D t+b t) (Icc 0 T) t) + (hD : ∀ t ∈ Icc 0 T, 0 < D t) + (hb : ∀ t ∈ Icc 0 T, b t ≤ L) : + ∀ t ∈ Icc 0 T, x t ≤ L*t := by + intro t ht + apply le_of_forall_pos_le_add + intro ε hε + apply image_le_of_deriv_right_lt_deriv_boundary + (fun s hs => (hd s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hd s ⟨hs.1,hs.2.le⟩)) + (B := fun s => L*s+ε) (B' := fun _ => L) + (by simp only [hx0,mul_zero,zero_add]; exact hε.le) + (fun s => by simpa using ((hasDerivAt_id s).const_mul L).add_const ε) + (fun s hs he => ?_) ht + have hxs : 0 < x s := by + rw [he] + exact add_pos_of_nonneg_of_pos (mul_nonneg hL hs.1) hε + have hn : -2*x s/D s < 0 := div_neg_of_neg_of_pos (by linarith) (hD s ⟨hs.1,hs.2.le⟩) + linarith [hb s ⟨hs.1,hs.2.le⟩] + +lemma damped_real_abs_barrier {T L : ℝ} (hL : 0 ≤ L) + (x b D : ℝ → ℝ) (hx0 : x 0=0) + (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt x + (-2*x t/D t+b t) (Icc 0 T) t) + (hD : ∀ t ∈ Icc 0 T, 0 < D t) + (hb : ∀ t ∈ Icc 0 T, |b t| ≤ L) : + ∀ t ∈ Icc 0 T, |x t| ≤ L*t := by + intro t ht + refine abs_le.mpr ⟨?_,damped_real_barrier hL x b D hx0 hd hD + (fun s hs => (le_abs_self _).trans (hb s hs)) t ht⟩ + have hn := damped_real_barrier hL (fun s => -x s) (fun s => -b s) D + (by simp [hx0]) (fun s hs => ?_) hD + (fun s hs => (neg_le_abs _).trans (hb s hs)) t ht + · linarith + · convert (hd s hs).neg using 1; ring + +lemma reverse_height_square_barrier {T L : ℝ} + (x y : ℝ → ℝ) + (hy : ∀ t ∈ Icc 0 T, 0 < y t) + (hd : ∀ t ∈ Icc 0 T, HasDerivWithinAt y + (2*y t/((x t)^2+(y t)^2)) (Icc 0 T) t) + (hx : ∀ t ∈ Icc 0 T, |x t| ≤ L*t) + (hsmall : ∀ t ∈ Icc 0 T, L^2*t ≤ 1) : + ∀ t ∈ Icc 0 T, 2*t ≤ (y t)^2 := by + intro t ht + have hh := image_le_of_deriv_right_lt_deriv_boundary + (f := fun s => -(y s)^2) (f' := fun s => -(4*(y s)^2/((x s)^2+(y s)^2))) + (fun s hs => ((hd s hs).pow 2).neg.continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs ?_) + (B := fun s => -2*s) (B' := fun _ => -2) + (by nlinarith [sq_nonneg (y 0)]) + (fun s => by simpa using (hasDerivAt_id s).const_mul (-2)) + (fun s hs he => ?_) ht + · linarith + · convert ((hd s ⟨hs.1,hs.2.le⟩).pow 2).neg using 1; dsimp; ring + · have hs' : s ∈ Icc 0 T := ⟨hs.1,hs.2.le⟩ + have hy2 : (y s)^2=2*s := by linarith + have hspos : 0 < s := by nlinarith [sq_pos_of_pos (hy s hs')] + have hxx : (x s)^2 ≤ L^2*s^2 := by + have ha := hx s hs' + have hb : |x s|^2 ≤ (L*s)^2 := pow_le_pow_left₀ (abs_nonneg _) ha 2 + simpa only [sq_abs,mul_pow] using hb + have hyp := hy s hs' + have hden : 0 < (x s)^2+(y s)^2 := by positivity + have hxs : (x s)^2 ≤ s := by + have hh := mul_le_mul_of_nonneg_right (hsmall s hs') hs.1 + nlinarith + have hp : 2 < 4*(y s)^2/((x s)^2+(y s)^2) := by + apply (lt_div_iff₀ hden).mpr + nlinarith + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ReverseTrajectoryExistence.lean b/lean/OAI/Probability/SLE/Loewner/ReverseTrajectoryExistence.lean index d52800911d4210c9198124528333a3e3ad312e97..27f848d9aaf7510b11d9312e9e497f0f0eb85cdf 100644 --- a/lean/OAI/Probability/SLE/Loewner/ReverseTrajectoryExistence.lean +++ b/lean/OAI/Probability/SLE/Loewner/ReverseTrajectoryExistence.lean @@ -1,48 +1,48 @@ -import OAI.Probability.SLE.Loewner.TargetInverseExistence - -namespace OAI - -/-! The unclipped reverse equation has a global trajectory from every -point of the upper half-plane, for every continuous real driver. -/ -noncomputable section -open Set Filter Complex _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -theorem exists_reverse_trajectory (u : ℝ → ℝ) (hu : Continuous u) - (w : ℂ) (hw : 0 < w.im) : - ∃ q : ℝ → ℂ, q 0=w ∧ - (∀ t, 0 ≤ t → w.im ≤ (q t).im) ∧ - ∀ t, 0 ≤ t → HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Ici 0) t := by - let a : ℝ := w.im/2 - have ha : 0 < a := half_pos hw - obtain ⟨Q,_,hQ0,hQd⟩ := exists_measurable_reverse_flow - (fun t (_ : Unit) => u t) (hu.measurable.comp measurable_fst) - (fun _ => hu) (fun _ => w) measurable_const ha - let q : ℝ → ℂ := fun t => Q t () - have hd (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt q - (reverseField a (u t) (q t)) (Ici 0) t := hQd () t ht - have hqc : ContinuousOn q (Ici 0) := fun t ht => (hd t ht).continuousWithinAt - have hm : MonotoneOn (fun t => (q t).im) (Ici 0) := by - apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Ici 0) - (Complex.continuous_im.comp_continuousOn hqc) - (fun t ht => (Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t - (hd t (interior_subset ht))).mono interior_subset) - intro t ht - exact reverseField_im_nonneg ha _ _ - have hq0 : q 0=w := hQ0 () - have hheight (t : ℝ) (ht : 0 ≤ t) : w.im ≤ (q t).im := by - rw [← hq0] - exact hm (show 0 ∈ Ici (0 : ℝ) from le_refl (0 : ℝ)) ht ht - refine ⟨q,hq0,hheight,?_⟩ - intro t ht - have h := hd t ht - rw [reverseField,reverseCenter_eq ((half_le_self hw.le).trans (hheight t ht))] at h - simpa only [div_eq_mul_inv] using h - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetInverseExistence + +namespace OAI + +/-! The unclipped reverse equation has a global trajectory from every +point of the upper half-plane, for every continuous real driver. -/ +noncomputable section +open Set Filter Complex _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +theorem exists_reverse_trajectory (u : ℝ → ℝ) (hu : Continuous u) + (w : ℂ) (hw : 0 < w.im) : + ∃ q : ℝ → ℂ, q 0=w ∧ + (∀ t, 0 ≤ t → w.im ≤ (q t).im) ∧ + ∀ t, 0 ≤ t → HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Ici 0) t := by + let a : ℝ := w.im/2 + have ha : 0 < a := half_pos hw + obtain ⟨Q,_,hQ0,hQd⟩ := exists_measurable_reverse_flow + (fun t (_ : Unit) => u t) (hu.measurable.comp measurable_fst) + (fun _ => hu) (fun _ => w) measurable_const ha + let q : ℝ → ℂ := fun t => Q t () + have hd (t : ℝ) (ht : 0 ≤ t) : HasDerivWithinAt q + (reverseField a (u t) (q t)) (Ici 0) t := hQd () t ht + have hqc : ContinuousOn q (Ici 0) := fun t ht => (hd t ht).continuousWithinAt + have hm : MonotoneOn (fun t => (q t).im) (Ici 0) := by + apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Ici 0) + (Complex.continuous_im.comp_continuousOn hqc) + (fun t ht => (Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t + (hd t (interior_subset ht))).mono interior_subset) + intro t ht + exact reverseField_im_nonneg ha _ _ + have hq0 : q 0=w := hQ0 () + have hheight (t : ℝ) (ht : 0 ≤ t) : w.im ≤ (q t).im := by + rw [← hq0] + exact hm (show 0 ∈ Ici (0 : ℝ) from le_refl (0 : ℝ)) ht ht + refine ⟨q,hq0,hheight,?_⟩ + intro t ht + have h := hd t ht + rw [reverseField,reverseCenter_eq ((half_le_self hw.le).trans (hheight t ht))] at h + simpa only [div_eq_mul_inv] using h + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/RightFieldDriverStability.lean b/lean/OAI/Probability/SLE/Loewner/RightFieldDriverStability.lean index f0d29c645e3665ceddf86403a2da4188e80b7be4..4ef95f1678c10f81ba968aa313c13cbf2f4e3599 100644 --- a/lean/OAI/Probability/SLE/Loewner/RightFieldDriverStability.lean +++ b/lean/OAI/Probability/SLE/Loewner/RightFieldDriverStability.lean @@ -1,72 +1,72 @@ -import OAI.Probability.SLE.NearRealContraction - -namespace OAI - -/-! Stability of the clipped right-half-plane flow under perturbations of -the driver. This also applies arbitrarily close to the real boundary. -/ -noncomputable section -open Set Filter Complex Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma rightCenter_driver_dist (a u v : ℝ) (z : ℂ) : - dist (rightCenter a u z) (rightCenter a v z) ≤ |u-v| := by - have he : rightCenter a u z-rightCenter a v z = - ((max a (z.re-u)-max a (z.re-v) : ℝ) : ℂ) := by - simp only [rightCenter,Complex.ofReal_sub] - ring - rw [dist_eq_norm,he,Complex.norm_real,Real.norm_eq_abs] - have hh := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul - (z.re-u) (z.re-v) - simpa only [Real.dist_eq,NNReal.coe_one,one_mul,id_eq,sub_sub_sub_cancel_left, - abs_sub_comm v u] using hh - -lemma rightField_driver_dist {a : ℝ} (ha : 0 < a) (u v : ℝ) (z : ℂ) : - dist (rightField a u z) (rightField a v z) ≤ 2*a⁻¹^2*|u-v| := by - have hu : ‖(rightCenter a u z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv] - exact inv_anti₀ ha (rightCenter_norm a u z) - have hv : ‖(rightCenter a v z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv] - exact inv_anti₀ ha (rightCenter_norm a v z) - rw [dist_eq_norm,rightField,rightField,← mul_sub,norm_mul,norm_ofNat, - inv_sub_inv' (rightCenter_ne_zero ha _ _) (rightCenter_ne_zero ha _ _), - norm_mul,norm_mul] - calc - _ ≤ 2*(a⁻¹*|u-v| * a⁻¹) := by - gcongr - simpa only [← dist_eq_norm,dist_comm] using rightCenter_driver_dist a u v z - _ = _ := by ring - -lemma right_flow_driver_distance {a : ℝ} (ha : 0 < a) - (U V : ℝ → ℝ) (Z Q : ℝ → ℂ) (T : ℝ) (_hT : 0 ≤ T) - (hZ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z - (rightField a (U t) (Z t)) (Icc 0 T) t) - (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q - (rightField a (V t) (Q t)) (Icc 0 T) t) - (δ : ℝ) (hinit : dist (Z 0) (Q 0) ≤ δ) - (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) : - ∀ t ∈ Icc 0 T, dist (Z t) (Q t) ≤ - gronwallBound δ (4*a⁻¹^2) (2*a⁻¹^2*δ) t := by - intro t ht - have he := dist_le_of_approx_trajectories_ODE - (K := (⟨4*a⁻¹^2,by positivity⟩ : ℝ≥0)) - (v := fun s => rightField a (U s)) (fun _ => rightField_lipschitz ha _) - (f := Z) (fun s hs => (hZ s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hZ s ⟨hs.1,hs.2.le⟩)) - (εf := 0) (fun _ _ => by simp) - (g := Q) (fun s hs => (hQ s hs).continuousWithinAt) - (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQ s ⟨hs.1,hs.2.le⟩)) - (εg := 2*a⁻¹^2*δ) (fun s hs => ?_) hinit t ht - · change dist (Z t) (Q t) ≤ gronwallBound δ (4*a⁻¹^2) (0+2*a⁻¹^2*δ) (t-0) at he - simpa only [zero_add,sub_zero] using he - · rw [dist_comm] - exact (rightField_driver_dist ha _ _ _).trans - (mul_le_mul_of_nonneg_left (hUV s (Ico_subset_Icc_self hs)) (by positivity)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.NearRealContraction + +namespace OAI + +/-! Stability of the clipped right-half-plane flow under perturbations of +the driver. This also applies arbitrarily close to the real boundary. -/ +noncomputable section +open Set Filter Complex Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma rightCenter_driver_dist (a u v : ℝ) (z : ℂ) : + dist (rightCenter a u z) (rightCenter a v z) ≤ |u-v| := by + have he : rightCenter a u z-rightCenter a v z = + ((max a (z.re-u)-max a (z.re-v) : ℝ) : ℂ) := by + simp only [rightCenter,Complex.ofReal_sub] + ring + rw [dist_eq_norm,he,Complex.norm_real,Real.norm_eq_abs] + have hh := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul + (z.re-u) (z.re-v) + simpa only [Real.dist_eq,NNReal.coe_one,one_mul,id_eq,sub_sub_sub_cancel_left, + abs_sub_comm v u] using hh + +lemma rightField_driver_dist {a : ℝ} (ha : 0 < a) (u v : ℝ) (z : ℂ) : + dist (rightField a u z) (rightField a v z) ≤ 2*a⁻¹^2*|u-v| := by + have hu : ‖(rightCenter a u z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv] + exact inv_anti₀ ha (rightCenter_norm a u z) + have hv : ‖(rightCenter a v z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv] + exact inv_anti₀ ha (rightCenter_norm a v z) + rw [dist_eq_norm,rightField,rightField,← mul_sub,norm_mul,norm_ofNat, + inv_sub_inv' (rightCenter_ne_zero ha _ _) (rightCenter_ne_zero ha _ _), + norm_mul,norm_mul] + calc + _ ≤ 2*(a⁻¹*|u-v| * a⁻¹) := by + gcongr + simpa only [← dist_eq_norm,dist_comm] using rightCenter_driver_dist a u v z + _ = _ := by ring + +lemma right_flow_driver_distance {a : ℝ} (ha : 0 < a) + (U V : ℝ → ℝ) (Z Q : ℝ → ℂ) (T : ℝ) (_hT : 0 ≤ T) + (hZ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Z + (rightField a (U t) (Z t)) (Icc 0 T) t) + (hQ : ∀ t ∈ Icc 0 T, HasDerivWithinAt Q + (rightField a (V t) (Q t)) (Icc 0 T) t) + (δ : ℝ) (hinit : dist (Z 0) (Q 0) ≤ δ) + (hUV : ∀ t ∈ Icc 0 T, |U t-V t| ≤ δ) : + ∀ t ∈ Icc 0 T, dist (Z t) (Q t) ≤ + gronwallBound δ (4*a⁻¹^2) (2*a⁻¹^2*δ) t := by + intro t ht + have he := dist_le_of_approx_trajectories_ODE + (K := (⟨4*a⁻¹^2,by positivity⟩ : ℝ≥0)) + (v := fun s => rightField a (U s)) (fun _ => rightField_lipschitz ha _) + (f := Z) (fun s hs => (hZ s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hZ s ⟨hs.1,hs.2.le⟩)) + (εf := 0) (fun _ _ => by simp) + (g := Q) (fun s hs => (hQ s hs).continuousWithinAt) + (fun s hs => hasDerivWithinAt_Ici_of_Icc hs (hQ s ⟨hs.1,hs.2.le⟩)) + (εg := 2*a⁻¹^2*δ) (fun s hs => ?_) hinit t ht + · change dist (Z t) (Q t) ≤ gronwallBound δ (4*a⁻¹^2) (0+2*a⁻¹^2*δ) (t-0) at he + simpa only [zero_add,sub_zero] using he + · rw [dist_comm] + exact (rightField_driver_dist ha _ _ _).trans + (mul_le_mul_of_nonneg_left (hUV s (Ico_subset_Icc_self hs)) (by positivity)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ScaleTraceLaw.lean b/lean/OAI/Probability/SLE/Loewner/ScaleTraceLaw.lean index 913c0d3ec22029a63a0499e01c092b8aeb66321e..8c7f46a97256688b94589b81c8bb7b8cfa44b1df 100644 --- a/lean/OAI/Probability/SLE/Loewner/ScaleTraceLaw.lean +++ b/lean/OAI/Probability/SLE/Loewner/ScaleTraceLaw.lean @@ -1,166 +1,166 @@ -import OAI.Probability.SLE.StoppedMeasures -import OAI.Probability.SLE.InteriorDiskProof - -namespace OAI - -/-! -Equality of actual continuous SLE trace laws under capacity scaling. The -Brownian driver laws agree by finite-dimensional uniqueness. Canonical -reverse-flow approximants transfer the measurable trace factor through that -law equality, so no scaling law for geometric events is assumed. --/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.ScaleTraceLaw - -local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ -local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -theorem brownian_continuous_path_law - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (B B' : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P) - (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hc' : ∀ ω, Continuous (fun t => B' t ω)) : - P.map (sourceDriverPath B hc) = P.map (sourceDriverPath B' hc') := by - have hb : Measurable (fun ω t => B t ω) := Measurable.of_eval hm - have hb' : Measurable (fun ω t => B' t ω) := Measurable.of_eval hm' - have hleft : IsProjectiveLimit (P.map (fun ω t => B t ω)) BrownianReal.projectiveFamily := by - intro I - rw [Measure.map_map (Finset.measurable_restrict I) hb] - exact (hB.hasLaw I).map_eq - have hright : IsProjectiveLimit (P.map (fun ω t => B' t ω)) BrownianReal.projectiveFamily := by - intro I - rw [Measure.map_map (Finset.measurable_restrict I) hb'] - exact (hB'.hasLaw I).map_eq - have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n))) - (hleft.unique hright) - have hem : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) := - Measurable.of_eval (fun _ => measurable_pi_apply _) - rw [Measure.map_map hem hb, Measure.map_map hem hb'] at he - apply source_path_evaluation_embedding.map_injective - rw [Measure.map_map source_path_evaluation_embedding.measurable - (source_path_measurable _ hm), - Measure.map_map source_path_evaluation_embedding.measurable (source_path_measurable _ hm')] - exact he - -/-- Equality of continuous driver laws gives equality of the actual trace -laws. Reverse-flow approximants identify the trace factor on the common law. -/ -theorem trace_path_law_of_driver_law - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B B' : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hc' : ∀ ω, Continuous (fun t => B' t ω)) - (γ γ' : Ω → ℝ≥0 → ℂ) (G F G' F' : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hw' : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B' t ω) (γ' ω) (G' ω) (F' ω)) - (hlaw : P.map (sourceDriverPath B hc) = P.map (sourceDriverPath B' hc')) : - P.map (sourceTracePath γ (fun ω => (hw ω).1)) = - P.map (sourceTracePath γ' (fun ω => (hw' ω).1)) := by - obtain ⟨Φ, hΦm, hΦ, _⟩ := sourceStoppedTraceLaw κ hκ hκ8 B hB hm hc γ G F hw - let bp := sourceDriverPath B hc - let bp' := sourceDriverPath B' hc' - let gp := sourceTracePath γ (fun ω => (hw ω).1) - let gp' := sourceTracePath γ' (fun ω => (hw' ω).1) - have hbm : Measurable bp := source_path_measurable _ hm - have hbm' : Measurable bp' := source_path_measurable _ hm' - choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) => - source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity) - let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1) - have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity - have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) := - tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, - Eventually.of_forall ha⟩ - have hlimit (B₀ : ℝ≥0 → Ω → ℝ) (hc₀ : ∀ ω, Continuous (fun t => B₀ t ω)) - (γ₀ : Ω → ℝ≥0 → ℂ) (G₀ F₀ : Ω → ℝ≥0 → ℂ → ℂ) - (hw₀ : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) - (γ₀ ω) (G₀ ω) (F₀ ω)) (ω : Ω) (t : ℝ≥0) : - Tendsto (fun n => Q t n t (sourceDriverPath B₀ hc₀ ω)) atTop (𝓝 (γ₀ ω t)) := by - have he (n : ℕ) : Q t n t (sourceDriverPath B₀ hc₀ ω) = - F₀ ω t ((Real.sqrt κ * B₀ t ω : ℝ) + (a n : ℂ) * Complex.I) := by - apply (hw₀ ω).reverse_flow_identifies t - ((Real.sqrt κ * B₀ t ω : ℝ) + (a n : ℂ) * Complex.I) (ha n) (by simp) - (fun s => Q t n s (sourceDriverPath B₀ hc₀ ω)) (hQ0 t n _) - intro s hs - exact (hQd t n _ s hs.1).mono Icc_subset_Ici_self - have ht := ((hw₀ ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy - simpa only [he, Function.comp_def] using ht - have hset (t : ℝ≥0) : MeasurableSet - {p : C(ℝ≥0, ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} := - measurableSet_tendsto (𝓝 (0 : ℂ)) (fun n => (hQm t n t).sub - ((continuous_eval_const t).measurable.comp hΦm)) - have hprime (t : ℝ≥0) : ∀ᵐ ω ∂P, Φ (bp' ω) t = γ' ω t := by - have ho : ∀ᵐ p ∂P.map bp, Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by - apply (ae_map_iff hbm.aemeasurable (hset t)).mpr - exact Eventually.of_forall (fun ω => by - have hh := hlimit B hc γ G F hw ω t - have he : Φ (bp ω) t = γ ω t := congrArg (fun p : C(ℝ≥0, ℂ) => p t) (hΦ ω) - rw [he] - exact tendsto_sub_nhds_zero_iff.mpr hh) - rw [hlaw] at ho - filter_upwards [(ae_map_iff hbm'.aemeasurable (hset t)).mp ho] with ω hω - exact tendsto_nhds_unique (tendsto_sub_nhds_zero_iff.mp hω) (hlimit B' hc' γ' G' F' hw' ω t) - have hpath : ∀ᵐ ω ∂P, Φ (bp' ω) = gp' ω := by - filter_upwards [ae_all_iff.mpr (fun n => hprime (denseSeq ℝ≥0 n))] with ω hω - apply DFunLike.coe_injective - exact (denseRange_denseSeq ℝ≥0).equalizer (Φ (bp' ω)).continuous (hw' ω).1 - (funext hω) - calc - P.map gp = (P.map bp).map Φ := by - rw [Measure.map_map hΦm hbm] - congr 1 - exact funext (fun ω => (hΦ ω).symm) - _ = (P.map bp').map Φ := by rw [hlaw] - _ = P.map (Φ ∘ bp') := Measure.map_map hΦm hbm' - _ = P.map gp' := Measure.map_congr hpath - -/-- The actual capacity-normalized scaling on continuous trace paths. -/ -def scalePath (c : ℝ≥0) (p : C(ℝ≥0, ℂ)) : C(ℝ≥0, ℂ) := - ⟨sourceScaleTrace c p, continuous_const.mul - (p.continuous.comp (continuous_id.div_const (c ^ 2)))⟩ - -theorem measurable_scalePath (c : ℝ≥0) : Measurable (scalePath c) := - source_path_measurable _ (fun t => measurable_const.mul - (continuous_eval_const (sourceScaleTime c t)).measurable) - -/-- The scaled actual capacity trace has exactly the original path law. -/ -theorem scaled_trace_path_law - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (c : ℝ≥0) (hcpos : 0 < c) : - P.map (fun ω => scalePath c (sourceTracePath γ (fun ω => (hw ω).1) ω)) = - P.map (sourceTracePath γ (fun ω => (hw ω).1)) := by - let Bc : ℝ≥0 → Ω → ℝ := fun t ω => (c : ℝ) * B (sourceScaleTime c t) ω - have hBc : IsBrownianReal Bc P := by - have hh := hB.smul (c := (c ^ 2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hcpos.ne')) - simpa only [Bc, sourceScaleTime, NNReal.coe_inv, NNReal.coe_pow, Real.sqrt_inv, - Real.sqrt_sq c.coe_nonneg, inv_inv, div_eq_mul_inv, mul_comm] using hh - have hmc : ∀ t, Measurable (Bc t) := fun t => measurable_const.mul (hm _) - have hcc : ∀ ω, Continuous (fun t => Bc t ω) := fun ω => - continuous_const.mul ((hc ω).comp (continuous_id.div_const (c ^ 2))) - have hwc : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * Bc t ω) - (sourceScaleTrace c (γ ω)) (sourceScaleChart c (G ω)) (sourceScaleChart c (F ω)) := by - intro ω - have hh := source_scale_witness c hcpos (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω) (hw ω) - simpa only [Bc, mul_left_comm] using hh - have hlaw := brownian_continuous_path_law B Bc hB hBc hm hmc hc hcc - have he := trace_path_law_of_driver_law κ hκ hκ8 B Bc hB hm hmc hc hcc - γ (fun ω => sourceScaleTrace c (γ ω)) G F (fun ω => sourceScaleChart c (G ω)) - (fun ω => sourceScaleChart c (F ω)) hw hwc hlaw - exact he.symm - -end SLEExactGauge.ScaleTraceLaw - -end - -end OAI +import OAI.Probability.SLE.StoppedMeasures +import OAI.Probability.SLE.InteriorDiskProof + +namespace OAI + +/-! +Equality of actual continuous SLE trace laws under capacity scaling. The +Brownian driver laws agree by finite-dimensional uniqueness. Canonical +reverse-flow approximants transfer the measurable trace factor through that +law equality, so no scaling law for geometric events is assumed. +-/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.ScaleTraceLaw + +local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ +local instance : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +theorem brownian_continuous_path_law + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (B B' : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) (hB' : IsBrownianReal B' P) + (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hc' : ∀ ω, Continuous (fun t => B' t ω)) : + P.map (sourceDriverPath B hc) = P.map (sourceDriverPath B' hc') := by + have hb : Measurable (fun ω t => B t ω) := Measurable.of_eval hm + have hb' : Measurable (fun ω t => B' t ω) := Measurable.of_eval hm' + have hleft : IsProjectiveLimit (P.map (fun ω t => B t ω)) BrownianReal.projectiveFamily := by + intro I + rw [Measure.map_map (Finset.measurable_restrict I) hb] + exact (hB.hasLaw I).map_eq + have hright : IsProjectiveLimit (P.map (fun ω t => B' t ω)) BrownianReal.projectiveFamily := by + intro I + rw [Measure.map_map (Finset.measurable_restrict I) hb'] + exact (hB'.hasLaw I).map_eq + have he := congrArg (Measure.map (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n))) + (hleft.unique hright) + have hem : Measurable (fun p : ℝ≥0 → ℝ => fun n => p (denseSeq ℝ≥0 n)) := + Measurable.of_eval (fun _ => measurable_pi_apply _) + rw [Measure.map_map hem hb, Measure.map_map hem hb'] at he + apply source_path_evaluation_embedding.map_injective + rw [Measure.map_map source_path_evaluation_embedding.measurable + (source_path_measurable _ hm), + Measure.map_map source_path_evaluation_embedding.measurable (source_path_measurable _ hm')] + exact he + +/-- Equality of continuous driver laws gives equality of the actual trace +laws. Reverse-flow approximants identify the trace factor on the common law. -/ +theorem trace_path_law_of_driver_law + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B B' : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hm' : ∀ t, Measurable (B' t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hc' : ∀ ω, Continuous (fun t => B' t ω)) + (γ γ' : Ω → ℝ≥0 → ℂ) (G F G' F' : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hw' : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B' t ω) (γ' ω) (G' ω) (F' ω)) + (hlaw : P.map (sourceDriverPath B hc) = P.map (sourceDriverPath B' hc')) : + P.map (sourceTracePath γ (fun ω => (hw ω).1)) = + P.map (sourceTracePath γ' (fun ω => (hw' ω).1)) := by + obtain ⟨Φ, hΦm, hΦ, _⟩ := sourceStoppedTraceLaw κ hκ hκ8 B hB hm hc γ G F hw + let bp := sourceDriverPath B hc + let bp' := sourceDriverPath B' hc' + let gp := sourceTracePath γ (fun ω => (hw ω).1) + let gp' := sourceTracePath γ' (fun ω => (hw' ω).1) + have hbm : Measurable bp := source_path_measurable _ hm + have hbm' : Measurable bp' := source_path_measurable _ hm' + choose Q hQm hQ0 hQd using fun (t : ℝ≥0) (n : ℕ) => + source_canonical_reverse_flow κ t (a := 1 / ((n : ℝ) + 1)) (by positivity) + let a (n : ℕ) : ℝ := 1 / ((n : ℝ) + 1) + have ha (n : ℕ) : 0 < a n := by dsimp [a]; positivity + have hy : Tendsto a atTop (𝓝[>] (0 : ℝ)) := + tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, + Eventually.of_forall ha⟩ + have hlimit (B₀ : ℝ≥0 → Ω → ℝ) (hc₀ : ∀ ω, Continuous (fun t => B₀ t ω)) + (γ₀ : Ω → ℝ≥0 → ℂ) (G₀ F₀ : Ω → ℝ≥0 → ℂ → ℂ) + (hw₀ : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B₀ t ω) + (γ₀ ω) (G₀ ω) (F₀ ω)) (ω : Ω) (t : ℝ≥0) : + Tendsto (fun n => Q t n t (sourceDriverPath B₀ hc₀ ω)) atTop (𝓝 (γ₀ ω t)) := by + have he (n : ℕ) : Q t n t (sourceDriverPath B₀ hc₀ ω) = + F₀ ω t ((Real.sqrt κ * B₀ t ω : ℝ) + (a n : ℂ) * Complex.I) := by + apply (hw₀ ω).reverse_flow_identifies t + ((Real.sqrt κ * B₀ t ω : ℝ) + (a n : ℂ) * Complex.I) (ha n) (by simp) + (fun s => Q t n s (sourceDriverPath B₀ hc₀ ω)) (hQ0 t n _) + intro s hs + exact (hQd t n _ s hs.1).mono Icc_subset_Ici_self + have ht := ((hw₀ ω).2.2.2.2.2.2.2.2.2.2.2.2 t).comp hy + simpa only [he, Function.comp_def] using ht + have hset (t : ℝ≥0) : MeasurableSet + {p : C(ℝ≥0, ℝ) | Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0)} := + measurableSet_tendsto (𝓝 (0 : ℂ)) (fun n => (hQm t n t).sub + ((continuous_eval_const t).measurable.comp hΦm)) + have hprime (t : ℝ≥0) : ∀ᵐ ω ∂P, Φ (bp' ω) t = γ' ω t := by + have ho : ∀ᵐ p ∂P.map bp, Tendsto (fun n => Q t n t p - Φ p t) atTop (𝓝 0) := by + apply (ae_map_iff hbm.aemeasurable (hset t)).mpr + exact Eventually.of_forall (fun ω => by + have hh := hlimit B hc γ G F hw ω t + have he : Φ (bp ω) t = γ ω t := congrArg (fun p : C(ℝ≥0, ℂ) => p t) (hΦ ω) + rw [he] + exact tendsto_sub_nhds_zero_iff.mpr hh) + rw [hlaw] at ho + filter_upwards [(ae_map_iff hbm'.aemeasurable (hset t)).mp ho] with ω hω + exact tendsto_nhds_unique (tendsto_sub_nhds_zero_iff.mp hω) (hlimit B' hc' γ' G' F' hw' ω t) + have hpath : ∀ᵐ ω ∂P, Φ (bp' ω) = gp' ω := by + filter_upwards [ae_all_iff.mpr (fun n => hprime (denseSeq ℝ≥0 n))] with ω hω + apply DFunLike.coe_injective + exact (denseRange_denseSeq ℝ≥0).equalizer (Φ (bp' ω)).continuous (hw' ω).1 + (funext hω) + calc + P.map gp = (P.map bp).map Φ := by + rw [Measure.map_map hΦm hbm] + congr 1 + exact funext (fun ω => (hΦ ω).symm) + _ = (P.map bp').map Φ := by rw [hlaw] + _ = P.map (Φ ∘ bp') := Measure.map_map hΦm hbm' + _ = P.map gp' := Measure.map_congr hpath + +/-- The actual capacity-normalized scaling on continuous trace paths. -/ +def scalePath (c : ℝ≥0) (p : C(ℝ≥0, ℂ)) : C(ℝ≥0, ℂ) := + ⟨sourceScaleTrace c p, continuous_const.mul + (p.continuous.comp (continuous_id.div_const (c ^ 2)))⟩ + +theorem measurable_scalePath (c : ℝ≥0) : Measurable (scalePath c) := + source_path_measurable _ (fun t => measurable_const.mul + (continuous_eval_const (sourceScaleTime c t)).measurable) + +/-- The scaled actual capacity trace has exactly the original path law. -/ +theorem scaled_trace_path_law + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (B : ℝ≥0 → Ω → ℝ) (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (c : ℝ≥0) (hcpos : 0 < c) : + P.map (fun ω => scalePath c (sourceTracePath γ (fun ω => (hw ω).1) ω)) = + P.map (sourceTracePath γ (fun ω => (hw ω).1)) := by + let Bc : ℝ≥0 → Ω → ℝ := fun t ω => (c : ℝ) * B (sourceScaleTime c t) ω + have hBc : IsBrownianReal Bc P := by + have hh := hB.smul (c := (c ^ 2)⁻¹) (inv_ne_zero (pow_ne_zero 2 hcpos.ne')) + simpa only [Bc, sourceScaleTime, NNReal.coe_inv, NNReal.coe_pow, Real.sqrt_inv, + Real.sqrt_sq c.coe_nonneg, inv_inv, div_eq_mul_inv, mul_comm] using hh + have hmc : ∀ t, Measurable (Bc t) := fun t => measurable_const.mul (hm _) + have hcc : ∀ ω, Continuous (fun t => Bc t ω) := fun ω => + continuous_const.mul ((hc ω).comp (continuous_id.div_const (c ^ 2))) + have hwc : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * Bc t ω) + (sourceScaleTrace c (γ ω)) (sourceScaleChart c (G ω)) (sourceScaleChart c (F ω)) := by + intro ω + have hh := source_scale_witness c hcpos (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω) (hw ω) + simpa only [Bc, mul_left_comm] using hh + have hlaw := brownian_continuous_path_law B Bc hB hBc hm hmc hc hcc + have he := trace_path_law_of_driver_law κ hκ hκ8 B Bc hB hm hmc hc hcc + γ (fun ω => sourceScaleTrace c (γ ω)) G F (fun ω => sourceScaleChart c (G ω)) + (fun ω => sourceScaleChart c (F ω)) hw hwc hlaw + exact he.symm + +end SLEExactGauge.ScaleTraceLaw + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/ShiftedHydrodynamicComposition.lean b/lean/OAI/Probability/SLE/Loewner/ShiftedHydrodynamicComposition.lean index eb7377807e69acb3903f49b6d483106bacb36bea..c9e548db1d8e5e8207b9abf1973cd19f8f3030c8 100644 --- a/lean/OAI/Probability/SLE/Loewner/ShiftedHydrodynamicComposition.lean +++ b/lean/OAI/Probability/SLE/Loewner/ShiftedHydrodynamicComposition.lean @@ -1,121 +1,121 @@ -import OAI.Probability.SLE.Loewner.HydrodynamicComposition -import Mathlib.Topology.Algebra.Group.ContinuousDiv - -namespace OAI - -/-! Hydrodynamic normalization under the translated chart compositions -used for a future slit. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma complex_cocompact_eventually_ne_zero : ∀ᶠ z : ℂ in cocompact ℂ, z≠0 := - (isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact - -lemma normalized_ratio_escape {l : Filter ℂ} {F : ℂ → ℂ} - (hl : l≤cocompact ℂ) (hF : Tendsto (fun z => F z/z) l (𝓝 1)) : - Tendsto F l (cocompact ℂ) := by - have hr : Tendsto (fun z => ‖F z/z‖) l (𝓝 1) := by - simpa only [norm_one] using hF.norm - have hn := Filter.Tendsto.pos_mul_atTop (by norm_num : (0:ℝ)<1) hr - (tendsto_norm_cocompact_atTop.mono_left hl) - have hnorm : Tendsto (fun z => ‖F z‖) l atTop := by - apply hn.congr' - filter_upwards [complex_cocompact_eventually_ne_zero.filter_mono hl] - with z hz - rw [norm_div,div_mul_cancel₀ _ (norm_ne_zero_iff.mpr hz)] - simpa only [Metric.cobounded_eq_cocompact] using tendsto_norm_atTop_iff_cobounded.mp hnorm - -lemma complex_inv_tendsto_cocompact_zero : - Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by - simpa only [Metric.cobounded_eq_cocompact] using - (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) - -theorem shifted_hydrodynamic_coefficient {G H : ℂ → ℂ} {c d : ℂ} (u : ℝ) - (hG : Tendsto (fun z => z*(G z-z)) (cocompact ℂ) (𝓝 c)) - (hH : Tendsto (fun z => z*(H z-z)) (cocompact ℂ) (𝓝 d)) : - Tendsto (fun z => z*((u:ℂ)+H (G z-(u:ℂ))-z)) (cocompact ℂ) (𝓝 (c+d)) := by - have hu : Tendsto (fun z : ℂ => (u:ℂ)/z) (cocompact ℂ) (𝓝 0) := by - simpa only [div_eq_mul_inv,mul_zero] using complex_inv_tendsto_cocompact_zero.const_mul (u:ℂ) - have hs : Tendsto (fun z => (G z-(u:ℂ))/z) (cocompact ℂ) (𝓝 1) := by - simpa only [sub_div,sub_zero] using (hydrodynamic_ratio_limit hG).sub hu - have he := normalized_ratio_escape le_rfl hs - have hq : Tendsto (fun z => z/(G z-(u:ℂ))) (cocompact ℂ) (𝓝 1) := by - simpa only [inv_div,inv_one] using hs.inv₀ one_ne_zero - have hh := (hq.mul (hH.comp he)).add hG - rw [one_mul,add_comm d c] at hh - apply hh.congr' - filter_upwards [he.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] - with z hz - dsimp only [Function.comp_apply] - field_simp [show G z-(u:ℂ)≠0 from hz] - ring - -lemma upper_translation_tendsto (u : ℝ) : - Tendsto (fun z : ℂ => z+(u:ℂ)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) := by - apply tendsto_inf.mpr - constructor - · exact (Homeomorph.addRight (u:ℂ)).isClosedEmbedding.tendsto_cocompact.mono_left inf_le_left - · apply tendsto_principal.mpr - rw [eventually_inf_principal] - exact Eventually.of_forall fun z hz => by - change 0<(z+(u:ℂ)).im - simpa only [add_im,ofReal_im,add_zero] using (show 0 F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) - (hJ : Tendsto (fun z => J z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) - (hJm : MapsTo J upperHalfPlane upperHalfPlane) : - Tendsto (fun z => F ((u:ℂ)+J (z-(u:ℂ)))/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - have hshift := upper_translation_tendsto (-u) - simp only [ofReal_neg,← sub_eq_add_neg] at hshift - have hi := complex_inv_tendsto_cocompact_zero.mono_left - (show cocompact ℂ ⊓ 𝓟 upperHalfPlane≤cocompact ℂ from inf_le_left) - have hu : Tendsto (fun z : ℂ => (u:ℂ)/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by - simpa only [div_eq_mul_inv,mul_zero] using hi.const_mul (u:ℂ) - have hratio : Tendsto (fun z : ℂ => (z-(u:ℂ))/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - have hh := (tendsto_const_nhds : Tendsto (fun _ : ℂ => (1:ℂ)) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)).sub hu - rw [sub_zero] at hh - apply hh.congr' - filter_upwards [complex_cocompact_eventually_ne_zero.filter_mono - (show cocompact ℂ ⊓ 𝓟 upperHalfPlane≤cocompact ℂ from inf_le_left)] with z hz - rw [sub_div,div_self (show z≠0 from hz)] - have hJratio := (hJ.comp hshift).mul hratio - have hmid : Tendsto (fun z : ℂ => ((u:ℂ)+J (z-(u:ℂ)))/z) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - have hh := hu.add hJratio - simp only [zero_add,mul_one] at hh - apply hh.congr' - filter_upwards [(hshift.mono_right inf_le_left).eventually - ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] with z hz - dsimp only [Function.comp_apply] - field_simp [show z-(u:ℂ)≠0 from hz] - have hmidEscape := normalized_ratio_escape inf_le_left hmid - have hmidUpper : Tendsto (fun z : ℂ => (u:ℂ)+J (z-(u:ℂ))) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓟 upperHalfPlane) := by - apply tendsto_principal.mpr - rw [eventually_inf_principal] - exact Eventually.of_forall fun z hz => by - have hsub : z-(u:ℂ)∈upperHalfPlane := by - simpa only [upperHalfPlane,mem_ofPred_eq,sub_im,ofReal_im,sub_zero] using hz - simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hJm hsub - have hh := (hF.comp (tendsto_inf.mpr ⟨hmidEscape,hmidUpper⟩)).mul hmid - rw [mul_one] at hh - apply hh.congr' - filter_upwards [hmidEscape.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] - with z hz - dsimp only [Function.comp_apply] - field_simp [show (u:ℂ)+J (z-(u:ℂ))≠0 from hz] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.HydrodynamicComposition +import Mathlib.Topology.Algebra.Group.ContinuousDiv + +namespace OAI + +/-! Hydrodynamic normalization under the translated chart compositions +used for a future slit. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma complex_cocompact_eventually_ne_zero : ∀ᶠ z : ℂ in cocompact ℂ, z≠0 := + (isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact + +lemma normalized_ratio_escape {l : Filter ℂ} {F : ℂ → ℂ} + (hl : l≤cocompact ℂ) (hF : Tendsto (fun z => F z/z) l (𝓝 1)) : + Tendsto F l (cocompact ℂ) := by + have hr : Tendsto (fun z => ‖F z/z‖) l (𝓝 1) := by + simpa only [norm_one] using hF.norm + have hn := Filter.Tendsto.pos_mul_atTop (by norm_num : (0:ℝ)<1) hr + (tendsto_norm_cocompact_atTop.mono_left hl) + have hnorm : Tendsto (fun z => ‖F z‖) l atTop := by + apply hn.congr' + filter_upwards [complex_cocompact_eventually_ne_zero.filter_mono hl] + with z hz + rw [norm_div,div_mul_cancel₀ _ (norm_ne_zero_iff.mpr hz)] + simpa only [Metric.cobounded_eq_cocompact] using tendsto_norm_atTop_iff_cobounded.mp hnorm + +lemma complex_inv_tendsto_cocompact_zero : + Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by + simpa only [Metric.cobounded_eq_cocompact] using + (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) + +theorem shifted_hydrodynamic_coefficient {G H : ℂ → ℂ} {c d : ℂ} (u : ℝ) + (hG : Tendsto (fun z => z*(G z-z)) (cocompact ℂ) (𝓝 c)) + (hH : Tendsto (fun z => z*(H z-z)) (cocompact ℂ) (𝓝 d)) : + Tendsto (fun z => z*((u:ℂ)+H (G z-(u:ℂ))-z)) (cocompact ℂ) (𝓝 (c+d)) := by + have hu : Tendsto (fun z : ℂ => (u:ℂ)/z) (cocompact ℂ) (𝓝 0) := by + simpa only [div_eq_mul_inv,mul_zero] using complex_inv_tendsto_cocompact_zero.const_mul (u:ℂ) + have hs : Tendsto (fun z => (G z-(u:ℂ))/z) (cocompact ℂ) (𝓝 1) := by + simpa only [sub_div,sub_zero] using (hydrodynamic_ratio_limit hG).sub hu + have he := normalized_ratio_escape le_rfl hs + have hq : Tendsto (fun z => z/(G z-(u:ℂ))) (cocompact ℂ) (𝓝 1) := by + simpa only [inv_div,inv_one] using hs.inv₀ one_ne_zero + have hh := (hq.mul (hH.comp he)).add hG + rw [one_mul,add_comm d c] at hh + apply hh.congr' + filter_upwards [he.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] + with z hz + dsimp only [Function.comp_apply] + field_simp [show G z-(u:ℂ)≠0 from hz] + ring + +lemma upper_translation_tendsto (u : ℝ) : + Tendsto (fun z : ℂ => z+(u:ℂ)) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) := by + apply tendsto_inf.mpr + constructor + · exact (Homeomorph.addRight (u:ℂ)).isClosedEmbedding.tendsto_cocompact.mono_left inf_le_left + · apply tendsto_principal.mpr + rw [eventually_inf_principal] + exact Eventually.of_forall fun z hz => by + change 0<(z+(u:ℂ)).im + simpa only [add_im,ofReal_im,add_zero] using (show 0 F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) + (hJ : Tendsto (fun z => J z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)) + (hJm : MapsTo J upperHalfPlane upperHalfPlane) : + Tendsto (fun z => F ((u:ℂ)+J (z-(u:ℂ)))/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + have hshift := upper_translation_tendsto (-u) + simp only [ofReal_neg,← sub_eq_add_neg] at hshift + have hi := complex_inv_tendsto_cocompact_zero.mono_left + (show cocompact ℂ ⊓ 𝓟 upperHalfPlane≤cocompact ℂ from inf_le_left) + have hu : Tendsto (fun z : ℂ => (u:ℂ)/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0) := by + simpa only [div_eq_mul_inv,mul_zero] using hi.const_mul (u:ℂ) + have hratio : Tendsto (fun z : ℂ => (z-(u:ℂ))/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + have hh := (tendsto_const_nhds : Tendsto (fun _ : ℂ => (1:ℂ)) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1)).sub hu + rw [sub_zero] at hh + apply hh.congr' + filter_upwards [complex_cocompact_eventually_ne_zero.filter_mono + (show cocompact ℂ ⊓ 𝓟 upperHalfPlane≤cocompact ℂ from inf_le_left)] with z hz + rw [sub_div,div_self (show z≠0 from hz)] + have hJratio := (hJ.comp hshift).mul hratio + have hmid : Tendsto (fun z : ℂ => ((u:ℂ)+J (z-(u:ℂ)))/z) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + have hh := hu.add hJratio + simp only [zero_add,mul_one] at hh + apply hh.congr' + filter_upwards [(hshift.mono_right inf_le_left).eventually + ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] with z hz + dsimp only [Function.comp_apply] + field_simp [show z-(u:ℂ)≠0 from hz] + have hmidEscape := normalized_ratio_escape inf_le_left hmid + have hmidUpper : Tendsto (fun z : ℂ => (u:ℂ)+J (z-(u:ℂ))) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓟 upperHalfPlane) := by + apply tendsto_principal.mpr + rw [eventually_inf_principal] + exact Eventually.of_forall fun z hz => by + have hsub : z-(u:ℂ)∈upperHalfPlane := by + simpa only [upperHalfPlane,mem_ofPred_eq,sub_im,ofReal_im,sub_zero] using hz + simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hJm hsub + have hh := (hF.comp (tendsto_inf.mpr ⟨hmidEscape,hmidUpper⟩)).mul hmid + rw [mul_one] at hh + apply hh.congr' + filter_upwards [hmidEscape.eventually ((isCompact_singleton (x := (0:ℂ))).compl_mem_cocompact)] + with z hz + dsimp only [Function.comp_apply] + field_simp [show (u:ℂ)+J (z-(u:ℂ))≠0 from hz] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitBoundaryBarrier.lean b/lean/OAI/Probability/SLE/Loewner/SlitBoundaryBarrier.lean index a6566d77e2215375f0233dbd6211a5cd55d5853f..0d5c7b5f78e3f33863a7a260f008140dcd944c0c 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitBoundaryBarrier.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitBoundaryBarrier.lean @@ -1,95 +1,95 @@ -import OAI.Probability.SLE.Loewner.InitialCapacity -import OAI.Probability.SLE.Loewner.TraceBoundaryBarrier - -namespace OAI - -/-! Compact, locally path-connected boundary barriers for static attached slits. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -def slitBoundaryBarrier (p : C(unitInterval,ℂ)) (R : ℝ) : Set ℂ := - range p ∪ (Complex.ofReal '' Icc (-R) R) - -lemma slitBoundaryBarrier_compact (p : C(unitInterval,ℂ)) (R : ℝ) : - IsCompact (slitBoundaryBarrier p R) := - (isCompact_range p.continuous).union (isCompact_Icc.image Complex.continuous_ofReal) - -lemma slitBoundaryBarrier_locallyPathConnected (p : C(unitInterval,ℂ)) - (R : ℝ) (hR : 0≤R) : LocallyPathConnectedSpace (slitBoundaryBarrier p R) := by - have : LocallyPathConnectedSpace unitInterval := - (isQuotientMap_projIcc (a := (0:ℝ)) (b := 1) (h := zero_le_one)).locallyPathConnectedSpace - let q : unitInterval → range p := fun t => ⟨p t,mem_range_self t⟩ - have hq : Continuous q := p.continuous.subtype_mk _ - have hs : Surjective q := by - rintro ⟨_,t,rfl⟩ - exact ⟨t,rfl⟩ - have : LocallyPathConnectedSpace (range p) := - (hq.isClosedMap.isQuotientMap hq hs).locallyPathConnectedSpace - have : LocallyPathConnectedSpace (Icc (-R) R) := - (isQuotientMap_projIcc (a := -R) (b := R) (h := by linarith)).locallyPathConnectedSpace - let r : Icc (-R) R → Complex.ofReal '' Icc (-R) R := fun x => ⟨x,⟨x,x.2,rfl⟩⟩ - have hr : Continuous r := (Complex.continuous_ofReal.comp continuous_subtype_val).subtype_mk _ - have hs : Surjective r := by - rintro ⟨_,x,hx,rfl⟩ - exact ⟨⟨x,hx⟩,rfl⟩ - have : LocallyPathConnectedSpace (Complex.ofReal '' Icc (-R) R) := - (hr.isClosedMap.isQuotientMap hr hs).locallyPathConnectedSpace - exact compact_union_locallyPathConnected (isCompact_range p.continuous) - (isCompact_Icc.image Complex.continuous_ofReal) - -lemma slitBoundaryBarrier_disjoint_domain (p : C(unitInterval,ℂ)) (R : ℝ) : - Disjoint (slitBoundaryBarrier p R) (upperHalfPlane \ range p) := by - apply disjoint_left.mpr - intro z hz hD - rcases hz with hz | ⟨x,_,rfl⟩ - · exact hD.2 hz - · have him := hD.1 - change 0<(x:ℂ).im at him - simp at him - -lemma frontier_mem_slitBoundaryBarrier (p : C(unitInterval,ℂ)) {R : ℝ} {z : ℂ} - (hz : z∈frontier (upperHalfPlane \ range p)) (hR : ‖z‖≤R) : - z∈slitBoundaryBarrier p R := by - by_cases hp : z∈range p - · exact Or.inl hp - right - have hsub : upperHalfPlane \ range p ⊆ {w : ℂ | 0≤w.im} := by - intro w hw - exact (show 0 simp [him] - -lemma slitBoundaryBarrier_uniform_paths (p : C(unitInterval,ℂ)) - (R : ℝ) (hR : 0≤R) (ε : ℝ) (hε : 0<ε) : - ∃ δ : ℝ, 0<δ ∧ ∀ x∈slitBoundaryBarrier p R, ∀ y∈slitBoundaryBarrier p R, - dist x y<δ → ∃ q : Path x y, (∀ s, q s∈slitBoundaryBarrier p R) ∧ - ∀ s, dist (q s) x<ε := by - have : CompactSpace (slitBoundaryBarrier p R) := - isCompact_iff_compactSpace.mp (slitBoundaryBarrier_compact p R) - have : LocallyPathConnectedSpace (slitBoundaryBarrier p R) := - slitBoundaryBarrier_locallyPathConnected p R hR - obtain ⟨δ,hδ,hpath⟩ := compact_locallyPathConnected_uniform_path - (A := slitBoundaryBarrier p R) ε hε - refine ⟨δ,hδ,fun x hx y hy hxy => ?_⟩ - obtain ⟨q,hq⟩ := hpath ⟨x,hx⟩ ⟨y,hy⟩ hxy - exact ⟨q.map continuous_subtype_val,fun s => (q s).property,hq⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialCapacity +import OAI.Probability.SLE.Loewner.TraceBoundaryBarrier + +namespace OAI + +/-! Compact, locally path-connected boundary barriers for static attached slits. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +def slitBoundaryBarrier (p : C(unitInterval,ℂ)) (R : ℝ) : Set ℂ := + range p ∪ (Complex.ofReal '' Icc (-R) R) + +lemma slitBoundaryBarrier_compact (p : C(unitInterval,ℂ)) (R : ℝ) : + IsCompact (slitBoundaryBarrier p R) := + (isCompact_range p.continuous).union (isCompact_Icc.image Complex.continuous_ofReal) + +lemma slitBoundaryBarrier_locallyPathConnected (p : C(unitInterval,ℂ)) + (R : ℝ) (hR : 0≤R) : LocallyPathConnectedSpace (slitBoundaryBarrier p R) := by + have : LocallyPathConnectedSpace unitInterval := + (isQuotientMap_projIcc (a := (0:ℝ)) (b := 1) (h := zero_le_one)).locallyPathConnectedSpace + let q : unitInterval → range p := fun t => ⟨p t,mem_range_self t⟩ + have hq : Continuous q := p.continuous.subtype_mk _ + have hs : Surjective q := by + rintro ⟨_,t,rfl⟩ + exact ⟨t,rfl⟩ + have : LocallyPathConnectedSpace (range p) := + (hq.isClosedMap.isQuotientMap hq hs).locallyPathConnectedSpace + have : LocallyPathConnectedSpace (Icc (-R) R) := + (isQuotientMap_projIcc (a := -R) (b := R) (h := by linarith)).locallyPathConnectedSpace + let r : Icc (-R) R → Complex.ofReal '' Icc (-R) R := fun x => ⟨x,⟨x,x.2,rfl⟩⟩ + have hr : Continuous r := (Complex.continuous_ofReal.comp continuous_subtype_val).subtype_mk _ + have hs : Surjective r := by + rintro ⟨_,x,hx,rfl⟩ + exact ⟨⟨x,hx⟩,rfl⟩ + have : LocallyPathConnectedSpace (Complex.ofReal '' Icc (-R) R) := + (hr.isClosedMap.isQuotientMap hr hs).locallyPathConnectedSpace + exact compact_union_locallyPathConnected (isCompact_range p.continuous) + (isCompact_Icc.image Complex.continuous_ofReal) + +lemma slitBoundaryBarrier_disjoint_domain (p : C(unitInterval,ℂ)) (R : ℝ) : + Disjoint (slitBoundaryBarrier p R) (upperHalfPlane \ range p) := by + apply disjoint_left.mpr + intro z hz hD + rcases hz with hz | ⟨x,_,rfl⟩ + · exact hD.2 hz + · have him := hD.1 + change 0<(x:ℂ).im at him + simp at him + +lemma frontier_mem_slitBoundaryBarrier (p : C(unitInterval,ℂ)) {R : ℝ} {z : ℂ} + (hz : z∈frontier (upperHalfPlane \ range p)) (hR : ‖z‖≤R) : + z∈slitBoundaryBarrier p R := by + by_cases hp : z∈range p + · exact Or.inl hp + right + have hsub : upperHalfPlane \ range p ⊆ {w : ℂ | 0≤w.im} := by + intro w hw + exact (show 0 simp [him] + +lemma slitBoundaryBarrier_uniform_paths (p : C(unitInterval,ℂ)) + (R : ℝ) (hR : 0≤R) (ε : ℝ) (hε : 0<ε) : + ∃ δ : ℝ, 0<δ ∧ ∀ x∈slitBoundaryBarrier p R, ∀ y∈slitBoundaryBarrier p R, + dist x y<δ → ∃ q : Path x y, (∀ s, q s∈slitBoundaryBarrier p R) ∧ + ∀ s, dist (q s) x<ε := by + have : CompactSpace (slitBoundaryBarrier p R) := + isCompact_iff_compactSpace.mp (slitBoundaryBarrier_compact p R) + have : LocallyPathConnectedSpace (slitBoundaryBarrier p R) := + slitBoundaryBarrier_locallyPathConnected p R hR + obtain ⟨δ,hδ,hpath⟩ := compact_locallyPathConnected_uniform_path + (A := slitBoundaryBarrier p R) ε hε + refine ⟨δ,hδ,fun x hx y hy hxy => ?_⟩ + obtain ⟨q,hq⟩ := hpath ⟨x,hx⟩ ⟨y,hy⟩ hxy + exact ⟨q.map continuous_subtype_val,fun s => (q s).property,hq⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitBoundaryFiber.lean b/lean/OAI/Probability/SLE/Loewner/SlitBoundaryFiber.lean index 93abd6e517021473a3d656527056bfb237849745..57f00cadb1348b460bb4f86f30b990c2a01d5da3 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitBoundaryFiber.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitBoundaryFiber.lean @@ -1,62 +1,62 @@ -import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryIdentity -import OAI.Probability.SLE.Loewner.SlitChartBoundaryExtension - -namespace OAI - -/-! No interval of real boundary parameters collapses to a single point -under the continuous inverse slit chart. -/ -noncomputable section -open Set Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma vertical_point_mem_ball (x : ℝ) {y r : ℝ} (hy : 0 he z hz) - have hlocal := halfPlane_boundary_identity hC hd x hr c (fun z hz hzi => by - have hre : |z.re-x| simp [hzi] - rw [hez] - exact hconst z.re ⟨by linarith,by linarith⟩) - let z₁ : ℂ := (x:ℂ)+((r/2:ℝ):ℂ)*I - let z₂ : ℂ := (x:ℂ)+((r/3:ℝ):ℂ)*I - have hz₁ : z₁∈upperHalfPlane := by - change 0 he z hz) + have hlocal := halfPlane_boundary_identity hC hd x hr c (fun z hz hzi => by + have hre : |z.re-x| simp [hzi] + rw [hez] + exact hconst z.re ⟨by linarith,by linarith⟩) + let z₁ : ℂ := (x:ℂ)+((r/2:ℝ):ℂ)*I + let z₂ : ℂ := (x:ℂ)+((r/3:ℝ):ℂ)*I + have hz₁ : z₁∈upperHalfPlane := by + change 0 2/(slitReflectedFlow p hi h0 hH T (x:ℂ) t- - (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))) := - continuous_const.div ((slitReflectedFlow_real_continuous p hi h0 hH T hx).sub - (Complex.continuous_ofReal.comp - ((slitCapacityDriver p hi h0 hH hT).continuous.comp continuous_real_toNNReal))) - (slitReflectedFlow_separated p hi h0 hH hT hx) - -lemma slitReflectedFlow_near_real_mem (p : C(unitInterval,ℂ)) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {x : ℝ} (hx : x≠0) : - ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), (x:ℂ)+(y:ℂ)*I∈upperHalfPlane \ range p := by - have hD : IsOpen (range p)ᶜ := (isCompact_range p.continuous).isClosed.isOpen_compl - have hxD : (x:ℂ)∈(range p)ᶜ := fun he => real_not_reflected_arc p h0 hH hx (Or.inl he) - have hv : Tendsto (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (x:ℂ)) := by - have hc : Continuous (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) := by fun_prop - simpa only [ofReal_zero,zero_mul,add_zero] using - (hc.tendsto 0).mono_left nhdsWithin_le_nhds - filter_upwards [self_mem_nhdsWithin,hv.eventually (hD.mem_nhds hxD)] with y hy hyD - exact ⟨by simpa [upperHalfPlane] using hy,hyD⟩ - -lemma slitReflectedFlow_near_real_kernel_bound (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) : - ∃ C : ℝ, ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), - ∀ t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ), - ‖2/(slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I) t- - (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))‖≤C := by - obtain ⟨m,hm,hgap⟩ := slitReflectedFlow_positive_gap p hi h0 hH hT hx - have hc := Metric.tendstoUniformly_iff.mp (slitReflectedFlow_uniform_limit p hi h0 hH T hx) - (m/2) (half_pos hm) - refine ⟨4/m,?_⟩ - filter_upwards [hc] with y hy t ht - let q := slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I) t - let r := slitReflectedFlow p hi h0 hH T (x:ℂ) t - let u : ℂ := (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ) - have hsmall : ‖r-q‖< m/2 := by simpa only [dist_eq_norm] using hy t - have htri := norm_sub_le_norm_sub_add_norm_sub r q u - have hlarge : m/2<‖q-u‖ := by linarith [hgap t ht] - have hnorm : 0<‖q-u‖ := (half_pos hm).trans hlarge - change ‖(2:ℂ)/(q-u)‖≤4/m - rw [norm_div,Complex.norm_ofNat] - apply (div_le_div_iff₀ hnorm hm).mpr - nlinarith - -theorem slitReflectedFlow_integral_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) - {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : - Tendsto (fun y : ℝ => ∫ u in 0..t, - 2/(slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I) u- - (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ))) (𝓝[>] (0:ℝ)) - (𝓝 (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T (x:ℂ) u- - (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))) := by - obtain ⟨C,hC⟩ := slitReflectedFlow_near_real_kernel_bound p hi h0 hH hT hx - have hmem := slitReflectedFlow_near_real_mem p h0 hH hx - simp_rw [intervalIntegral.integral_of_le ht.1] - apply tendsto_integral_filter_of_dominated_convergence (fun _ => C) - · filter_upwards [hmem] with y hy - exact (slitReflectedFlow_interior_kernel_continuous p hi h0 hH hT hy).aestronglyMeasurable - · filter_upwards [hC] with y hy - filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u hu - exact hy u ⟨hu.1.le,hu.2.trans ht.2⟩ - · exact integrable_const _ - · filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u _hu - exact tendsto_const_nhds.div - (((slitReflectedFlow_uniform_limit p hi h0 hH T hx).tendsto_at u).sub_const _) - (slitReflectedFlow_separated p hi h0 hH hT hx u) - -theorem slitReflectedFlow_boundary_integral (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) - {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : - (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T (x:ℂ) u- - (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))= - slitReflectedFlow p hi h0 hH T (x:ℂ) t-(x:ℂ) := by - have hv : Tendsto (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (x:ℂ)) := by - have hc : Continuous (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) := by fun_prop - simpa only [ofReal_zero,zero_mul,add_zero] using - (hc.tendsto 0).mono_left nhdsWithin_le_nhds - apply tendsto_nhds_unique (slitReflectedFlow_integral_tendsto p hi h0 hH hT hx ht) - apply (((slitReflectedFlow_uniform_limit p hi h0 hH T hx).tendsto_at t).sub hv).congr' - filter_upwards [slitReflectedFlow_near_real_mem p h0 hH hx] with y hy - exact (slitReflectedFlow_interior_integral p hi h0 hH hT hy ht).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitReflectedInterior + +namespace OAI + +/-! The uniformly separated boundary coordinates admit passage through +the exact interior Loewner integral. -/ +noncomputable section +open Set Filter Function Complex Metric MeasureTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma slitReflectedFlow_real_kernel_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) : + Continuous (fun t : ℝ => 2/(slitReflectedFlow p hi h0 hH T (x:ℂ) t- + (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))) := + continuous_const.div ((slitReflectedFlow_real_continuous p hi h0 hH T hx).sub + (Complex.continuous_ofReal.comp + ((slitCapacityDriver p hi h0 hH hT).continuous.comp continuous_real_toNNReal))) + (slitReflectedFlow_separated p hi h0 hH hT hx) + +lemma slitReflectedFlow_near_real_mem (p : C(unitInterval,ℂ)) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {x : ℝ} (hx : x≠0) : + ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), (x:ℂ)+(y:ℂ)*I∈upperHalfPlane \ range p := by + have hD : IsOpen (range p)ᶜ := (isCompact_range p.continuous).isClosed.isOpen_compl + have hxD : (x:ℂ)∈(range p)ᶜ := fun he => real_not_reflected_arc p h0 hH hx (Or.inl he) + have hv : Tendsto (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (x:ℂ)) := by + have hc : Continuous (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) := by fun_prop + simpa only [ofReal_zero,zero_mul,add_zero] using + (hc.tendsto 0).mono_left nhdsWithin_le_nhds + filter_upwards [self_mem_nhdsWithin,hv.eventually (hD.mem_nhds hxD)] with y hy hyD + exact ⟨by simpa [upperHalfPlane] using hy,hyD⟩ + +lemma slitReflectedFlow_near_real_kernel_bound (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) : + ∃ C : ℝ, ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), + ∀ t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ), + ‖2/(slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I) t- + (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))‖≤C := by + obtain ⟨m,hm,hgap⟩ := slitReflectedFlow_positive_gap p hi h0 hH hT hx + have hc := Metric.tendstoUniformly_iff.mp (slitReflectedFlow_uniform_limit p hi h0 hH T hx) + (m/2) (half_pos hm) + refine ⟨4/m,?_⟩ + filter_upwards [hc] with y hy t ht + let q := slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I) t + let r := slitReflectedFlow p hi h0 hH T (x:ℂ) t + let u : ℂ := (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ) + have hsmall : ‖r-q‖< m/2 := by simpa only [dist_eq_norm] using hy t + have htri := norm_sub_le_norm_sub_add_norm_sub r q u + have hlarge : m/2<‖q-u‖ := by linarith [hgap t ht] + have hnorm : 0<‖q-u‖ := (half_pos hm).trans hlarge + change ‖(2:ℂ)/(q-u)‖≤4/m + rw [norm_div,Complex.norm_ofNat] + apply (div_le_div_iff₀ hnorm hm).mpr + nlinarith + +theorem slitReflectedFlow_integral_tendsto (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) + {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : + Tendsto (fun y : ℝ => ∫ u in 0..t, + 2/(slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I) u- + (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ))) (𝓝[>] (0:ℝ)) + (𝓝 (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T (x:ℂ) u- + (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))) := by + obtain ⟨C,hC⟩ := slitReflectedFlow_near_real_kernel_bound p hi h0 hH hT hx + have hmem := slitReflectedFlow_near_real_mem p h0 hH hx + simp_rw [intervalIntegral.integral_of_le ht.1] + apply tendsto_integral_filter_of_dominated_convergence (fun _ => C) + · filter_upwards [hmem] with y hy + exact (slitReflectedFlow_interior_kernel_continuous p hi h0 hH hT hy).aestronglyMeasurable + · filter_upwards [hC] with y hy + filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u hu + exact hy u ⟨hu.1.le,hu.2.trans ht.2⟩ + · exact integrable_const _ + · filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u _hu + exact tendsto_const_nhds.div + (((slitReflectedFlow_uniform_limit p hi h0 hH T hx).tendsto_at u).sub_const _) + (slitReflectedFlow_separated p hi h0 hH hT hx u) + +theorem slitReflectedFlow_boundary_integral (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) + {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : + (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T (x:ℂ) u- + (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))= + slitReflectedFlow p hi h0 hH T (x:ℂ) t-(x:ℂ) := by + have hv : Tendsto (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (x:ℂ)) := by + have hc : Continuous (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) := by fun_prop + simpa only [ofReal_zero,zero_mul,add_zero] using + (hc.tendsto 0).mono_left nhdsWithin_le_nhds + apply tendsto_nhds_unique (slitReflectedFlow_integral_tendsto p hi h0 hH hT hx ht) + apply (((slitReflectedFlow_uniform_limit p hi h0 hH T hx).tendsto_at t).sub hv).congr' + filter_upwards [slitReflectedFlow_near_real_mem p h0 hH hx] with y hy + exact (slitReflectedFlow_interior_integral p hi h0 hH hT hy ht).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitBoundaryTip.lean b/lean/OAI/Probability/SLE/Loewner/SlitBoundaryTip.lean index 0cc4f0a2845928542d16217876d8e127cb1e35a8..8aacdc27feb6b18134a52c1c86800b95053ecdb1 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitBoundaryTip.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitBoundaryTip.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Loewner.InitialDrivingTip - -namespace OAI - -/-! The static inverse boundary extension takes the actual driver to -the prescribed tip. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -theorem exists_initial_inverse_boundary_tip (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {s : unitInterval} (hs : s<1) : - ∃ B : ℂ → ℂ, ContinuousOn B {z : ℂ | 0 ≤ z.im} ∧ - (∀ z∈upperHalfPlane, B z=initialInverseChart p hi h0 hH s z) ∧ - B (initialDrivingValue p hi h0 hH s hs:ℂ)=p s := by - by_cases hs0 : s=0 - · subst s - refine ⟨id,continuous_id.continuousOn,?_,?_⟩ - · intro z _ - simp only [initialInverseChart_zero,id_eq] - · simp only [id_eq,initialDrivingValue_zero,ofReal_zero,h0] - obtain ⟨B,hB,hBF⟩ := (initialSlitChart p hi h0 hH s hs0).exists_boundary_extension - have he (z : ℂ) (hz : z∈upperHalfPlane) : B z=initialInverseChart p hi h0 hH s z := by - simpa only [initialInverseChart,dite_eq_right hs0] using hBF z hz - refine ⟨B,hB,he,?_⟩ - let u := initialDrivingValue p hi h0 hH s hs - have hc : Continuous (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) := by fun_prop - have hv : Tendsto (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (u:ℂ)) := by - simpa only [ofReal_zero,zero_mul,add_zero] using - (hc.tendsto 0).mono_left (show 𝓝[>] (0:ℝ) ≤ 𝓝 (0:ℝ) from nhdsWithin_le_nhds) - have hm : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), (u:ℂ)+(y:ℂ)*I∈{z : ℂ | 0 ≤ z.im} := by - filter_upwards [self_mem_nhdsWithin] with y hy - simpa using hy.le - have hb := (hB (u:ℂ) (by simp)).tendsto.comp (tendsto_nhdsWithin_iff.mpr ⟨hv,hm⟩) - apply tendsto_nhds_unique hb - apply (initialInverse_radial_driver_limit p hi h0 hH hs).congr' - filter_upwards [self_mem_nhdsWithin] with y hy - exact (he _ (by simpa [upperHalfPlane] using hy)).symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialDrivingTip + +namespace OAI + +/-! The static inverse boundary extension takes the actual driver to +the prescribed tip. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +theorem exists_initial_inverse_boundary_tip (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {s : unitInterval} (hs : s<1) : + ∃ B : ℂ → ℂ, ContinuousOn B {z : ℂ | 0 ≤ z.im} ∧ + (∀ z∈upperHalfPlane, B z=initialInverseChart p hi h0 hH s z) ∧ + B (initialDrivingValue p hi h0 hH s hs:ℂ)=p s := by + by_cases hs0 : s=0 + · subst s + refine ⟨id,continuous_id.continuousOn,?_,?_⟩ + · intro z _ + simp only [initialInverseChart_zero,id_eq] + · simp only [id_eq,initialDrivingValue_zero,ofReal_zero,h0] + obtain ⟨B,hB,hBF⟩ := (initialSlitChart p hi h0 hH s hs0).exists_boundary_extension + have he (z : ℂ) (hz : z∈upperHalfPlane) : B z=initialInverseChart p hi h0 hH s z := by + simpa only [initialInverseChart,dite_eq_right hs0] using hBF z hz + refine ⟨B,hB,he,?_⟩ + let u := initialDrivingValue p hi h0 hH s hs + have hc : Continuous (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) := by fun_prop + have hv : Tendsto (fun y : ℝ => (u:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (u:ℂ)) := by + simpa only [ofReal_zero,zero_mul,add_zero] using + (hc.tendsto 0).mono_left (show 𝓝[>] (0:ℝ) ≤ 𝓝 (0:ℝ) from nhdsWithin_le_nhds) + have hm : ∀ᶠ y : ℝ in 𝓝[>] (0:ℝ), (u:ℂ)+(y:ℂ)*I∈{z : ℂ | 0 ≤ z.im} := by + filter_upwards [self_mem_nhdsWithin] with y hy + simpa using hy.le + have hb := (hB (u:ℂ) (by simp)).tendsto.comp (tendsto_nhdsWithin_iff.mpr ⟨hv,hm⟩) + apply tendsto_nhds_unique hb + apply (initialInverse_radial_driver_limit p hi h0 hH hs).congr' + filter_upwards [self_mem_nhdsWithin] with y hy + exact (he _ (by simpa [upperHalfPlane] using hy)).symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitCapacityChart.lean b/lean/OAI/Probability/SLE/Loewner/SlitCapacityChart.lean index 6476184977aed22e51a25c1ff43f0c15138546c5..9fd8ba2124baa4c612e1dde4ce02fdd02a3db6d4 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitCapacityChart.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitCapacityChart.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Loewner.HydrodynamicCoefficient - -namespace OAI - -/-! Nonnegative capacity and height comparison for the exact normalized -chart of a finite simple attached slit. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -theorem simple_slit_exists_nonnegative_capacity_chart - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ∃ (G F : ℂ → ℂ) (c : ℝ), 0≤c ∧ - BijOn G (upperHalfPlane \ range p) upperHalfPlane ∧ - DifferentiableOn ℂ G (upperHalfPlane \ range p) ∧ - DifferentiableOn ℂ F upperHalfPlane ∧ - MapsTo F upperHalfPlane (upperHalfPlane \ range p) ∧ - (∀ z ∈ upperHalfPlane \ range p, F (G z)=z) ∧ - (∀ z ∈ upperHalfPlane, G (F z)=z) ∧ - Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) ∧ - Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) ∧ - (∀ z ∈ upperHalfPlane \ range p, (G z).im ≤ z.im) := by - obtain ⟨f,a,ha,hb,hf,hf0,hfa,hs,hu⟩ := - reflected_slit_exists_normalized_chart p hi h0 hH - have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union - ((isCompact_range p.continuous).image Complex.continuous_conj) - have hD : invertedExterior (reflectedArcSet p) ∈ 𝓝 (0:ℂ) := - (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) - have hreal := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD - (slitInfinityFactor_reflection hD hs a)) - let G := slitHydrodynamicMap f a - have hG : BijOn G (upperHalfPlane \ range p) upperHalfPlane := by - apply ((positive_real_affine_halfPlane ha (deriv (slitInfinityFactor f a) 0).re).comp - (exteriorJoukowski_bijOn.comp (slitExteriorMap_bijOn p h0 hH hs hu))).congr - intro z _ - change (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- - ((deriv (slitInfinityFactor f a) 0).re:ℂ)= - (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)-deriv (slitInfinityFactor f a) 0 - rw [← hreal] - have hdG : DifferentiableOn ℂ G (upperHalfPlane \ range p) := - ((exteriorJoukowski_differentiableOn.comp - (slitExteriorMap_differentiableOn p h0 hH hf hb.injOn hf0) - (slitExteriorMap_bijOn p h0 hH hs hu).mapsTo).const_mul (a:ℂ)).sub_const _ - let F := invFunOn G (upperHalfPlane \ range p) - have ho : IsOpen (upperHalfPlane \ range p) := - isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed - have hdF : DifferentiableOn ℂ F upperHalfPlane := by - simpa only [F,hG.image_eq] using hdG.invFunOn ho hG.injOn - have hmF : MapsTo F upperHalfPlane (upperHalfPlane \ range p) := hG.surjOn.mapsTo_invFunOn - have hl : ∀ z ∈ upperHalfPlane \ range p, F (G z)=z := hG.injOn.leftInvOn_invFunOn - have hr : ∀ z ∈ upperHalfPlane, G (F z)=z := hG.surjOn.rightInvOn_invFunOn - have hnorm := slit_inverse_tendsto_ratio p h0 hH ha hf hf0 hfa hs F hl - have hheight := normalized_inverse_height hG.mapsTo hdF - (hmF.mono_right sdiff_subset) hl hnorm - let c := (deriv (dslope (slitInfinityFactor f a) 0) 0).re - have hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) := by - have hh := slitExteriorMap_laurent_limit (hf.analyticAt hD) hf0 ha.ne' hfa - have he := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD - (dslope_reflection hD (slitInfinityFactor_reflection hD hs a))) - rw [he] at hh - exact hh - have hc0 : 0≤c := by - apply hydrodynamic_coefficient_nonneg hc - rw [eventually_inf_principal] - filter_upwards [(isCompact_range p.continuous).compl_mem_cocompact] with z hz - exact fun hzH => hheight z ⟨hzH,hz⟩ - exact ⟨G,F,c,hc0,hG,hdG,hdF,hmF,hl,hr,hc,hnorm,hheight⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.HydrodynamicCoefficient + +namespace OAI + +/-! Nonnegative capacity and height comparison for the exact normalized +chart of a finite simple attached slit. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +theorem simple_slit_exists_nonnegative_capacity_chart + (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + ∃ (G F : ℂ → ℂ) (c : ℝ), 0≤c ∧ + BijOn G (upperHalfPlane \ range p) upperHalfPlane ∧ + DifferentiableOn ℂ G (upperHalfPlane \ range p) ∧ + DifferentiableOn ℂ F upperHalfPlane ∧ + MapsTo F upperHalfPlane (upperHalfPlane \ range p) ∧ + (∀ z ∈ upperHalfPlane \ range p, F (G z)=z) ∧ + (∀ z ∈ upperHalfPlane, G (F z)=z) ∧ + Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) ∧ + Tendsto (fun z : ℂ => F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) ∧ + (∀ z ∈ upperHalfPlane \ range p, (G z).im ≤ z.im) := by + obtain ⟨f,a,ha,hb,hf,hf0,hfa,hs,hu⟩ := + reflected_slit_exists_normalized_chart p hi h0 hH + have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj) + have hD : invertedExterior (reflectedArcSet p) ∈ 𝓝 (0:ℂ) := + (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) + have hreal := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD + (slitInfinityFactor_reflection hD hs a)) + let G := slitHydrodynamicMap f a + have hG : BijOn G (upperHalfPlane \ range p) upperHalfPlane := by + apply ((positive_real_affine_halfPlane ha (deriv (slitInfinityFactor f a) 0).re).comp + (exteriorJoukowski_bijOn.comp (slitExteriorMap_bijOn p h0 hH hs hu))).congr + intro z _ + change (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- + ((deriv (slitInfinityFactor f a) 0).re:ℂ)= + (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)-deriv (slitInfinityFactor f a) 0 + rw [← hreal] + have hdG : DifferentiableOn ℂ G (upperHalfPlane \ range p) := + ((exteriorJoukowski_differentiableOn.comp + (slitExteriorMap_differentiableOn p h0 hH hf hb.injOn hf0) + (slitExteriorMap_bijOn p h0 hH hs hu).mapsTo).const_mul (a:ℂ)).sub_const _ + let F := invFunOn G (upperHalfPlane \ range p) + have ho : IsOpen (upperHalfPlane \ range p) := + isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed + have hdF : DifferentiableOn ℂ F upperHalfPlane := by + simpa only [F,hG.image_eq] using hdG.invFunOn ho hG.injOn + have hmF : MapsTo F upperHalfPlane (upperHalfPlane \ range p) := hG.surjOn.mapsTo_invFunOn + have hl : ∀ z ∈ upperHalfPlane \ range p, F (G z)=z := hG.injOn.leftInvOn_invFunOn + have hr : ∀ z ∈ upperHalfPlane, G (F z)=z := hG.surjOn.rightInvOn_invFunOn + have hnorm := slit_inverse_tendsto_ratio p h0 hH ha hf hf0 hfa hs F hl + have hheight := normalized_inverse_height hG.mapsTo hdF + (hmF.mono_right sdiff_subset) hl hnorm + let c := (deriv (dslope (slitInfinityFactor f a) 0) 0).re + have hc : Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) := by + have hh := slitExteriorMap_laurent_limit (hf.analyticAt hD) hf0 ha.ne' hfa + have he := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD + (dslope_reflection hD (slitInfinityFactor_reflection hD hs a))) + rw [he] at hh + exact hh + have hc0 : 0≤c := by + apply hydrodynamic_coefficient_nonneg hc + rw [eventually_inf_principal] + filter_upwards [(isCompact_range p.continuous).compl_mem_cocompact] with z hz + exact fun hzH => hheight z ⟨hzH,hz⟩ + exact ⟨G,F,c,hc0,hG,hdG,hdF,hmF,hl,hr,hc,hnorm,hheight⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitCapacityQuotient.lean b/lean/OAI/Probability/SLE/Loewner/SlitCapacityQuotient.lean index 3288b8c8c721b6eb620579dadb01bc7227b6ba3d..8a1768a36913d3425ae2b2e801cf5842144a8b99 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitCapacityQuotient.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitCapacityQuotient.lean @@ -1,33 +1,33 @@ -import OAI.Probability.SLE.Loewner.SlitCapacityRemainder - -namespace OAI - -/-! Dividing by the positive, exact capacity gives the first-order -Loewner kernel with a radius-controlled error. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -theorem SimpleSlitChart.forward_quotient_kernel {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 (not_lt_of_ge hw) (reflectedArcSet_norm_lt hp h) - have hG (w : ℂ) (hw : R≤‖w‖) : DifferentiableAt ℂ Q.chart.forward w := - (Q.forward_differentiable h0).differentiableAt (hK.isClosed.isOpen_compl.mem_nhds (hnot w hw)) - have hs (w : ℂ) (hw : R≤‖w‖) : w.im*(Q.chart.forward w-w).im≤0 := - Q.height_displacement_sign (hnot w hw) - have hb := exterior_capacity_remainder hR Q.chart.capacity_pos hG Q.chart.coefficient hs hz - have hnorm : R≤‖z‖ := by linarith - have hzP : z∈upperHalfPlane \ range p := ⟨hzH,fun h => hnot z hnorm (Or.inl h)⟩ - obtain ⟨he,_,hc⟩ := P.unique_on_domain Q.chart - rwa [← he hzP,← hc] at hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ExteriorCapacityRemainder +import OAI.Probability.SLE.Loewner.ReflectedExteriorHeight + +namespace OAI + +/-! The local Loewner kernel estimate for an arbitrary simple attached +slit, with error proportional to its exact capacity. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +lemma reflectedArcSet_norm_lt {p : C(unitInterval,ℂ)} {R : ℝ} + (hp : ∀ t : unitInterval, ‖p t‖ (not_lt_of_ge hw) (reflectedArcSet_norm_lt hp h) + have hG (w : ℂ) (hw : R≤‖w‖) : DifferentiableAt ℂ Q.chart.forward w := + (Q.forward_differentiable h0).differentiableAt (hK.isClosed.isOpen_compl.mem_nhds (hnot w hw)) + have hs (w : ℂ) (hw : R≤‖w‖) : w.im*(Q.chart.forward w-w).im≤0 := + Q.height_displacement_sign (hnot w hw) + have hb := exterior_capacity_remainder hR Q.chart.capacity_pos hG Q.chart.coefficient hs hz + have hnorm : R≤‖z‖ := by linarith + have hzP : z∈upperHalfPlane \ range p := ⟨hzH,fun h => hnot z hnorm (Or.inl h)⟩ + obtain ⟨he,_,hc⟩ := P.unique_on_domain Q.chart + rwa [← he hzP,← hc] at hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryBasics.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryBasics.lean index 6754925e2be18d430ce8d4e0d27bf9b59b07d7a0..aa5ab01f130fb31104c663fdf2b0f9de93ac5d41 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryBasics.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryBasics.lean @@ -1,38 +1,38 @@ -import OAI.Probability.SLE.Loewner.SlitBoundaryBarrier -import OAI.Probability.SLE.Loewner.InverseBoundaryBound - -namespace OAI - -/-! Analytic and compactness facts for the actual normalized static slit charts. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.isOpen_domain {p : C(unitInterval,ℂ)} (_P : SimpleSlitChart p) : - IsOpen (upperHalfPlane \ range p) := - isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed - -lemma SimpleSlitChart.inverse_injOn {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) : - InjOn P.inverse upperHalfPlane := by - intro z hz w hw he - have hh := congrArg P.forward he - simpa only [P.right_inverse z hz,P.right_inverse w hw] using hh - -lemma SimpleSlitChart.inverse_im_ge {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) - {z : ℂ} (hz : z∈upperHalfPlane) : z.im≤(P.inverse z).im := by - have hh := P.height_le (P.inverse z) (P.inverse_mapsTo hz) - simpa only [P.right_inverse z hz] using hh - -lemma SimpleSlitChart.inverse_bounded_over_ball {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (R : ℝ) : - ∃ M : ℝ, ∀ w∈upperHalfPlane, ‖w‖≤R → ‖P.inverse w‖≤M := - inverse_bound_of_hydrodynamic P.inverse_mapsTo P.right_inverse - (P.coefficient.mono_left inf_le_left) R - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitBoundaryBarrier +import OAI.Probability.SLE.Loewner.InverseBoundaryBound + +namespace OAI + +/-! Analytic and compactness facts for the actual normalized static slit charts. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.isOpen_domain {p : C(unitInterval,ℂ)} (_P : SimpleSlitChart p) : + IsOpen (upperHalfPlane \ range p) := + isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed + +lemma SimpleSlitChart.inverse_injOn {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) : + InjOn P.inverse upperHalfPlane := by + intro z hz w hw he + have hh := congrArg P.forward he + simpa only [P.right_inverse z hz,P.right_inverse w hw] using hh + +lemma SimpleSlitChart.inverse_im_ge {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) + {z : ℂ} (hz : z∈upperHalfPlane) : z.im≤(P.inverse z).im := by + have hh := P.height_le (P.inverse z) (P.inverse_mapsTo hz) + simpa only [P.right_inverse z hz] using hh + +lemma SimpleSlitChart.inverse_bounded_over_ball {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (R : ℝ) : + ∃ M : ℝ, ∀ w∈upperHalfPlane, ‖w‖≤R → ‖P.inverse w‖≤M := + inverse_bound_of_hydrodynamic P.inverse_mapsTo P.right_inverse + (P.coefficient.mono_left inf_le_left) R + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryExtension.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryExtension.lean index b25fcd83f14a9cb5af957083e1d41c42f85a67d4..3ca0732dc5d44eccbec37fb7c640d45dc51281aa 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryExtension.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryExtension.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Loewner.SlitChartBoundaryShrink -import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryExtension - -namespace OAI - -/-! Continuous finite boundary values of the normalized static slit inverse. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.finite_real_boundary_limit - {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) (x : ℝ) : - ∃ z : ℂ, Tendsto (P.inverse) (𝓝[upperHalfPlane] (x:ℂ)) (𝓝 z) := by - have hx : (x:ℂ)∈closure upperHalfPlane := - closedHalfPlane_subset_closure_upperHalfPlane (by simp) - have : NeBot (𝓝[upperHalfPlane] (x:ℂ)) := mem_closure_iff_nhdsWithin_neBot.mp hx - apply cauchy_map_iff_exists_tendsto.mp - apply Metric.cauchy_iff.mpr - refine ⟨inferInstance,fun ε hε => ?_⟩ - obtain ⟨r,hr,z,hsmall⟩ := P.small_boundary_halfdisk x (ε/3) (by positivity) - refine ⟨P.inverse '' (upperHalfPlane∩ball (x:ℂ) r),?_,?_⟩ - · change (P.inverse) ⁻¹' (P.inverse '' (upperHalfPlane∩ball (x:ℂ) r))∈𝓝[upperHalfPlane] (x:ℂ) - filter_upwards [self_mem_nhdsWithin, - mem_nhdsWithin_of_mem_nhds (ball_mem_nhds (x:ℂ) hr)] with w hwH hwr - exact ⟨w,⟨hwH,hwr⟩,rfl⟩ - · rintro a ⟨u,hu,rfl⟩ b ⟨v,hv,rfl⟩ - have hsu := hsmall u hu.1 hu.2 - have hsv := hsmall v hv.1 hv.2 - have htri := dist_triangle (P.inverse u) z (P.inverse v) - rw [dist_comm z (P.inverse v)] at htri - linarith - -theorem SimpleSlitChart.exists_boundary_extension - {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) : - ∃ Fbar : ℂ → ℂ, ContinuousOn Fbar {z : ℂ | 0≤z.im} ∧ - ∀ z∈upperHalfPlane, Fbar z=P.inverse z := - exists_halfPlane_boundary_extension P.inverse_differentiable.continuousOn - P.finite_real_boundary_limit - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartBoundaryShrink +import OAI.Probability.SLE.Geometry.HalfPlaneBoundaryExtension + +namespace OAI + +/-! Continuous finite boundary values of the normalized static slit inverse. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.finite_real_boundary_limit + {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) (x : ℝ) : + ∃ z : ℂ, Tendsto (P.inverse) (𝓝[upperHalfPlane] (x:ℂ)) (𝓝 z) := by + have hx : (x:ℂ)∈closure upperHalfPlane := + closedHalfPlane_subset_closure_upperHalfPlane (by simp) + have : NeBot (𝓝[upperHalfPlane] (x:ℂ)) := mem_closure_iff_nhdsWithin_neBot.mp hx + apply cauchy_map_iff_exists_tendsto.mp + apply Metric.cauchy_iff.mpr + refine ⟨inferInstance,fun ε hε => ?_⟩ + obtain ⟨r,hr,z,hsmall⟩ := P.small_boundary_halfdisk x (ε/3) (by positivity) + refine ⟨P.inverse '' (upperHalfPlane∩ball (x:ℂ) r),?_,?_⟩ + · change (P.inverse) ⁻¹' (P.inverse '' (upperHalfPlane∩ball (x:ℂ) r))∈𝓝[upperHalfPlane] (x:ℂ) + filter_upwards [self_mem_nhdsWithin, + mem_nhdsWithin_of_mem_nhds (ball_mem_nhds (x:ℂ) hr)] with w hwH hwr + exact ⟨w,⟨hwH,hwr⟩,rfl⟩ + · rintro a ⟨u,hu,rfl⟩ b ⟨v,hv,rfl⟩ + have hsu := hsmall u hu.1 hu.2 + have hsv := hsmall v hv.1 hv.2 + have htri := dist_triangle (P.inverse u) z (P.inverse v) + rw [dist_comm z (P.inverse v)] at htri + linarith + +theorem SimpleSlitChart.exists_boundary_extension + {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) : + ∃ Fbar : ℂ → ℂ, ContinuousOn Fbar {z : ℂ | 0≤z.im} ∧ + ∀ z∈upperHalfPlane, Fbar z=P.inverse z := + exists_halfPlane_boundary_extension P.inverse_differentiable.continuousOn + P.finite_real_boundary_limit + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryShrink.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryShrink.lean index c50a2a944a7f5bf8613f9bb26dbed4b5ea0d9645..009700bb9cd5398eefc16d2282f65b99c889a2cd 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryShrink.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartBoundaryShrink.lean @@ -1,75 +1,75 @@ -import OAI.Probability.SLE.Loewner.SlitChartClosingPath -import OAI.Probability.SLE.Loewner.InverseBoundaryShrink - -namespace OAI - -/-! Small inverse half-disks for static slit charts, from the closed boundary arcs. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.inverse_arc_traps_radial_point - {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) - (x r ρ θ : ℝ) (hρ : 0<ρ) (hρr : ρ (hqb u).2 - · exact hε - · exact hcb - · exact fun u => (hqb u).1 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartClosingPath +import OAI.Probability.SLE.Loewner.InverseBoundaryShrink + +namespace OAI + +/-! Small inverse half-disks for static slit charts, from the closed boundary arcs. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.inverse_arc_traps_radial_point + {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) + (x r ρ θ : ℝ) (hρ : 0<ρ) (hρr : ρ (hqb u).2 + · exact hε + · exact hcb + · exact fun u => (hqb u).1 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartClosingPath.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartClosingPath.lean index c50decd6d7df553941c31d4a84909f1225172ef5..95673be0bdd6d514b53e0a32ef79f7998569b772 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartClosingPath.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartClosingPath.lean @@ -1,75 +1,75 @@ -import OAI.Probability.SLE.Loewner.SlitChartSmallSemicircles - -namespace OAI - -/-! A small image semicircle can be closed by a small path outside the -surviving domain. The closing path lies on the static slit and the real boundary. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma SimpleSlitChart.exists_small_inverse_closing_path - {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) (x Q ε : ℝ) (hε : 0<ε) : - ∃ (v : ℝ) (c : ℝ → ℂ) (p : Path (c Real.pi) (c 0)), v (disjoint_left.mp (slitBoundaryBarrier_disjoint_domain arc R)) (hpB s) hD⟩ - have htri := dist_triangle (p s) (c Real.pi) (c 0) - have hps := hp s - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartSmallSemicircles + +namespace OAI + +/-! A small image semicircle can be closed by a small path outside the +surviving domain. The closing path lies on the static slit and the real boundary. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma SimpleSlitChart.exists_small_inverse_closing_path + {arc : C(unitInterval,ℂ)} (P : SimpleSlitChart arc) (x Q ε : ℝ) (hε : 0<ε) : + ∃ (v : ℝ) (c : ℝ → ℂ) (p : Path (c Real.pi) (c 0)), v (disjoint_left.mp (slitBoundaryBarrier_disjoint_domain arc R)) (hpB s) hD⟩ + have htri := dist_triangle (p s) (c Real.pi) (c 0) + have hps := hp s + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartDisplacement.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartDisplacement.lean index de5f5cab6f14f46a68b50d8bbc78387639355e55..ffdfa323e7ced071d7364c06a01dcc8e6163e92a 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartDisplacement.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartDisplacement.lean @@ -1,58 +1,58 @@ -import OAI.Probability.SLE.Loewner.SlitChartBoundaryBasics -import OAI.Probability.SLE.Loewner.HalfPlaneDisplacementBound - -namespace OAI - -/-! Quantitative comparison of the actual normalized inverse slit charts. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.inverse_displacement {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) - {z : ℂ} (hz : z∈upperHalfPlane) : ‖P.inverse z-z‖≤P.capacity/z.im := by - apply nonnegative_im_hydrodynamic_bound (P.inverse_differentiable.sub differentiableOn_id) - (fun w hw => by - change 0 ≤ (P.inverse w-w).im - simpa only [Complex.sub_im] using sub_nonneg.mpr (P.inverse_im_ge hw)) - P.capacity_pos.le - (inverse_hydrodynamic_coefficient P.coefficient P.inverse_normalized P.right_inverse) hz - -lemma SimpleSlitChart.forward_displacement {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) - {z : ℂ} (hz : z∈upperHalfPlane \ range p) : - ‖P.forward z-z‖≤P.capacity/(P.forward z).im := by - have hh := P.inverse_displacement (P.bijOn.mapsTo hz) - simpa only [P.left_inverse z hz,norm_sub_rev] using hh - -lemma SimpleSlitChart.inverse_comparison {p q : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (Q : SimpleSlitChart q) (hpq : range p ⊆ range q) - {z : ℂ} (hz : z∈upperHalfPlane) : - ‖P.forward (Q.inverse z)-z‖≤(Q.capacity-P.capacity)/z.im := by - have hdom : upperHalfPlane \ range q ⊆ upperHalfPlane \ range p := - fun _ hw => ⟨hw.1,fun hp => hw.2 (hpq hp)⟩ - have hm := Q.inverse_mapsTo.mono_right hdom - have hd := P.forward_differentiable.comp Q.inverse_differentiable hm - have hn := hydrodynamic_composition_ratio P.coefficient Q.inverse_normalized - apply nonnegative_im_hydrodynamic_bound (hd.sub differentiableOn_id) - (fun w hw => by - change 0 ≤ (P.forward (Q.inverse w)-w).im - simpa only [Function.comp_apply,Complex.sub_im] using sub_nonneg.mpr - (normalized_halfPlane_height hd (P.bijOn.mapsTo.comp hm) hn hw)) - (sub_nonneg.mpr (P.capacity_le_of_range_subset Q hpq)) - · have hh := hydrodynamic_composition_coefficient P.coefficient Q.coefficient - Q.inverse_normalized Q.right_inverse - have he : -((Q.capacity-P.capacity:ℝ):ℂ) = (P.capacity:ℂ)-Q.capacity := by - push_cast - ring - change Tendsto (fun w : ℂ => w*(P.forward (Q.inverse w)-w)) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-((Q.capacity-P.capacity:ℝ):ℂ))) - rw [he] - exact hh - · exact hz - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartBoundaryBasics +import OAI.Probability.SLE.Loewner.HalfPlaneDisplacementBound + +namespace OAI + +/-! Quantitative comparison of the actual normalized inverse slit charts. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.inverse_displacement {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) + {z : ℂ} (hz : z∈upperHalfPlane) : ‖P.inverse z-z‖≤P.capacity/z.im := by + apply nonnegative_im_hydrodynamic_bound (P.inverse_differentiable.sub differentiableOn_id) + (fun w hw => by + change 0 ≤ (P.inverse w-w).im + simpa only [Complex.sub_im] using sub_nonneg.mpr (P.inverse_im_ge hw)) + P.capacity_pos.le + (inverse_hydrodynamic_coefficient P.coefficient P.inverse_normalized P.right_inverse) hz + +lemma SimpleSlitChart.forward_displacement {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) + {z : ℂ} (hz : z∈upperHalfPlane \ range p) : + ‖P.forward z-z‖≤P.capacity/(P.forward z).im := by + have hh := P.inverse_displacement (P.bijOn.mapsTo hz) + simpa only [P.left_inverse z hz,norm_sub_rev] using hh + +lemma SimpleSlitChart.inverse_comparison {p q : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (Q : SimpleSlitChart q) (hpq : range p ⊆ range q) + {z : ℂ} (hz : z∈upperHalfPlane) : + ‖P.forward (Q.inverse z)-z‖≤(Q.capacity-P.capacity)/z.im := by + have hdom : upperHalfPlane \ range q ⊆ upperHalfPlane \ range p := + fun _ hw => ⟨hw.1,fun hp => hw.2 (hpq hp)⟩ + have hm := Q.inverse_mapsTo.mono_right hdom + have hd := P.forward_differentiable.comp Q.inverse_differentiable hm + have hn := hydrodynamic_composition_ratio P.coefficient Q.inverse_normalized + apply nonnegative_im_hydrodynamic_bound (hd.sub differentiableOn_id) + (fun w hw => by + change 0 ≤ (P.forward (Q.inverse w)-w).im + simpa only [Function.comp_apply,Complex.sub_im] using sub_nonneg.mpr + (normalized_halfPlane_height hd (P.bijOn.mapsTo.comp hm) hn hw)) + (sub_nonneg.mpr (P.capacity_le_of_range_subset Q hpq)) + · have hh := hydrodynamic_composition_coefficient P.coefficient Q.coefficient + Q.inverse_normalized Q.right_inverse + have he : -((Q.capacity-P.capacity:ℝ):ℂ) = (P.capacity:ℂ)-Q.capacity := by + push_cast + ring + change Tendsto (fun w : ℂ => w*(P.forward (Q.inverse w)-w)) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-((Q.capacity-P.capacity:ℝ):ℂ))) + rw [he] + exact hh + · exact hz + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartExteriorComparison.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartExteriorComparison.lean index 7a4422657cc341b71987059e4022a217b32c4f34..80ae716c87225e38744cfc7ea2a3f32d9623c53c 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartExteriorComparison.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartExteriorComparison.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Loewner.SlitChartDisplacement -import OAI.Probability.SLE.Loewner.CapacityRadiusBound - -namespace OAI - -/-! A common exterior half-disk controls all normalized charts of slits inside it. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma radiusHullInverse_map {R : ℝ} (hR : 0 radiusHullMap_inverse hR hw) - have hc : 0≤R^2-P.capacity := sub_nonneg.mpr (P.capacity_le_radius_sq hR hp) - have hh := nonnegative_im_hydrodynamic_bound (hd.sub differentiableOn_id) - (fun w hw => by - change 0 ≤ (P.forward (radiusHullInverse R w)-w).im - simpa only [comp_apply,sub_im] using sub_nonneg.mpr (hheight w hw)) hc - (by - change Tendsto (fun w : ℂ => w*(P.forward (radiusHullInverse R w)-w)) - (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-((R^2-P.capacity:ℝ):ℂ))) - have he : -((R^2-P.capacity:ℝ):ℂ) = (P.capacity:ℂ)-(R:ℂ)^2 := by - push_cast - ring - rw [he] - exact hcoef) - ((radiusHullMap_bijOn hR).mapsTo hz) - have hle := hheight _ ((radiusHullMap_bijOn hR).mapsTo hz) - change ‖P.forward (radiusHullInverse R (radiusHullMap R z))-radiusHullMap R z‖ ≤ - (R^2-P.capacity)/(radiusHullMap R z).im at hh - change (radiusHullMap R z).im ≤ (P.forward (radiusHullInverse R (radiusHullMap R z))).im at hle - rw [radiusHullInverse_map hR hz] at hh hle - exact ⟨hle,hh⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartDisplacement +import OAI.Probability.SLE.Loewner.CapacityRadiusBound + +namespace OAI + +/-! A common exterior half-disk controls all normalized charts of slits inside it. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +lemma radiusHullInverse_map {R : ℝ} (hR : 0 radiusHullMap_inverse hR hw) + have hc : 0≤R^2-P.capacity := sub_nonneg.mpr (P.capacity_le_radius_sq hR hp) + have hh := nonnegative_im_hydrodynamic_bound (hd.sub differentiableOn_id) + (fun w hw => by + change 0 ≤ (P.forward (radiusHullInverse R w)-w).im + simpa only [comp_apply,sub_im] using sub_nonneg.mpr (hheight w hw)) hc + (by + change Tendsto (fun w : ℂ => w*(P.forward (radiusHullInverse R w)-w)) + (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 (-((R^2-P.capacity:ℝ):ℂ))) + have he : -((R^2-P.capacity:ℝ):ℂ) = (P.capacity:ℂ)-(R:ℂ)^2 := by + push_cast + ring + rw [he] + exact hcoef) + ((radiusHullMap_bijOn hR).mapsTo hz) + have hle := hheight _ ((radiusHullMap_bijOn hR).mapsTo hz) + change ‖P.forward (radiusHullInverse R (radiusHullMap R z))-radiusHullMap R z‖ ≤ + (R^2-P.capacity)/(radiusHullMap R z).im at hh + change (radiusHullMap R z).im ≤ (P.forward (radiusHullInverse R (radiusHullMap R z))).im at hle + rw [radiusHullInverse_map hR hz] at hh hle + exact ⟨hle,hh⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartFarHeight.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartFarHeight.lean index a377a4413f0fd198e13ac49a5f397ae26840ab7a..e5058509ee1e68a924f35f1544414b4e823ab7f2 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartFarHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartFarHeight.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Loewner.SlitChartExteriorComparison - -namespace OAI - -/-! A uniform far-field displacement bound for all slits in one fixed disk. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology - -namespace SLEExactGauge - -lemma radiusHullMap_im_mul_normSq {R : ℝ} (hR : 0 hb.injOn.leftInvOn_invFunOn hz - let A := P.forward ∘ F - let B := G ∘ P.inverse - have hA : DifferentiableOn ℂ A upperHalfPlane := P.forward_differentiable.comp hF hb.mapsTo - have hB : DifferentiableOn ℂ B upperHalfPlane := hG.comp P.inverse_differentiable P.inverse_mapsTo - have hAm : MapsTo A upperHalfPlane upperHalfPlane := P.bijOn.mapsTo.comp hb.mapsTo - have hBm : MapsTo B upperHalfPlane upperHalfPlane := hGm.comp P.inverse_mapsTo - have hBA : ∀ z∈upperHalfPlane, B (A z)=z := by - intro z hz - dsimp only [A,B,Function.comp_apply] - rw [P.left_inverse _ (hb.mapsTo hz),hGF z hz] - have hAbound {z : ℂ} (hz : 2*R≤z.im) : ‖A z-z‖≤(3*R^2+C)/z.im := by - have hy : 0 A (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0) := by - apply vertical_displacement_zero_of_bound - filter_upwards [hev] with n hn - simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero] - using hAbound hn - have hB0 : Tendsto (fun n : ℕ => B (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0) := by - apply vertical_displacement_zero_of_bound - filter_upwards [hev] with n hn - simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero] - using hBbound hn - have hAid := vertical_halfPlane_left_inverse_eq_id hA hB hAm hBm hBA - (vertical_normalization_of_displacement hA0) (vertical_normalization_of_displacement hB0) hA0 - intro z hz - have he := P.left_inverse (F z) (hb.mapsTo hz) - rw [show P.forward (F z)=z from hAid hz] at he - exact he.symm - -theorem initialInverse_limit_eq_final (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - (P : SimpleSlitChart p) {t : ℕ → unitInterval} {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) - F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) : - EqOn F P.inverse upperHalfPlane := by - obtain ⟨R0,hR0⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le - let R := max R0 1 - have hR : 0 initialInverse_limit_im_ge p hi h0 hH hlim hz) - hR hp - (fun _ hz => initialInverse_limit_uniform_displacement p hi h0 hH hlim hz) - (fun _ hz => initialInverse_limit_inverse_far_height p hi h0 hH hF hlim hR hp hz) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialLimitInverse +import OAI.Probability.SLE.Loewner.VerticalDisplacementLimit + +namespace OAI + +/-! Identification of a terminal inverse-chart limit with the normalized +static chart. Only the preserved height and far-field bounds are used. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.eq_inverse_of_vertical_bounds {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hb : BijOn F upperHalfPlane (upperHalfPlane \ range p)) + (him : ∀ z∈upperHalfPlane, z.im≤(F z).im) + {C R : ℝ} (hR : 0 hb.injOn.leftInvOn_invFunOn hz + let A := P.forward ∘ F + let B := G ∘ P.inverse + have hA : DifferentiableOn ℂ A upperHalfPlane := P.forward_differentiable.comp hF hb.mapsTo + have hB : DifferentiableOn ℂ B upperHalfPlane := hG.comp P.inverse_differentiable P.inverse_mapsTo + have hAm : MapsTo A upperHalfPlane upperHalfPlane := P.bijOn.mapsTo.comp hb.mapsTo + have hBm : MapsTo B upperHalfPlane upperHalfPlane := hGm.comp P.inverse_mapsTo + have hBA : ∀ z∈upperHalfPlane, B (A z)=z := by + intro z hz + dsimp only [A,B,Function.comp_apply] + rw [P.left_inverse _ (hb.mapsTo hz),hGF z hz] + have hAbound {z : ℂ} (hz : 2*R≤z.im) : ‖A z-z‖≤(3*R^2+C)/z.im := by + have hy : 0 A (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0) := by + apply vertical_displacement_zero_of_bound + filter_upwards [hev] with n hn + simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero] + using hAbound hn + have hB0 : Tendsto (fun n : ℕ => B (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0) := by + apply vertical_displacement_zero_of_bound + filter_upwards [hev] with n hn + simpa only [verticalInfinity,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero] + using hBbound hn + have hAid := vertical_halfPlane_left_inverse_eq_id hA hB hAm hBm hBA + (vertical_normalization_of_displacement hA0) (vertical_normalization_of_displacement hB0) hA0 + intro z hz + have he := P.left_inverse (F z) (hb.mapsTo hz) + rw [show P.forward (F z)=z from hAid hz] at he + exact he.symm + +theorem initialInverse_limit_eq_final (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + (P : SimpleSlitChart p) {t : ℕ → unitInterval} {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hlim : TendstoLocallyUniformlyOn (fun n => initialInverseChart p hi h0 hH (t n)) + F atTop upperHalfPlane) (ht : Tendsto t atTop (𝓝 1)) : + EqOn F P.inverse upperHalfPlane := by + obtain ⟨R0,hR0⟩ := (isCompact_range p.continuous).isBounded.exists_norm_le + let R := max R0 1 + have hR : 0 initialInverse_limit_im_ge p hi h0 hH hlim hz) + hR hp + (fun _ hz => initialInverse_limit_uniform_displacement p hi h0 hH hlim hz) + (fun _ hz => initialInverse_limit_inverse_far_height p hi h0 hH hF hlim hR hp hz) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartProper.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartProper.lean index 18f8e290eaa352faa8989d6d28f0ccce90bac7ee..fd2386dc93e5e537bead1ae4d7653ddf51ff0cb9 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartProper.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartProper.lean @@ -1,37 +1,37 @@ -import OAI.Probability.SLE.Loewner.SlitChartBoundaryBasics - -namespace OAI - -/-! Normalization at infinity prevents bounded physical points from having -unbounded chart coordinates, including when they approach the slit. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.forward_bounded_over_ball {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (R : ℝ) : - ∃ M : ℝ, ∀ z∈upperHalfPlane \ range p, ‖z‖≤R → ‖P.forward z‖≤M := by - have he := Metric.tendsto_nhds.mp P.inverse_normalized (1/2) (by norm_num) - rw [eventually_inf_principal,← Metric.cobounded_eq_cocompact] at he - obtain ⟨N,_,hN⟩ := Filter.hasBasis_cobounded_norm.eventually_iff.mp he - refine ⟨max (max N 1) (2*R),fun z hz hzR => ?_⟩ - by_cases hb : ‖P.forward z‖ ≤ max N 1 - · exact hb.trans (le_max_left _ _) - have hb' := lt_of_not_ge hb - have hnorm : 0<‖P.forward z‖ := by linarith [le_max_right N (1:ℝ)] - have hsmall := hN ((le_max_left N (1:ℝ)).trans hb'.le) (P.bijOn.mapsTo hz) - rw [dist_eq_norm,P.left_inverse z hz] at hsmall - have htri := norm_le_norm_sub_add (1:ℂ) (z/P.forward z) - rw [norm_one,norm_sub_rev,norm_div] at htri - have hprod : (1/2:ℝ)*‖P.forward z‖<‖z‖ := by - apply (lt_div_iff₀ hnorm).mp - linarith - exact (show ‖P.forward z‖≤2*R by linarith).trans (le_max_right _ _) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartBoundaryBasics + +namespace OAI + +/-! Normalization at infinity prevents bounded physical points from having +unbounded chart coordinates, including when they approach the slit. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.forward_bounded_over_ball {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (R : ℝ) : + ∃ M : ℝ, ∀ z∈upperHalfPlane \ range p, ‖z‖≤R → ‖P.forward z‖≤M := by + have he := Metric.tendsto_nhds.mp P.inverse_normalized (1/2) (by norm_num) + rw [eventually_inf_principal,← Metric.cobounded_eq_cocompact] at he + obtain ⟨N,_,hN⟩ := Filter.hasBasis_cobounded_norm.eventually_iff.mp he + refine ⟨max (max N 1) (2*R),fun z hz hzR => ?_⟩ + by_cases hb : ‖P.forward z‖ ≤ max N 1 + · exact hb.trans (le_max_left _ _) + have hb' := lt_of_not_ge hb + have hnorm : 0<‖P.forward z‖ := by linarith [le_max_right N (1:ℝ)] + have hsmall := hN ((le_max_left N (1:ℝ)).trans hb'.le) (P.bijOn.mapsTo hz) + rw [dist_eq_norm,P.left_inverse z hz] at hsmall + have htri := norm_le_norm_sub_add (1:ℂ) (z/P.forward z) + rw [norm_one,norm_sub_rev,norm_div] at htri + have hprod : (1/2:ℝ)*‖P.forward z‖<‖z‖ := by + apply (lt_div_iff₀ hnorm).mp + linarith + exact (show ‖P.forward z‖≤2*R by linarith).trans (le_max_right _ _) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartRangeUniqueness.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartRangeUniqueness.lean index b75bc7472a2ccc9a6e9a34cdf8404f71f9bc09aa..d46650f1b81a59a28b73a66776c28b7df2eae699 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartRangeUniqueness.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartRangeUniqueness.lean @@ -1,38 +1,38 @@ -import OAI.Probability.SLE.Loewner.SlitChartUniqueness - -namespace OAI - -/-! Normalized charts depend only on the slit range, including when its -parameter has been changed. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.eq_of_range_eq {p q : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (Q : SimpleSlitChart q) (hr : range p=range q) : - P.capacity=Q.capacity ∧ EqOn P.inverse Q.inverse upperHalfPlane ∧ - EqOn P.forward Q.forward (upperHalfPlane \ range p) := by - have hc : P.capacity=Q.capacity := le_antisymm - (P.capacity_le_of_range_subset Q hr.le) (Q.capacity_le_of_range_subset P hr.ge) - have hcomp (z : ℂ) (hz : z∈upperHalfPlane) : P.forward (Q.inverse z)=z := by - have hb := P.inverse_comparison Q hr.le hz - rw [hc,sub_self,zero_div] at hb - exact sub_eq_zero.mp (norm_le_zero_iff.mp hb) - have hi : EqOn P.inverse Q.inverse upperHalfPlane := by - intro z hz - have hh := P.left_inverse (Q.inverse z) (by simpa only [hr] using Q.inverse_mapsTo hz) - rw [hcomp z hz] at hh - exact hh - refine ⟨hc,hi,?_⟩ - intro z hz - have hzQ : z∈upperHalfPlane \ range q := by simpa only [hr] using hz - apply Q.inverse_injOn (P.bijOn.mapsTo hz) (Q.bijOn.mapsTo hzQ) - rw [← hi (P.bijOn.mapsTo hz),P.left_inverse z hz,Q.left_inverse z hzQ] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartUniqueness + +namespace OAI + +/-! Normalized charts depend only on the slit range, including when its +parameter has been changed. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.eq_of_range_eq {p q : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (Q : SimpleSlitChart q) (hr : range p=range q) : + P.capacity=Q.capacity ∧ EqOn P.inverse Q.inverse upperHalfPlane ∧ + EqOn P.forward Q.forward (upperHalfPlane \ range p) := by + have hc : P.capacity=Q.capacity := le_antisymm + (P.capacity_le_of_range_subset Q hr.le) (Q.capacity_le_of_range_subset P hr.ge) + have hcomp (z : ℂ) (hz : z∈upperHalfPlane) : P.forward (Q.inverse z)=z := by + have hb := P.inverse_comparison Q hr.le hz + rw [hc,sub_self,zero_div] at hb + exact sub_eq_zero.mp (norm_le_zero_iff.mp hb) + have hi : EqOn P.inverse Q.inverse upperHalfPlane := by + intro z hz + have hh := P.left_inverse (Q.inverse z) (by simpa only [hr] using Q.inverse_mapsTo hz) + rw [hcomp z hz] at hh + exact hh + refine ⟨hc,hi,?_⟩ + intro z hz + have hzQ : z∈upperHalfPlane \ range q := by simpa only [hr] using hz + apply Q.inverse_injOn (P.bijOn.mapsTo hz) (Q.bijOn.mapsTo hzQ) + rw [← hi (P.bijOn.mapsTo hz),P.left_inverse z hz,Q.left_inverse z hzQ] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartSmallSemicircles.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartSmallSemicircles.lean index 99e2218ebee15367a5d8f8f16d260c588851bbea..23a99972c927547dd93dbe2de8954a17cdb2f178 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartSmallSemicircles.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartSmallSemicircles.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Loewner.SlitChartStripEnergy -import OAI.Probability.SLE.Geometry.ShrinkingStripEnergy -import OAI.Probability.SLE.Loewner.InverseBoundaryArc - -namespace OAI - -/-! Arbitrarily small closed image semicircles at each real boundary point. -The endpoint values are obtained from finite energy, not from an assumed -continuous extension of the inverse map. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma SimpleSlitChart.inverse_log_holomorphic - {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (x : ℝ) : - DifferentiableOn ℂ (fun z => P.inverse ((x:ℂ)+Complex.exp z)) - {z | 0 P.inverse ((x:ℂ)+Complex.exp z) - have hf := P.inverse_log_holomorphic x - obtain ⟨v,hv,hd,hsmall⟩ := exists_small_holomorphic_section hf - (P.inverse_strip_dirichlet_finite x 0) Q ε hε - let w : ℝ → ℂ := fun θ => (x:ℂ)+Complex.exp ((v:ℂ)+(θ:ℂ)*Complex.I) - have hS : IsOpen {z : ℂ | 0 P.inverse (w θ)) θ := by - change DifferentiableAt ℝ (f ∘ fun θ : ℝ => (v:ℂ)+(θ:ℂ)*Complex.I) θ - have hz : (v:ℂ)+(θ:ℂ)*Complex.I∈{z : ℂ | 0 - simp [w,upperHalfPlane,Complex.exp_add,Complex.exp_ofReal_mul_I,← Complex.ofReal_exp] - have hend (s : ℝ) (hs : s∈Icc 0 Real.pi) (hs' : s=0 ∨ s=Real.pi) : - c s∈frontier ((upperHalfPlane \ range p)) := - inverse_arc_endpoint_mem_frontier P.isOpen_domain - P.forward_differentiable.continuousOn P.bijOn.mapsTo P.inverse_mapsTo - P.right_inverse hwc hc Real.pi_pos hs hwin (hwend s hs') he - refine ⟨v,c,hv,hc,he,?_,hend 0 ⟨le_rfl,Real.pi_pos.le⟩ (Or.inl rfl), - hend Real.pi ⟨Real.pi_pos.le,le_rfl⟩ (Or.inr rfl),?_,?_⟩ - · apply continuous_angular_oscillation_on_endpoints hc ε - intro θ hθ φ hφ - rw [he hθ,he hφ] - exact (hsmall θ hθ φ hφ).le - · intro θ hθ - rw [he hθ] - exact P.inverse_mapsTo (hwin hθ) - · intro θ hθ φ hφ heq - rw [he hθ,he hφ] at heq - change P.inverse (w θ)=P.inverse (w φ) at heq - have hwθ := P.right_inverse (w θ) (hwin hθ) - have hwφ := P.right_inverse (w φ) (hwin hφ) - have hww : w θ=w φ := by rw [heq,hwφ] at hwθ; exact hwθ.symm - have harg := exp_injective_strip - (x₁ := (v:ℂ)+(θ:ℂ)*Complex.I) (x₂ := (v:ℂ)+(φ:ℂ)*Complex.I) - (by simpa using hθ) (by simpa using hφ) (add_left_cancel hww) - have him := congrArg Complex.im harg - simpa using him - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartStripEnergy +import OAI.Probability.SLE.Geometry.ShrinkingStripEnergy +import OAI.Probability.SLE.Loewner.InverseBoundaryArc + +namespace OAI + +/-! Arbitrarily small closed image semicircles at each real boundary point. +The endpoint values are obtained from finite energy, not from an assumed +continuous extension of the inverse map. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma SimpleSlitChart.inverse_log_holomorphic + {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (x : ℝ) : + DifferentiableOn ℂ (fun z => P.inverse ((x:ℂ)+Complex.exp z)) + {z | 0 P.inverse ((x:ℂ)+Complex.exp z) + have hf := P.inverse_log_holomorphic x + obtain ⟨v,hv,hd,hsmall⟩ := exists_small_holomorphic_section hf + (P.inverse_strip_dirichlet_finite x 0) Q ε hε + let w : ℝ → ℂ := fun θ => (x:ℂ)+Complex.exp ((v:ℂ)+(θ:ℂ)*Complex.I) + have hS : IsOpen {z : ℂ | 0 P.inverse (w θ)) θ := by + change DifferentiableAt ℝ (f ∘ fun θ : ℝ => (v:ℂ)+(θ:ℂ)*Complex.I) θ + have hz : (v:ℂ)+(θ:ℂ)*Complex.I∈{z : ℂ | 0 + simp [w,upperHalfPlane,Complex.exp_add,Complex.exp_ofReal_mul_I,← Complex.ofReal_exp] + have hend (s : ℝ) (hs : s∈Icc 0 Real.pi) (hs' : s=0 ∨ s=Real.pi) : + c s∈frontier ((upperHalfPlane \ range p)) := + inverse_arc_endpoint_mem_frontier P.isOpen_domain + P.forward_differentiable.continuousOn P.bijOn.mapsTo P.inverse_mapsTo + P.right_inverse hwc hc Real.pi_pos hs hwin (hwend s hs') he + refine ⟨v,c,hv,hc,he,?_,hend 0 ⟨le_rfl,Real.pi_pos.le⟩ (Or.inl rfl), + hend Real.pi ⟨Real.pi_pos.le,le_rfl⟩ (Or.inr rfl),?_,?_⟩ + · apply continuous_angular_oscillation_on_endpoints hc ε + intro θ hθ φ hφ + rw [he hθ,he hφ] + exact (hsmall θ hθ φ hφ).le + · intro θ hθ + rw [he hθ] + exact P.inverse_mapsTo (hwin hθ) + · intro θ hθ φ hφ heq + rw [he hθ,he hφ] at heq + change P.inverse (w θ)=P.inverse (w φ) at heq + have hwθ := P.right_inverse (w θ) (hwin hθ) + have hwφ := P.right_inverse (w φ) (hwin hφ) + have hww : w θ=w φ := by rw [heq,hwφ] at hwθ; exact hwθ.symm + have harg := exp_injective_strip + (x₁ := (v:ℂ)+(θ:ℂ)*Complex.I) (x₂ := (v:ℂ)+(φ:ℂ)*Complex.I) + (by simpa using hθ) (by simpa using hφ) (add_left_cancel hww) + have him := congrArg Complex.im harg + simpa using him + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartStripEnergy.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartStripEnergy.lean index 969cb37272624d9dc3272e11f91a9ab256833623..a3a185aa4a3e7132defab32939259f188a680f0d 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartStripEnergy.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartStripEnergy.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Loewner.SlitChartBoundaryBasics -import OAI.Probability.SLE.Loewner.InverseDirichletEnergy -import OAI.Probability.SLE.DyadicStrips - -namespace OAI - -/-! Finite boundary-strip energy for static normalized slit charts. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma SimpleSlitChart.inverse_strip_dirichlet_finite - {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (x q : ℝ) : - (∫⁻ z in {z : ℂ | z.re P.inverse ((x:ℂ)+Complex.exp w)) z‖^2))<∞ := by - let S : Set ℂ := {z | z.re (x:ℂ)+Complex.exp z - let f : ℂ → ℂ := fun z => P.inverse (A z) - have hS : IsOpen S := (isOpen_lt Complex.continuous_re continuous_const).inter - ((isOpen_lt continuous_const Complex.continuous_im).inter - (isOpen_lt Complex.continuous_im continuous_const)) - have hAH : MapsTo A S upperHalfPlane := by - intro z hz - change 0<((x:ℂ)+Complex.exp z).im - have he : 0<(Complex.exp z).im := exp_strip_halfplane hz.2 - simpa only [Complex.add_im,Complex.ofReal_im,zero_add] using he - have hf : DifferentiableOn ℂ f S := P.inverse_differentiable.comp - ((differentiable_const (x:ℂ)).add Complex.differentiable_exp).differentiableOn hAH - have hi : InjOn f S := by - intro z hz w hw' he - have hAz := P.right_inverse (A z) (hAH hz) - have hAw := P.right_inverse (A w) (hAH hw') - have heA : A z=A w := by - rw [show P.inverse (A z)=P.inverse (A w) from he,hAw] at hAz - exact hAz.symm - exact exp_injective_strip hz.2 hw'.2 (add_left_cancel heA) - have harea : (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))=volume (f '' S) := by - have hd (z : ℂ) (hz : z∈S) := - (hf.differentiableAt (hS.mem_nhds hz)).hasDerivAt - have he := lintegral_abs_det_fderiv_eq_addHaar_image volume hS.measurableSet - (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi - simpa only [complex_derivative_real_det,abs_pow,abs_norm] using he - obtain ⟨M,hM⟩ := P.inverse_bounded_over_ball (‖(x:ℂ)‖+Real.exp q) - have himage : f '' S ⊆ closedBall (0:ℂ) M := by - rintro _ ⟨z,hz,rfl⟩ - rw [mem_closedBall,dist_zero_right] - apply hM (A z) (hAH hz) - calc - ‖A z‖ ≤ ‖(x:ℂ)‖+‖Complex.exp z‖ := norm_add_le _ _ - _ ≤ ‖(x:ℂ)‖+Real.exp q := by - rw [Complex.norm_exp] - exact add_le_add le_rfl (Real.exp_le_exp.mpr hz.1.le) - change (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))<∞ - rw [harea] - exact (measure_mono himage).trans_lt (isCompact_closedBall (0:ℂ) M).measure_lt_top - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartBoundaryBasics +import OAI.Probability.SLE.Loewner.InverseDirichletEnergy +import OAI.Probability.SLE.DyadicStrips + +namespace OAI + +/-! Finite boundary-strip energy for static normalized slit charts. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma SimpleSlitChart.inverse_strip_dirichlet_finite + {p : C(unitInterval,ℂ)} (P : SimpleSlitChart p) (x q : ℝ) : + (∫⁻ z in {z : ℂ | z.re P.inverse ((x:ℂ)+Complex.exp w)) z‖^2))<∞ := by + let S : Set ℂ := {z | z.re (x:ℂ)+Complex.exp z + let f : ℂ → ℂ := fun z => P.inverse (A z) + have hS : IsOpen S := (isOpen_lt Complex.continuous_re continuous_const).inter + ((isOpen_lt continuous_const Complex.continuous_im).inter + (isOpen_lt Complex.continuous_im continuous_const)) + have hAH : MapsTo A S upperHalfPlane := by + intro z hz + change 0<((x:ℂ)+Complex.exp z).im + have he : 0<(Complex.exp z).im := exp_strip_halfplane hz.2 + simpa only [Complex.add_im,Complex.ofReal_im,zero_add] using he + have hf : DifferentiableOn ℂ f S := P.inverse_differentiable.comp + ((differentiable_const (x:ℂ)).add Complex.differentiable_exp).differentiableOn hAH + have hi : InjOn f S := by + intro z hz w hw' he + have hAz := P.right_inverse (A z) (hAH hz) + have hAw := P.right_inverse (A w) (hAH hw') + have heA : A z=A w := by + rw [show P.inverse (A z)=P.inverse (A w) from he,hAw] at hAz + exact hAz.symm + exact exp_injective_strip hz.2 hw'.2 (add_left_cancel heA) + have harea : (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))=volume (f '' S) := by + have hd (z : ℂ) (hz : z∈S) := + (hf.differentiableAt (hS.mem_nhds hz)).hasDerivAt + have he := lintegral_abs_det_fderiv_eq_addHaar_image volume hS.measurableSet + (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) hi + simpa only [complex_derivative_real_det,abs_pow,abs_norm] using he + obtain ⟨M,hM⟩ := P.inverse_bounded_over_ball (‖(x:ℂ)‖+Real.exp q) + have himage : f '' S ⊆ closedBall (0:ℂ) M := by + rintro _ ⟨z,hz,rfl⟩ + rw [mem_closedBall,dist_zero_right] + apply hM (A z) (hAH hz) + calc + ‖A z‖ ≤ ‖(x:ℂ)‖+‖Complex.exp z‖ := norm_add_le _ _ + _ ≤ ‖(x:ℂ)‖+Real.exp q := by + rw [Complex.norm_exp] + exact add_le_add le_rfl (Real.exp_le_exp.mpr hz.1.le) + change (∫⁻ z in S, ENNReal.ofReal (‖deriv f z‖^2))<∞ + rw [harea] + exact (measure_mono himage).trans_lt (isCompact_closedBall (0:ℂ) M).measure_lt_top + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartUniqueness.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartUniqueness.lean index 49422ccdfe4e916a2e6230b1fe04e3c76a53648f..cb6a8cdad2804bcfdc5cf4f0c125e253d80ab3f8 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartUniqueness.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartUniqueness.lean @@ -1,37 +1,37 @@ -import OAI.Probability.SLE.Loewner.SlitChartDisplacement - -namespace OAI - -/-! The normalized static slit chart and its capacity do not depend on a choice. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -lemma SimpleSlitChart.unique_on_domain {p : C(unitInterval,ℂ)} - (P Q : SimpleSlitChart p) : - EqOn P.forward Q.forward (upperHalfPlane \ range p) ∧ - EqOn P.inverse Q.inverse upperHalfPlane ∧ P.capacity=Q.capacity := by - have hc : P.capacity=Q.capacity := le_antisymm - (P.capacity_le_of_range_subset Q Subset.rfl) - (Q.capacity_le_of_range_subset P Subset.rfl) - have hcomp (z : ℂ) (hz : z∈upperHalfPlane) : P.forward (Q.inverse z)=z := by - have hb := P.inverse_comparison Q Subset.rfl hz - rw [hc,sub_self,zero_div] at hb - exact sub_eq_zero.mp (norm_le_zero_iff.mp hb) - have hi : EqOn P.inverse Q.inverse upperHalfPlane := by - intro z hz - have hh := P.left_inverse (Q.inverse z) (Q.inverse_mapsTo hz) - rw [hcomp z hz] at hh - exact hh - refine ⟨?_,hi,hc⟩ - intro z hz - apply Q.inverse_injOn (P.bijOn.mapsTo hz) (Q.bijOn.mapsTo hz) - rw [← hi (P.bijOn.mapsTo hz),P.left_inverse z hz,Q.left_inverse z hz] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartDisplacement + +namespace OAI + +/-! The normalized static slit chart and its capacity do not depend on a choice. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.unique_on_domain {p : C(unitInterval,ℂ)} + (P Q : SimpleSlitChart p) : + EqOn P.forward Q.forward (upperHalfPlane \ range p) ∧ + EqOn P.inverse Q.inverse upperHalfPlane ∧ P.capacity=Q.capacity := by + have hc : P.capacity=Q.capacity := le_antisymm + (P.capacity_le_of_range_subset Q Subset.rfl) + (Q.capacity_le_of_range_subset P Subset.rfl) + have hcomp (z : ℂ) (hz : z∈upperHalfPlane) : P.forward (Q.inverse z)=z := by + have hb := P.inverse_comparison Q Subset.rfl hz + rw [hc,sub_self,zero_div] at hb + exact sub_eq_zero.mp (norm_le_zero_iff.mp hb) + have hi : EqOn P.inverse Q.inverse upperHalfPlane := by + intro z hz + have hh := P.left_inverse (Q.inverse z) (Q.inverse_mapsTo hz) + rw [hcomp z hz] at hh + exact hh + refine ⟨?_,hi,hc⟩ + intro z hz + apply Q.inverse_injOn (P.bijOn.mapsTo hz) (Q.bijOn.mapsTo hz) + rw [← hi (P.bijOn.mapsTo hz),P.left_inverse z hz,Q.left_inverse z hz] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitChartVerticalBounds.lean b/lean/OAI/Probability/SLE/Loewner/SlitChartVerticalBounds.lean index dcc7865c41b343e71ad1b30338622498d17f47fe..759597f2d83c1f3074f465ece2ad42a2e9c6c448 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitChartVerticalBounds.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitChartVerticalBounds.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Loewner.SlitChartExteriorComparison - -namespace OAI - -/-! A common interior base point for normalized forward charts of bounded slits. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma radiusHullMap_two_radius {R : ℝ} (hR : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t)= - concatenateSlit (slitCapacityHorizon p hi h0 hH T) - (slitCapacityCurve p hi h0 hH T) (fun t => Target.trace 1 (Target.linearDriver 0) t) - B (initialDrivingValue p hi h0 hH T hT) := by - funext t - let A := slitCapacityHorizon p hi h0 hH T - let u := initialDrivingValue p hi h0 hH T hT - change Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t= - concatenateSlit A (slitCapacityCurve p hi h0 hH T) - (fun s => Target.trace 1 (Target.linearDriver 0) s) B u t - by_cases ht : t ≤ A - · exact (slitCapacity_raw_trace_eq p hi h0 hH hT ht).trans (by - simp only [concatenateSlit,ite_eq_left ht]) - · have hAt : A 0) (fun s hs => (Target.linearDriver_derivative 0 s hs.1).mono Icc_subset_Ici_self) - (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) - have hw : (u:ℂ)+Target.trace 1 (Target.linearDriver 0) (t-A : ℝ≥0)∈upperHalfPlane := by - simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hη - have he := Target.trace_constant_tail (slitCapacityDriver p hi h0 hH hT) A (t-A) hS u - (fun s hs => slitCapacityDriver_after p hi h0 hH hT hs) - rw [←NNReal.coe_add,add_tsub_cancel_of_le hAt.le] at he - rw [he,concatenateSlit,ite_eq_right ht,hB _ hw] - rw [slitCapacityInverse_eq_raw p hi h0 hH hT le_rfl hw] - change initialInverseChart p hi h0 hH (slitCapacityTime p hi h0 hH T A) _=_ - rw [slitCapacityTime_after p hi h0 hH T le_rfl] - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} - -theorem RegularSLEPresentation.slitCapacity_raw_geometry - (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT0 : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t - Continuous β ∧ β 0=0 ∧ Injective β ∧ (∀ t, 0 Target.trace 1 (Target.linearDriver 0) t - let β := slitCapacityCurve p hi h0 hH T - have hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue (Target.linearDriver 0)) 0 - (Ici 0) t := fun t ht => Target.linearDriver_derivative 0 t ht - have hb : ∀ t : ℝ, 0 ≤ t → |(0:ℝ)| ≤ (0:ℝ≥0) := by intros; simp - have hηc : Continuous η := Target.continuous_smooth_raw_trace _ (fun _ => 0) 0 hd hb - have hη0 : η 0=0 := Target.smooth_trace_zero _ (fun _ => 0) 0 hd hb - have hηi : Injective η := a.smooth_raw_trace_injective hκ _ (fun _ => 0) 0 hd hb - have hηH (t : ℝ≥0) (ht : 0 0) - (fun s hs => (hd s hs.1).mono Icc_subset_Ici_self) (show (0:ℝ) ≤ 0 from le_rfl) - (fun _ _ => by simp) - have hηclosed (t : ℝ≥0) : 0 ≤ (η t).im := by - by_cases ht : t=0 - · simp only [ht,hη0,zero_im,le_refl] - · exact (show 0<(η t).im from hηH t (pos_iff_ne_zero.mpr ht)).le - obtain ⟨B,hBc,hBF,hBu⟩ := exists_initial_inverse_boundary_tip p hi h0 hH hT - have hBc' : ContinuousOn B closedHalfPlane := hBc - have hβc : Continuous β := slitCapacityCurve_continuous p hi h0 hH T - have hβ0 : β 0=0 := slitCapacityCurve_zero p hi h0 hH T - have hβi : InjOn β (Icc 0 A) := slitCapacityCurve_injOn p hi h0 hH T - have hβH (t : ℝ≥0) (ht : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t)=f := - slitCapacity_raw_trace_concat p hi h0 hH hT B hBF - have hg : Continuous f ∧ f 0=0 ∧ Injective f ∧ (∀ t, 0 (hBmap hz).1.1), - concatenateSlit_free_complement A β η B u hβc hηc hβ0 hη0 hβi hηi hβH hηH - hBc' hBA hBi hBmap⟩ - simpa only [←hf] using hg - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitConstantDriverTail +import OAI.Probability.SLE.Loewner.SlitBoundaryTip +import OAI.Probability.SLE.Loewner.RawConstantTailTrace +import OAI.Probability.SLE.Geometry.ConcatenatedSlitGeometry +import OAI.Probability.SLE.Loewner.SmoothRawGeometry + +namespace OAI + +/-! The exact finite slit followed by its vertical image is the raw trace +of the constructed, globally continuous driver. -/ +noncomputable section +open Set Filter Function Complex _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem slitCapacity_raw_trace_concat (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) (B : ℂ → ℂ) + (hB : ∀ z∈upperHalfPlane, B z=initialInverseChart p hi h0 hH T z) : + (fun t : ℝ≥0 => Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t)= + concatenateSlit (slitCapacityHorizon p hi h0 hH T) + (slitCapacityCurve p hi h0 hH T) (fun t => Target.trace 1 (Target.linearDriver 0) t) + B (initialDrivingValue p hi h0 hH T hT) := by + funext t + let A := slitCapacityHorizon p hi h0 hH T + let u := initialDrivingValue p hi h0 hH T hT + change Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t= + concatenateSlit A (slitCapacityCurve p hi h0 hH T) + (fun s => Target.trace 1 (Target.linearDriver 0) s) B u t + by_cases ht : t ≤ A + · exact (slitCapacity_raw_trace_eq p hi h0 hH hT ht).trans (by + simp only [concatenateSlit,ite_eq_left ht]) + · have hAt : A 0) (fun s hs => (Target.linearDriver_derivative 0 s hs.1).mono Icc_subset_Ici_self) + (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) + have hw : (u:ℂ)+Target.trace 1 (Target.linearDriver 0) (t-A : ℝ≥0)∈upperHalfPlane := by + simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hη + have he := Target.trace_constant_tail (slitCapacityDriver p hi h0 hH hT) A (t-A) hS u + (fun s hs => slitCapacityDriver_after p hi h0 hH hT hs) + rw [←NNReal.coe_add,add_tsub_cancel_of_le hAt.le] at he + rw [he,concatenateSlit,ite_eq_right ht,hB _ hw] + rw [slitCapacityInverse_eq_raw p hi h0 hH hT le_rfl hw] + change initialInverseChart p hi h0 hH (slitCapacityTime p hi h0 hH T A) _=_ + rw [slitCapacityTime_after p hi h0 hH T le_rfl] + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} + +theorem RegularSLEPresentation.slitCapacity_raw_geometry + (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) + (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT0 : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t + Continuous β ∧ β 0=0 ∧ Injective β ∧ (∀ t, 0 Target.trace 1 (Target.linearDriver 0) t + let β := slitCapacityCurve p hi h0 hH T + have hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue (Target.linearDriver 0)) 0 + (Ici 0) t := fun t ht => Target.linearDriver_derivative 0 t ht + have hb : ∀ t : ℝ, 0 ≤ t → |(0:ℝ)| ≤ (0:ℝ≥0) := by intros; simp + have hηc : Continuous η := Target.continuous_smooth_raw_trace _ (fun _ => 0) 0 hd hb + have hη0 : η 0=0 := Target.smooth_trace_zero _ (fun _ => 0) 0 hd hb + have hηi : Injective η := a.smooth_raw_trace_injective hκ _ (fun _ => 0) 0 hd hb + have hηH (t : ℝ≥0) (ht : 0 0) + (fun s hs => (hd s hs.1).mono Icc_subset_Ici_self) (show (0:ℝ) ≤ 0 from le_rfl) + (fun _ _ => by simp) + have hηclosed (t : ℝ≥0) : 0 ≤ (η t).im := by + by_cases ht : t=0 + · simp only [ht,hη0,zero_im,le_refl] + · exact (show 0<(η t).im from hηH t (pos_iff_ne_zero.mpr ht)).le + obtain ⟨B,hBc,hBF,hBu⟩ := exists_initial_inverse_boundary_tip p hi h0 hH hT + have hBc' : ContinuousOn B closedHalfPlane := hBc + have hβc : Continuous β := slitCapacityCurve_continuous p hi h0 hH T + have hβ0 : β 0=0 := slitCapacityCurve_zero p hi h0 hH T + have hβi : InjOn β (Icc 0 A) := slitCapacityCurve_injOn p hi h0 hH T + have hβH (t : ℝ≥0) (ht : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t)=f := + slitCapacity_raw_trace_concat p hi h0 hH hT B hBF + have hg : Continuous f ∧ f 0=0 ∧ Injective f ∧ (∀ t, 0 (hBmap hz).1.1), + concatenateSlit_free_complement A β η B u hβc hηc hβ0 hη0 hβi hηi hβH hηH + hBc' hBA hBi hBmap⟩ + simpa only [←hf] using hg + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitExteriorChart.lean b/lean/OAI/Probability/SLE/Loewner/SlitExteriorChart.lean index 03e6fede1e02577474266bec48869e5d234c60b4..f823b545c101d039a2ce5bb90d61cd8c3a3f4067 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitExteriorChart.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitExteriorChart.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Geometry.NormalizedReflectedSlit -import OAI.Probability.SLE.Geometry.ConjugateInversionChart - -namespace OAI - -/-! The normalized reflected Riemann chart gives an analytic chart from -an arbitrary simple attached slit domain onto the upper exterior disk. -/ -noncomputable section -open Set Complex Function Metric -open scoped ComplexConjugate - -namespace SLEExactGauge - -def slitExteriorMap (f : ℂ → ℂ) (z : ℂ) : ℂ := (f z⁻¹)⁻¹ - -lemma inverse_mem_reflected_inverted - (p : C(unitInterval,ℂ)) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {z : ℂ} (hz : z ∈ upperHalfPlane \ range p) : - z⁻¹ ∈ invertedExterior (reflectedArcSet p) := by - have hm := (conjugateInversion_slit_bijOn p h0 hH).mapsTo hz - have hc := invertedExterior_conj (reflectedArcSet_conj p) hm.1 - simpa only [conjugateInversion,map_inv₀,conj_conj] using hc - -lemma slitExteriorMap_eq_conjugate - (p : C(unitInterval,ℂ)) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {f : ℂ → ℂ} - (hs : ∀ z ∈ invertedExterior (reflectedArcSet p), f (conj z)=conj (f z)) : - EqOn (conjugateInversion ∘ f ∘ conjugateInversion) (slitExteriorMap f) - (upperHalfPlane \ range p) := by - intro z hz - have he := hs z⁻¹ (inverse_mem_reflected_inverted p h0 hH hz) - change (conj (f ((conj z)⁻¹)))⁻¹ = (f z⁻¹)⁻¹ - rw [← map_inv₀ conj z,he,conj_conj] - -lemma slitExteriorMap_bijOn - (p : C(unitInterval,ℂ)) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {f : ℂ → ℂ} - (hs : ∀ z ∈ invertedExterior (reflectedArcSet p), f (conj z)=conj (f z)) - (hb : BijOn f (invertedExterior (reflectedArcSet p) ∩ upperHalfPlane) - (ball 0 1 ∩ upperHalfPlane)) : - BijOn (slitExteriorMap f) (upperHalfPlane \ range p) exteriorUpperDisk := by - exact ((conjugateInversion_disk_upper_bijOn.comp hb).comp - (conjugateInversion_slit_bijOn p h0 hH)).congr - (slitExteriorMap_eq_conjugate p h0 hH hs) - -lemma slitExteriorMap_differentiableOn - (p : C(unitInterval,ℂ)) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (invertedExterior (reflectedArcSet p))) - (hi : InjOn f (invertedExterior (reflectedArcSet p))) (hf0 : f 0=0) : - DifferentiableOn ℂ (slitExteriorMap f) (upperHalfPlane \ range p) := by - have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union - ((isCompact_range p.continuous).image Complex.continuous_conj) - intro z hz - have hz0 : z ≠ 0 := by - intro he - have hzH := hz.1 - simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hzH - have hm := inverse_mem_reflected_inverted p h0 hH hz - have hdf := hf.differentiableAt ((invertedExterior_isOpen hK).mem_nhds hm) - have hfi : f z⁻¹ ≠ 0 := by - intro he - have heq := hi hm (zero_mem_invertedExterior _) (he.trans hf0.symm) - exact inv_ne_zero hz0 heq - exact ((hdf.comp z (differentiableAt_id.inv hz0)).inv hfi).differentiableWithinAt - -theorem simple_slit_exists_exterior_chart - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ∃ (f : ℂ → ℂ) (a : ℝ), 0 < a ∧ - BijOn (slitExteriorMap f) (upperHalfPlane \ range p) exteriorUpperDisk ∧ - DifferentiableOn ℂ (slitExteriorMap f) (upperHalfPlane \ range p) ∧ - f 0=0 ∧ deriv f 0=(a:ℂ) ∧ - DifferentiableOn ℂ f (invertedExterior (reflectedArcSet p)) := by - obtain ⟨f,a,ha,hb,hf,hf0,hfa,hs,hu⟩ := - reflected_slit_exists_normalized_chart p hi h0 hH - exact ⟨f,a,ha,slitExteriorMap_bijOn p h0 hH hs hu, - slitExteriorMap_differentiableOn p h0 hH hf hb.injOn hf0,hf0,hfa,hf⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.NormalizedReflectedSlit +import OAI.Probability.SLE.Geometry.ConjugateInversionChart + +namespace OAI + +/-! The normalized reflected Riemann chart gives an analytic chart from +an arbitrary simple attached slit domain onto the upper exterior disk. -/ +noncomputable section +open Set Complex Function Metric +open scoped ComplexConjugate + +namespace SLEExactGauge + +def slitExteriorMap (f : ℂ → ℂ) (z : ℂ) : ℂ := (f z⁻¹)⁻¹ + +lemma inverse_mem_reflected_inverted + (p : C(unitInterval,ℂ)) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {z : ℂ} (hz : z ∈ upperHalfPlane \ range p) : + z⁻¹ ∈ invertedExterior (reflectedArcSet p) := by + have hm := (conjugateInversion_slit_bijOn p h0 hH).mapsTo hz + have hc := invertedExterior_conj (reflectedArcSet_conj p) hm.1 + simpa only [conjugateInversion,map_inv₀,conj_conj] using hc + +lemma slitExteriorMap_eq_conjugate + (p : C(unitInterval,ℂ)) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {f : ℂ → ℂ} + (hs : ∀ z ∈ invertedExterior (reflectedArcSet p), f (conj z)=conj (f z)) : + EqOn (conjugateInversion ∘ f ∘ conjugateInversion) (slitExteriorMap f) + (upperHalfPlane \ range p) := by + intro z hz + have he := hs z⁻¹ (inverse_mem_reflected_inverted p h0 hH hz) + change (conj (f ((conj z)⁻¹)))⁻¹ = (f z⁻¹)⁻¹ + rw [← map_inv₀ conj z,he,conj_conj] + +lemma slitExteriorMap_bijOn + (p : C(unitInterval,ℂ)) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {f : ℂ → ℂ} + (hs : ∀ z ∈ invertedExterior (reflectedArcSet p), f (conj z)=conj (f z)) + (hb : BijOn f (invertedExterior (reflectedArcSet p) ∩ upperHalfPlane) + (ball 0 1 ∩ upperHalfPlane)) : + BijOn (slitExteriorMap f) (upperHalfPlane \ range p) exteriorUpperDisk := by + exact ((conjugateInversion_disk_upper_bijOn.comp hb).comp + (conjugateInversion_slit_bijOn p h0 hH)).congr + (slitExteriorMap_eq_conjugate p h0 hH hs) + +lemma slitExteriorMap_differentiableOn + (p : C(unitInterval,ℂ)) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (invertedExterior (reflectedArcSet p))) + (hi : InjOn f (invertedExterior (reflectedArcSet p))) (hf0 : f 0=0) : + DifferentiableOn ℂ (slitExteriorMap f) (upperHalfPlane \ range p) := by + have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj) + intro z hz + have hz0 : z ≠ 0 := by + intro he + have hzH := hz.1 + simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hzH + have hm := inverse_mem_reflected_inverted p h0 hH hz + have hdf := hf.differentiableAt ((invertedExterior_isOpen hK).mem_nhds hm) + have hfi : f z⁻¹ ≠ 0 := by + intro he + have heq := hi hm (zero_mem_invertedExterior _) (he.trans hf0.symm) + exact inv_ne_zero hz0 heq + exact ((hdf.comp z (differentiableAt_id.inv hz0)).inv hfi).differentiableWithinAt + +theorem simple_slit_exists_exterior_chart + (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + ∃ (f : ℂ → ℂ) (a : ℝ), 0 < a ∧ + BijOn (slitExteriorMap f) (upperHalfPlane \ range p) exteriorUpperDisk ∧ + DifferentiableOn ℂ (slitExteriorMap f) (upperHalfPlane \ range p) ∧ + f 0=0 ∧ deriv f 0=(a:ℂ) ∧ + DifferentiableOn ℂ f (invertedExterior (reflectedArcSet p)) := by + obtain ⟨f,a,ha,hb,hf,hf0,hfa,hs,hu⟩ := + reflected_slit_exists_normalized_chart p hi h0 hH + exact ⟨f,a,ha,slitExteriorMap_bijOn p h0 hH hs hu, + slitExteriorMap_differentiableOn p h0 hH hf hb.injOn hf0,hf0,hfa,hf⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitForwardBoundaryHeight.lean b/lean/OAI/Probability/SLE/Loewner/SlitForwardBoundaryHeight.lean index 49194d7be5fb7eb8a0f6a714743c8ed73f13fdc0..2f507e7657c6886a101c73d48cd4d3724c12e198 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitForwardBoundaryHeight.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitForwardBoundaryHeight.lean @@ -1,74 +1,74 @@ -import OAI.Probability.SLE.Loewner.SlitChartProper -import OAI.Probability.SLE.Geometry.ComplexEndpointCluster - -namespace OAI - -/-! A physical sequence approaching the slit boundary has forward-chart -height tending to zero. -/ -noncomputable section -open Set Filter Function Metric Complex -open scoped Topology - -namespace SLEExactGauge - -theorem SimpleSlitChart.forward_height_tendsto_zero {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) {z : ℕ → ℂ} {b : ℂ} - (hz : Tendsto z atTop (𝓝 b)) - (hzD : ∀ n, z n∈upperHalfPlane \ range p) (hb : b∉upperHalfPlane \ range p) : - Tendsto (fun n => (P.forward (z n)).im) atTop (𝓝 0) := by - obtain ⟨R,hR⟩ := (Metric.isBounded_range_of_tendsto z hz).exists_norm_le - obtain ⟨M,hM⟩ := P.forward_bounded_over_ball R - apply tendsto_of_subseq_tendsto - intro φ hφ - obtain ⟨w,_,ψ,hψ,hw⟩ := (isCompact_closedBall (0:ℂ) M).tendsto_subseq (fun n => by - rw [mem_closedBall,dist_zero_right] - exact hM (z (φ n)) (hzD _) (hR _ (mem_range_self _))) - have hwim : 0≤w.im := ge_of_tendsto (continuous_im.continuousAt.tendsto.comp hw) - (Eventually.of_forall fun n => (show 0<(P.forward (z (φ (ψ n)))).im from - P.bijOn.mapsTo (hzD _)).le) - have hezero : w.im=0 := by - by_contra hne - have hwH : w∈upperHalfPlane := lt_of_le_of_ne hwim (Ne.symm hne) - have hback := ((P.inverse_differentiable.continuousOn w hwH).continuousAt - (isOpen_upperHalfPlane.mem_nhds hwH)).tendsto.comp hw - have he : P.inverse w=b := by - apply tendsto_nhds_unique hback - have hzb := (hz.comp hφ).comp hψ.tendsto_atTop - apply hzb.congr - intro n - exact (P.left_inverse _ (hzD _)).symm - exact hb (he ▸ P.inverse_mapsTo hwH) - refine ⟨ψ,?_⟩ - have hh := continuous_im.continuousAt.tendsto.comp hw - change Tendsto (fun n => (P.forward (z (φ (ψ n)))).im) atTop (𝓝 w.im) at hh - rwa [hezero] at hh - -theorem SimpleSlitChart.forward_height_tendsto_zero_filter {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) {α : Type*} {l : Filter α} [l.IsCountablyGenerated] - {z : α → ℂ} {b : ℂ} (hz : Tendsto z l (𝓝 b)) - (hzD : ∀ᶠ x in l, z x∈upperHalfPlane \ range p) - (hb : b∉upperHalfPlane \ range p) : - Tendsto (fun x => (P.forward (z x)).im) l (𝓝 0) := by - classical - apply tendsto_iff_seq_tendsto.mpr - intro u hu - let w : ℕ → ℂ := fun n => - if z (u n)∈upperHalfPlane \ range p then z (u n) else P.inverse I - have he : w =ᶠ[atTop] fun n => z (u n) := by - filter_upwards [hu.eventually hzD] with n hn - exact ite_eq_left hn - have hw : Tendsto w atTop (𝓝 b) := (hz.comp hu).congr' he.symm - have hwD : ∀ n, w n∈upperHalfPlane \ range p := by - intro n - dsimp only [w] - split_ifs with hn - · exact hn - · exact P.inverse_mapsTo (by change 0<(I:ℂ).im; norm_num) - exact (P.forward_height_tendsto_zero hw hwD hb).congr' - (he.mono fun n hn => congrArg (fun v => (P.forward v).im) hn) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitChartProper +import OAI.Probability.SLE.Geometry.ComplexEndpointCluster + +namespace OAI + +/-! A physical sequence approaching the slit boundary has forward-chart +height tending to zero. -/ +noncomputable section +open Set Filter Function Metric Complex +open scoped Topology + +namespace SLEExactGauge + +theorem SimpleSlitChart.forward_height_tendsto_zero {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) {z : ℕ → ℂ} {b : ℂ} + (hz : Tendsto z atTop (𝓝 b)) + (hzD : ∀ n, z n∈upperHalfPlane \ range p) (hb : b∉upperHalfPlane \ range p) : + Tendsto (fun n => (P.forward (z n)).im) atTop (𝓝 0) := by + obtain ⟨R,hR⟩ := (Metric.isBounded_range_of_tendsto z hz).exists_norm_le + obtain ⟨M,hM⟩ := P.forward_bounded_over_ball R + apply tendsto_of_subseq_tendsto + intro φ hφ + obtain ⟨w,_,ψ,hψ,hw⟩ := (isCompact_closedBall (0:ℂ) M).tendsto_subseq (fun n => by + rw [mem_closedBall,dist_zero_right] + exact hM (z (φ n)) (hzD _) (hR _ (mem_range_self _))) + have hwim : 0≤w.im := ge_of_tendsto (continuous_im.continuousAt.tendsto.comp hw) + (Eventually.of_forall fun n => (show 0<(P.forward (z (φ (ψ n)))).im from + P.bijOn.mapsTo (hzD _)).le) + have hezero : w.im=0 := by + by_contra hne + have hwH : w∈upperHalfPlane := lt_of_le_of_ne hwim (Ne.symm hne) + have hback := ((P.inverse_differentiable.continuousOn w hwH).continuousAt + (isOpen_upperHalfPlane.mem_nhds hwH)).tendsto.comp hw + have he : P.inverse w=b := by + apply tendsto_nhds_unique hback + have hzb := (hz.comp hφ).comp hψ.tendsto_atTop + apply hzb.congr + intro n + exact (P.left_inverse _ (hzD _)).symm + exact hb (he ▸ P.inverse_mapsTo hwH) + refine ⟨ψ,?_⟩ + have hh := continuous_im.continuousAt.tendsto.comp hw + change Tendsto (fun n => (P.forward (z (φ (ψ n)))).im) atTop (𝓝 w.im) at hh + rwa [hezero] at hh + +theorem SimpleSlitChart.forward_height_tendsto_zero_filter {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) {α : Type*} {l : Filter α} [l.IsCountablyGenerated] + {z : α → ℂ} {b : ℂ} (hz : Tendsto z l (𝓝 b)) + (hzD : ∀ᶠ x in l, z x∈upperHalfPlane \ range p) + (hb : b∉upperHalfPlane \ range p) : + Tendsto (fun x => (P.forward (z x)).im) l (𝓝 0) := by + classical + apply tendsto_iff_seq_tendsto.mpr + intro u hu + let w : ℕ → ℂ := fun n => + if z (u n)∈upperHalfPlane \ range p then z (u n) else P.inverse I + have he : w =ᶠ[atTop] fun n => z (u n) := by + filter_upwards [hu.eventually hzD] with n hn + exact ite_eq_left hn + have hw : Tendsto w atTop (𝓝 b) := (hz.comp hu).congr' he.symm + have hwD : ∀ n, w n∈upperHalfPlane \ range p := by + intro n + dsimp only [w] + split_ifs with hn + · exact hn + · exact P.inverse_mapsTo (by change 0<(I:ℂ).im; norm_num) + exact (P.forward_height_tendsto_zero hw hwD hb).congr' + (he.mono fun n hn => congrArg (fun v => (P.forward v).im) hn) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitForwardEndpoint.lean b/lean/OAI/Probability/SLE/Loewner/SlitForwardEndpoint.lean index d067765b4e9a38ad07ff3ac55d3ea5a21ad92007..e678bbe31455182a81eddbfc8109b64e01b8fcd6 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitForwardEndpoint.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitForwardEndpoint.lean @@ -1,54 +1,54 @@ -import OAI.Probability.SLE.Loewner.SlitForwardBoundaryHeight -import OAI.Probability.SLE.Geometry.ComplexEndpointLimit -import OAI.Probability.SLE.Loewner.SlitBoundaryFiber - -namespace OAI - -/-! A continuous curve in a slit domain approaching its boundary has a -real forward-chart limit. No uniqueness of arbitrary boundary fibers is -assumed: the connected cluster set and noncollapse of intervals suffice. -/ -noncomputable section -open Set Filter Function Metric Complex -open scoped Topology - -namespace SLEExactGauge - -theorem SimpleSlitChart.exists_forward_endpoint_limit {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) {z : ℝ → ℂ} {b : ℂ} {D : ℝ} - (hD : 0] (0:ℝ)) (𝓝 b)) - (hzD : MapsTo z (Ioo 0 D) (upperHalfPlane \ range p)) - (hb : b∉upperHalfPlane \ range p) : - ∃ a : ℝ, Tendsto (fun t => P.forward (z t)) (𝓝[>] (0:ℝ)) (𝓝 (a:ℂ)) := by - have hsmall : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), t∈Ioo 0 D := - Ioo_mem_nhdsGT hD - have hvalid : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), z t∈upperHalfPlane \ range p := - hsmall.mono hzD - have hnorm : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), ‖z t‖≤‖b‖+1 := by - exact (hz.norm.eventually (gt_mem_nhds (by linarith : ‖b‖<‖b‖+1))).mono - (fun _ h => h.le) - obtain ⟨M,hM⟩ := P.forward_bounded_over_ball (‖b‖+1) - have hbound : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), ‖P.forward (z t)‖≤M := by - filter_upwards [hvalid,hnorm] with t ht htn - exact hM _ ht htn - obtain ⟨B,hBC,hBe⟩ := P.exists_boundary_extension - have himage : Tendsto (fun t => B (P.forward (z t))) (𝓝[>] (0:ℝ)) (𝓝 b) := by - apply hz.congr' - filter_upwards [hvalid] with t ht - exact (hBe _ (P.bijOn.mapsTo ht)).trans (P.left_inverse _ ht) |>.symm - obtain ⟨a,ha,_⟩ := exists_complex_endpoint_limit hD - (P.forward_differentiable.continuousOn.comp hzC hzD) hbound - (hvalid.mono fun _ ht => (show 0<(P.forward _).im from P.bijOn.mapsTo ht).le) - (P.forward_height_tendsto_zero_filter hz hvalid hb) hBC himage (by - intro a d had hconst - apply P.boundary_not_constant_on_interval hBC hBe ((a+d)/2) - (show 0<(d-a)/2 by linarith) b - intro y hy - exact hconst y ⟨by linarith [hy.1],by linarith [hy.2]⟩) - exact ⟨a,ha⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitForwardBoundaryHeight +import OAI.Probability.SLE.Geometry.ComplexEndpointLimit +import OAI.Probability.SLE.Loewner.SlitBoundaryFiber + +namespace OAI + +/-! A continuous curve in a slit domain approaching its boundary has a +real forward-chart limit. No uniqueness of arbitrary boundary fibers is +assumed: the connected cluster set and noncollapse of intervals suffice. -/ +noncomputable section +open Set Filter Function Metric Complex +open scoped Topology + +namespace SLEExactGauge + +theorem SimpleSlitChart.exists_forward_endpoint_limit {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) {z : ℝ → ℂ} {b : ℂ} {D : ℝ} + (hD : 0] (0:ℝ)) (𝓝 b)) + (hzD : MapsTo z (Ioo 0 D) (upperHalfPlane \ range p)) + (hb : b∉upperHalfPlane \ range p) : + ∃ a : ℝ, Tendsto (fun t => P.forward (z t)) (𝓝[>] (0:ℝ)) (𝓝 (a:ℂ)) := by + have hsmall : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), t∈Ioo 0 D := + Ioo_mem_nhdsGT hD + have hvalid : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), z t∈upperHalfPlane \ range p := + hsmall.mono hzD + have hnorm : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), ‖z t‖≤‖b‖+1 := by + exact (hz.norm.eventually (gt_mem_nhds (by linarith : ‖b‖<‖b‖+1))).mono + (fun _ h => h.le) + obtain ⟨M,hM⟩ := P.forward_bounded_over_ball (‖b‖+1) + have hbound : ∀ᶠ t : ℝ in 𝓝[>] (0:ℝ), ‖P.forward (z t)‖≤M := by + filter_upwards [hvalid,hnorm] with t ht htn + exact hM _ ht htn + obtain ⟨B,hBC,hBe⟩ := P.exists_boundary_extension + have himage : Tendsto (fun t => B (P.forward (z t))) (𝓝[>] (0:ℝ)) (𝓝 b) := by + apply hz.congr' + filter_upwards [hvalid] with t ht + exact (hBe _ (P.bijOn.mapsTo ht)).trans (P.left_inverse _ ht) |>.symm + obtain ⟨a,ha,_⟩ := exists_complex_endpoint_limit hD + (P.forward_differentiable.continuousOn.comp hzC hzD) hbound + (hvalid.mono fun _ ht => (show 0<(P.forward _).im from P.bijOn.mapsTo ht).le) + (P.forward_height_tendsto_zero_filter hz hvalid hb) hBC himage (by + intro a d had hconst + apply P.boundary_not_constant_on_interval hBC hBe ((a+d)/2) + (show 0<(d-a)/2 by linarith) b + intro y hy + exact hconst y ⟨by linarith [hy.1],by linarith [hy.2]⟩) + exact ⟨a,ha⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitHydrodynamicChart.lean b/lean/OAI/Probability/SLE/Loewner/SlitHydrodynamicChart.lean index b7932f0dc8aae46a3d1d095a77e628f73f55f620..eeef8a5cc6850e6be0b39345b872daad98b638cb 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitHydrodynamicChart.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitHydrodynamicChart.lean @@ -1,98 +1,98 @@ -import OAI.Probability.SLE.Geometry.ExteriorLaurent - -namespace OAI - -/-! Static hydrodynamic normalization for the complement of an arbitrary -simple attached arc. No time parametrization or Loewner equation is asserted. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -lemma positive_real_affine_halfPlane {a : ℝ} (ha : 0 (a:ℂ)*z-(b:ℂ)) upperHalfPlane upperHalfPlane := by - have ha0 : (a:ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne' - refine ⟨?_,?_,?_⟩ - · intro z hz - change 0 < ((a:ℂ)*z-(b:ℂ)).im - simpa only [sub_im,mul_im,ofReal_re,ofReal_im,zero_mul,add_zero,sub_zero] - using mul_pos ha hz - · intro z _ w _ he - apply mul_left_cancel₀ ha0 - simpa only [sub_add_cancel] using congrArg (fun x : ℂ => x+(b:ℂ)) he - · intro z hz - refine ⟨(z+(b:ℂ))/(a:ℂ),?_,?_⟩ - · change 0 < ((z+(b:ℂ))/(a:ℂ)).im - have he : ((z+(b:ℂ))/(a:ℂ)).im=z.im/a := by - rw [div_ofReal_im,add_im,ofReal_im,add_zero] - rw [he] - exact div_pos hz ha - · field_simp - ring - -lemma eq_ofReal_re_of_conj_eq {z : ℂ} (hz : conj z=z) : z=(z.re:ℂ) := by - have hi := congrArg Complex.im hz - simp only [conj_im] at hi - apply Complex.ext - · simp - · simp only [ofReal_im] - linarith - -def slitHydrodynamicMap (f : ℂ → ℂ) (a : ℝ) (z : ℂ) : ℂ := - (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)-deriv (slitInfinityFactor f a) 0 - -theorem simple_slit_exists_hydrodynamic_chart - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : - ∃ (G F : ℂ → ℂ) (c : ℝ), - BijOn G (upperHalfPlane \ range p) upperHalfPlane ∧ - DifferentiableOn ℂ G (upperHalfPlane \ range p) ∧ - DifferentiableOn ℂ F upperHalfPlane ∧ - (∀ z ∈ upperHalfPlane \ range p, F (G z)=z) ∧ - (∀ z ∈ upperHalfPlane, G (F z)=z) ∧ - Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) := by - obtain ⟨f,a,ha,hb,hf,hf0,hfa,hs,hu⟩ := - reflected_slit_exists_normalized_chart p hi h0 hH - have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union - ((isCompact_range p.continuous).image Complex.continuous_conj) - have hD : invertedExterior (reflectedArcSet p) ∈ 𝓝 (0:ℂ) := - (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) - have hfa' : AnalyticAt ℂ f 0 := hf.analyticAt hD - have hsH := slitInfinityFactor_reflection hD hs a - have hreal := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD hsH) - have hsQ := dslope_reflection hD hsH - have hc := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD hsQ) - let G := slitHydrodynamicMap f a - have hG : BijOn G (upperHalfPlane \ range p) upperHalfPlane := by - have hh := (positive_real_affine_halfPlane ha (deriv (slitInfinityFactor f a) 0).re).comp - (exteriorJoukowski_bijOn.comp (slitExteriorMap_bijOn p h0 hH hs hu)) - apply hh.congr - intro z _ - change (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- - ((deriv (slitInfinityFactor f a) 0).re:ℂ)= - (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)-deriv (slitInfinityFactor f a) 0 - rw [← hreal] - have hdiff : DifferentiableOn ℂ G (upperHalfPlane \ range p) := by - have he := exteriorJoukowski_differentiableOn.comp - (slitExteriorMap_differentiableOn p h0 hH hf hb.injOn hf0) - (slitExteriorMap_bijOn p h0 hH hs hu).mapsTo - exact (he.const_mul (a:ℂ)).sub_const _ - have ho : IsOpen (upperHalfPlane \ range p) := - isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed - let F := invFunOn G (upperHalfPlane \ range p) - have hF : DifferentiableOn ℂ F upperHalfPlane := by - have hh := hdiff.invFunOn ho hG.injOn - simpa only [F,hG.image_eq] using hh - refine ⟨G,F,(deriv (dslope (slitInfinityFactor f a) 0) 0).re,hG,hdiff,hF, - fun z hz => hG.injOn.leftInvOn_invFunOn hz, - fun z hz => hG.surjOn.rightInvOn_invFunOn hz,?_⟩ - have ht := slitExteriorMap_laurent_limit hfa' hf0 ha.ne' hfa - rw [hc] at ht - exact ht - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ExteriorLaurent + +namespace OAI + +/-! Static hydrodynamic normalization for the complement of an arbitrary +simple attached arc. No time parametrization or Loewner equation is asserted. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +lemma positive_real_affine_halfPlane {a : ℝ} (ha : 0 (a:ℂ)*z-(b:ℂ)) upperHalfPlane upperHalfPlane := by + have ha0 : (a:ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne' + refine ⟨?_,?_,?_⟩ + · intro z hz + change 0 < ((a:ℂ)*z-(b:ℂ)).im + simpa only [sub_im,mul_im,ofReal_re,ofReal_im,zero_mul,add_zero,sub_zero] + using mul_pos ha hz + · intro z _ w _ he + apply mul_left_cancel₀ ha0 + simpa only [sub_add_cancel] using congrArg (fun x : ℂ => x+(b:ℂ)) he + · intro z hz + refine ⟨(z+(b:ℂ))/(a:ℂ),?_,?_⟩ + · change 0 < ((z+(b:ℂ))/(a:ℂ)).im + have he : ((z+(b:ℂ))/(a:ℂ)).im=z.im/a := by + rw [div_ofReal_im,add_im,ofReal_im,add_zero] + rw [he] + exact div_pos hz ha + · field_simp + ring + +lemma eq_ofReal_re_of_conj_eq {z : ℂ} (hz : conj z=z) : z=(z.re:ℂ) := by + have hi := congrArg Complex.im hz + simp only [conj_im] at hi + apply Complex.ext + · simp + · simp only [ofReal_im] + linarith + +def slitHydrodynamicMap (f : ℂ → ℂ) (a : ℝ) (z : ℂ) : ℂ := + (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)-deriv (slitInfinityFactor f a) 0 + +theorem simple_slit_exists_hydrodynamic_chart + (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) : + ∃ (G F : ℂ → ℂ) (c : ℝ), + BijOn G (upperHalfPlane \ range p) upperHalfPlane ∧ + DifferentiableOn ℂ G (upperHalfPlane \ range p) ∧ + DifferentiableOn ℂ F upperHalfPlane ∧ + (∀ z ∈ upperHalfPlane \ range p, F (G z)=z) ∧ + (∀ z ∈ upperHalfPlane, G (F z)=z) ∧ + Tendsto (fun z : ℂ => z*(G z-z)) (cocompact ℂ) (𝓝 (c:ℂ)) := by + obtain ⟨f,a,ha,hb,hf,hf0,hfa,hs,hu⟩ := + reflected_slit_exists_normalized_chart p hi h0 hH + have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj) + have hD : invertedExterior (reflectedArcSet p) ∈ 𝓝 (0:ℂ) := + (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) + have hfa' : AnalyticAt ℂ f 0 := hf.analyticAt hD + have hsH := slitInfinityFactor_reflection hD hs a + have hreal := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD hsH) + have hsQ := dslope_reflection hD hsH + have hc := eq_ofReal_re_of_conj_eq (derivative_real_of_reflection hD hsQ) + let G := slitHydrodynamicMap f a + have hG : BijOn G (upperHalfPlane \ range p) upperHalfPlane := by + have hh := (positive_real_affine_halfPlane ha (deriv (slitInfinityFactor f a) 0).re).comp + (exteriorJoukowski_bijOn.comp (slitExteriorMap_bijOn p h0 hH hs hu)) + apply hh.congr + intro z _ + change (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)- + ((deriv (slitInfinityFactor f a) 0).re:ℂ)= + (a:ℂ)*exteriorJoukowski (slitExteriorMap f z)-deriv (slitInfinityFactor f a) 0 + rw [← hreal] + have hdiff : DifferentiableOn ℂ G (upperHalfPlane \ range p) := by + have he := exteriorJoukowski_differentiableOn.comp + (slitExteriorMap_differentiableOn p h0 hH hf hb.injOn hf0) + (slitExteriorMap_bijOn p h0 hH hs hu).mapsTo + exact (he.const_mul (a:ℂ)).sub_const _ + have ho : IsOpen (upperHalfPlane \ range p) := + isOpen_upperHalfPlane.sdiff (isCompact_range p.continuous).isClosed + let F := invFunOn G (upperHalfPlane \ range p) + have hF : DifferentiableOn ℂ F upperHalfPlane := by + have hh := hdiff.invFunOn ho hG.injOn + simpa only [F,hG.image_eq] using hh + refine ⟨G,F,(deriv (dslope (slitInfinityFactor f a) 0) 0).re,hG,hdiff,hF, + fun z hz => hG.injOn.leftInvOn_invFunOn hz, + fun z hz => hG.surjOn.rightInvOn_invFunOn hz,?_⟩ + have ht := slitExteriorMap_laurent_limit hfa' hf0 ha.ne' hfa + rw [hc] at ht + exact ht + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitInitialEvolution.lean b/lean/OAI/Probability/SLE/Loewner/SlitInitialEvolution.lean index 3b25825f82a49d3de088dd4a04984e74025a7a54..065822da9f3a1d39f0256d1fbae608becc33f43e 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitInitialEvolution.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitInitialEvolution.lean @@ -1,45 +1,45 @@ -import OAI.Probability.SLE.Loewner.SlitClockRestriction - -namespace OAI - -/-! Every point surviving to a chosen physical time follows the exact -Loewner equation on the whole interval up to that time. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem slitCapacityForward_derivative_upto (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {t : ℝ≥0} - (ht : t ≤ slitCapacityHorizon p hi h0 hH T) {z : ℂ} - (hz : z∈upperHalfPlane \ range (initialSubarc p (slitCapacityTime p hi h0 hH T t))) - {u : ℝ} (hu : u∈Icc 0 (t:ℝ)) : - HasDerivWithinAt (fun v : ℝ => slitCapacityForward p hi h0 hH T v.toNNReal z) - (2/(slitCapacityForward p hi h0 hH T u.toNNReal z- - (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ))) (Icc 0 (t:ℝ)) u := by - let S := slitCapacityTime p hi h0 hH T t - have hST : S ≤ T := slitCapacityTime_le p hi h0 hH T t - have hS : S<1 := hST.trans_lt hT - have hHS : slitCapacityHorizon p hi h0 hH S=t := slitCapacityHorizon_time p hi h0 hH T ht - have hu' : u.toNNReal ≤ slitCapacityHorizon p hi h0 hH S := by - rw [hHS] - exact Real.toNNReal_le_iff_le_coe.mpr hu.2 - have hd := slitCapacityForward_prefix_derivative p hi h0 hH hS hz hu' - rw [hHS,Real.coe_toNNReal u hu.1] at hd - have hG (v : ℝ) (hv : v∈Icc 0 (t:ℝ)) : - slitCapacityForward p hi h0 hH T v.toNNReal z= - slitCapacityForward p hi h0 hH S v.toNNReal z := by - rw [slitCapacityForward_restrict p hi h0 hH hST] - rw [hHS] - exact Real.toNNReal_le_iff_le_coe.mpr hv.2 - have hres := hd.congr hG (hG u hu) - rw [← hG u hu,← slitCapacityDriver_restrict p hi h0 hH hST hT hu'] at hres - exact hres - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitClockRestriction + +namespace OAI + +/-! Every point surviving to a chosen physical time follows the exact +Loewner equation on the whole interval up to that time. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem slitCapacityForward_derivative_upto (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {t : ℝ≥0} + (ht : t ≤ slitCapacityHorizon p hi h0 hH T) {z : ℂ} + (hz : z∈upperHalfPlane \ range (initialSubarc p (slitCapacityTime p hi h0 hH T t))) + {u : ℝ} (hu : u∈Icc 0 (t:ℝ)) : + HasDerivWithinAt (fun v : ℝ => slitCapacityForward p hi h0 hH T v.toNNReal z) + (2/(slitCapacityForward p hi h0 hH T u.toNNReal z- + (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ))) (Icc 0 (t:ℝ)) u := by + let S := slitCapacityTime p hi h0 hH T t + have hST : S ≤ T := slitCapacityTime_le p hi h0 hH T t + have hS : S<1 := hST.trans_lt hT + have hHS : slitCapacityHorizon p hi h0 hH S=t := slitCapacityHorizon_time p hi h0 hH T ht + have hu' : u.toNNReal ≤ slitCapacityHorizon p hi h0 hH S := by + rw [hHS] + exact Real.toNNReal_le_iff_le_coe.mpr hu.2 + have hd := slitCapacityForward_prefix_derivative p hi h0 hH hS hz hu' + rw [hHS,Real.coe_toNNReal u hu.1] at hd + have hG (v : ℝ) (hv : v∈Icc 0 (t:ℝ)) : + slitCapacityForward p hi h0 hH T v.toNNReal z= + slitCapacityForward p hi h0 hH S v.toNNReal z := by + rw [slitCapacityForward_restrict p hi h0 hH hST] + rw [hHS] + exact Real.toNNReal_le_iff_le_coe.mpr hv.2 + have hres := hd.congr hG (hG u hu) + rw [← hG u hu,← slitCapacityDriver_restrict p hi h0 hH hST hT hu'] at hres + exact hres + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitInverseAtInfinity.lean b/lean/OAI/Probability/SLE/Loewner/SlitInverseAtInfinity.lean index 8c93eaf4e3ce5c9c4264cc26f7b822cbd01a2ff7..677a1f1da6d2ceeee4e1159e7859d0e6ac0ded1f 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitInverseAtInfinity.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitInverseAtInfinity.lean @@ -1,82 +1,82 @@ -import OAI.Probability.SLE.Geometry.ReciprocalInverseLimit - -namespace OAI - -/-! Identification of the analytic inverse germ with the physical slit -inverse for all sufficiently large points of the half-plane. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -lemma slitHydrodynamicMap_conjugateInversion - {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) {f : ℂ → ℂ} (hf0 : f 0=0) - (hs : ∀ z ∈ D, f (conj z)=conj (f z)) (a : ℝ) - {x : ℂ} (hx : x ∈ D) (hx0 : x ≠ 0) : - slitHydrodynamicMap f a (conjugateInversion x)= - conjugateInversion (slitReciprocalMap f a x) := by - have hh := slitReciprocalMap_identity hf0 a (conjugateInversion_ne_zero hx0) - have hq := slitReciprocalMap_reflection hD hs a x hx - simp only [conjugateInversion,inv_inv] at hh - rw [hq] at hh - exact hh.symm - -theorem slit_inverse_tendsto_ratio - (p : C(unitInterval,ℂ)) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) - {f : ℂ → ℂ} {a : ℝ} (ha : 0 F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by - have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union - ((isCompact_range p.continuous).image Complex.continuous_conj) - have hD : invertedExterior (reflectedArcSet p) ∈ 𝓝 (0:ℂ) := - (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) - obtain ⟨r,ε,hε,hr,hr0,hrd,hupper,hl,hh⟩ := - slitReciprocalMap_exists_upper_inverse hD (hf.analyticAt hD) ha.ne' hfa hs - have hrT : Tendsto r (𝓝 (0:ℂ)) (𝓝 0) := by simpa only [hr0] using hr.continuousAt.tendsto - have hM := conjugateInversion_tendsto_zero - have hrD := (hrT.comp hM).eventually hD - have heq := hM.eventually hh - have hsmall := hM.eventually (ball_mem_nhds (0:ℂ) hε) - have hMb := conjugateInversion_slit_bijOn p h0 hH - have he : (fun z : ℂ => F z/z) =ᶠ[cocompact ℂ ⊓ 𝓟 upperHalfPlane] - (fun z => conjugateInversion (r (conjugateInversion z))/z) := by - change ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, - F z/z=conjugateInversion (r (conjugateInversion z))/z - rw [eventually_inf_principal] - filter_upwards [hrD,heq,hsmall] with z hzD hzq hzb - intro hzH - have hzD' : r (conjugateInversion z) ∈ invertedExterior (reflectedArcSet p) := by - simpa only [Function.comp_apply,invertedExterior,mem_ofPred_eq] using hzD - have hMz := conjugateInversion_upper hzH - have hrH := hupper ⟨hzb,hMz⟩ - have hrne : r (conjugateInversion z) ≠ 0 := by - intro hzero - have hi := congrArg Complex.im hzero - simp only [zero_im] at hi - exact hrH.ne' hi - have hx : conjugateInversion (r (conjugateInversion z)) ∈ upperHalfPlane \ range p := by - obtain ⟨v,hv,hvr⟩ := hMb.surjOn ⟨hzD',hrH⟩ - have he : v=conjugateInversion (r (conjugateInversion z)) := by - rw [← hvr,conjugateInversion_involutive] - exact he ▸ hv - have hG := slitHydrodynamicMap_conjugateInversion hD hf0 hs a hzD' hrne - rw [hzq,conjugateInversion_involutive] at hG - have hF := hleft _ hx - rw [hG] at hF - rw [hF] - have ht := (reciprocal_inverse_tendsto_ratio - (by simpa only [hrd] using hr.differentiableAt.hasDerivAt) hr0).mono_left - (show cocompact ℂ ⊓ 𝓟 upperHalfPlane ≤ cocompact ℂ from inf_le_left) - exact ht.congr' he.symm - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ReciprocalInverseLimit + +namespace OAI + +/-! Identification of the analytic inverse germ with the physical slit +inverse for all sufficiently large points of the half-plane. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +lemma slitHydrodynamicMap_conjugateInversion + {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) {f : ℂ → ℂ} (hf0 : f 0=0) + (hs : ∀ z ∈ D, f (conj z)=conj (f z)) (a : ℝ) + {x : ℂ} (hx : x ∈ D) (hx0 : x ≠ 0) : + slitHydrodynamicMap f a (conjugateInversion x)= + conjugateInversion (slitReciprocalMap f a x) := by + have hh := slitReciprocalMap_identity hf0 a (conjugateInversion_ne_zero hx0) + have hq := slitReciprocalMap_reflection hD hs a x hx + simp only [conjugateInversion,inv_inv] at hh + rw [hq] at hh + exact hh.symm + +theorem slit_inverse_tendsto_ratio + (p : C(unitInterval,ℂ)) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t ≠ 0 → p t ∈ upperHalfPlane) + {f : ℂ → ℂ} {a : ℝ} (ha : 0 F z/z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 1) := by + have hK : IsCompact (reflectedArcSet p) := (isCompact_range p.continuous).union + ((isCompact_range p.continuous).image Complex.continuous_conj) + have hD : invertedExterior (reflectedArcSet p) ∈ 𝓝 (0:ℂ) := + (invertedExterior_isOpen hK).mem_nhds (zero_mem_invertedExterior _) + obtain ⟨r,ε,hε,hr,hr0,hrd,hupper,hl,hh⟩ := + slitReciprocalMap_exists_upper_inverse hD (hf.analyticAt hD) ha.ne' hfa hs + have hrT : Tendsto r (𝓝 (0:ℂ)) (𝓝 0) := by simpa only [hr0] using hr.continuousAt.tendsto + have hM := conjugateInversion_tendsto_zero + have hrD := (hrT.comp hM).eventually hD + have heq := hM.eventually hh + have hsmall := hM.eventually (ball_mem_nhds (0:ℂ) hε) + have hMb := conjugateInversion_slit_bijOn p h0 hH + have he : (fun z : ℂ => F z/z) =ᶠ[cocompact ℂ ⊓ 𝓟 upperHalfPlane] + (fun z => conjugateInversion (r (conjugateInversion z))/z) := by + change ∀ᶠ z in cocompact ℂ ⊓ 𝓟 upperHalfPlane, + F z/z=conjugateInversion (r (conjugateInversion z))/z + rw [eventually_inf_principal] + filter_upwards [hrD,heq,hsmall] with z hzD hzq hzb + intro hzH + have hzD' : r (conjugateInversion z) ∈ invertedExterior (reflectedArcSet p) := by + simpa only [Function.comp_apply,invertedExterior,mem_ofPred_eq] using hzD + have hMz := conjugateInversion_upper hzH + have hrH := hupper ⟨hzb,hMz⟩ + have hrne : r (conjugateInversion z) ≠ 0 := by + intro hzero + have hi := congrArg Complex.im hzero + simp only [zero_im] at hi + exact hrH.ne' hi + have hx : conjugateInversion (r (conjugateInversion z)) ∈ upperHalfPlane \ range p := by + obtain ⟨v,hv,hvr⟩ := hMb.surjOn ⟨hzD',hrH⟩ + have he : v=conjugateInversion (r (conjugateInversion z)) := by + rw [← hvr,conjugateInversion_involutive] + exact he ▸ hv + have hG := slitHydrodynamicMap_conjugateInversion hD hf0 hs a hzD' hrne + rw [hzq,conjugateInversion_involutive] at hG + have hF := hleft _ hx + rw [hG] at hF + rw [hF] + have ht := (reciprocal_inverse_tendsto_ratio + (by simpa only [hrd] using hr.differentiableAt.hasDerivAt) hr0).mono_left + (show cocompact ℂ ⊓ 𝓟 upperHalfPlane ≤ cocompact ℂ from inf_le_left) + exact ht.congr' he.symm + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitInverseGerm.lean b/lean/OAI/Probability/SLE/Loewner/SlitInverseGerm.lean index 030f86bda545cf44733ce32a8f1dfbfe749e7aa4..3a2b3d1812486741911b3ab59fe4570660377d97 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitInverseGerm.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitInverseGerm.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Geometry.StrictRealLocalChart - -namespace OAI - -/-! The analytic inverse germ at infinity respects reflection and the -upper side. This is the local branch for the actual slit inverse. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -lemma local_inverse_reflection {q r : ℂ → ℂ} - (hr : ContinuousAt r 0) (hr0 : r 0=0) - (hs : ∀ᶠ z in 𝓝 (0:ℂ), q (conj z)=conj (q z)) - (hl : ∀ᶠ z in 𝓝 (0:ℂ), r (q z)=z) - (hh : ∀ᶠ z in 𝓝 (0:ℂ), q (r z)=z) : - ∀ᶠ z in 𝓝 (0:ℂ), r (conj z)=conj (r z) := by - have ht : Tendsto r (𝓝 (0:ℂ)) (𝓝 0) := by simpa only [hr0] using hr.tendsto - have hc : Tendsto (fun z => conj (r z)) (𝓝 (0:ℂ)) (𝓝 0) := by - convert! (Complex.continuous_conj.tendsto 0).comp ht using 1 - simp - filter_upwards [ht.eventually hs,hc.eventually hl,hh] with z hsz hlz hhz - rwa [hsz,hhz] at hlz - -theorem slitReciprocalMap_exists_upper_inverse - {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) {f : ℂ → ℂ} {a : ℝ} - (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) - (hs : ∀ z ∈ D, f (conj z)=conj (f z)) : - ∃ (r : ℂ → ℂ) (ε : ℝ), 0<ε ∧ AnalyticAt ℂ r 0 ∧ r 0=0 ∧ deriv r 0=1 ∧ - MapsTo r (ball 0 ε ∩ upperHalfPlane) upperHalfPlane ∧ - (∀ᶠ z in 𝓝 (0:ℂ), r (slitReciprocalMap f a z)=z) ∧ - (∀ᶠ z in 𝓝 (0:ℂ), slitReciprocalMap f a (r z)=z) := by - obtain ⟨r,hr,hr0,hrd,hl,hh⟩ := slitReciprocalMap_exists_local_inverse hf ha hd - have hsym : ∀ᶠ z in 𝓝 (0:ℂ), slitReciprocalMap f a (conj z)= - conj (slitReciprocalMap f a z) := by - filter_upwards [hD] with z hz - exact slitReciprocalMap_reflection hD hs a z hz - have hrs := local_inverse_reflection hr.continuousAt hr0 hsym hl hh - obtain ⟨S,hS,hsS⟩ := hrs.exists_mem - have hstrict : HasStrictDerivAt r 1 0 := by simpa only [hrd] using hr.hasStrictDerivAt - obtain ⟨ε,hε,hupper⟩ := strict_real_germ_maps_upper hS hstrict hsS - exact ⟨r,ε,hε,hr,hr0,hrd,hupper,hl,hh⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.StrictRealLocalChart + +namespace OAI + +/-! The analytic inverse germ at infinity respects reflection and the +upper side. This is the local branch for the actual slit inverse. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +lemma local_inverse_reflection {q r : ℂ → ℂ} + (hr : ContinuousAt r 0) (hr0 : r 0=0) + (hs : ∀ᶠ z in 𝓝 (0:ℂ), q (conj z)=conj (q z)) + (hl : ∀ᶠ z in 𝓝 (0:ℂ), r (q z)=z) + (hh : ∀ᶠ z in 𝓝 (0:ℂ), q (r z)=z) : + ∀ᶠ z in 𝓝 (0:ℂ), r (conj z)=conj (r z) := by + have ht : Tendsto r (𝓝 (0:ℂ)) (𝓝 0) := by simpa only [hr0] using hr.tendsto + have hc : Tendsto (fun z => conj (r z)) (𝓝 (0:ℂ)) (𝓝 0) := by + convert! (Complex.continuous_conj.tendsto 0).comp ht using 1 + simp + filter_upwards [ht.eventually hs,hc.eventually hl,hh] with z hsz hlz hhz + rwa [hsz,hhz] at hlz + +theorem slitReciprocalMap_exists_upper_inverse + {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) {f : ℂ → ℂ} {a : ℝ} + (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) + (hs : ∀ z ∈ D, f (conj z)=conj (f z)) : + ∃ (r : ℂ → ℂ) (ε : ℝ), 0<ε ∧ AnalyticAt ℂ r 0 ∧ r 0=0 ∧ deriv r 0=1 ∧ + MapsTo r (ball 0 ε ∩ upperHalfPlane) upperHalfPlane ∧ + (∀ᶠ z in 𝓝 (0:ℂ), r (slitReciprocalMap f a z)=z) ∧ + (∀ᶠ z in 𝓝 (0:ℂ), slitReciprocalMap f a (r z)=z) := by + obtain ⟨r,hr,hr0,hrd,hl,hh⟩ := slitReciprocalMap_exists_local_inverse hf ha hd + have hsym : ∀ᶠ z in 𝓝 (0:ℂ), slitReciprocalMap f a (conj z)= + conj (slitReciprocalMap f a z) := by + filter_upwards [hD] with z hz + exact slitReciprocalMap_reflection hD hs a z hz + have hrs := local_inverse_reflection hr.continuousAt hr0 hsym hl hh + obtain ⟨S,hS,hsS⟩ := hrs.exists_mem + have hstrict : HasStrictDerivAt r 1 0 := by simpa only [hrd] using hr.hasStrictDerivAt + obtain ⟨ε,hε,hupper⟩ := strict_real_germ_maps_upper hS hstrict hsS + exact ⟨r,ε,hε,hr,hr0,hrd,hupper,hl,hh⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitPhysicalClock.lean b/lean/OAI/Probability/SLE/Loewner/SlitPhysicalClock.lean index d27e3f3e502e517b25916a69162e60cddae83986..62c9c0513372eae5d983fd2bc9da218349601bea 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitPhysicalClock.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitPhysicalClock.lean @@ -1,98 +1,98 @@ -import OAI.Probability.SLE.Loewner.InitialDrivingTip -import OAI.Probability.SLE.MomentGauge - -namespace OAI - -/-! The exact capacity-two parametrization of a finite prescribed slit. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -noncomputable def slitCapacityHorizon (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) : ℝ≥0 := - ⟨initialCapacity p hi h0 hH T/2,div_nonneg (initialCapacity_nonneg p hi h0 hH T) (by norm_num)⟩ - -noncomputable def slitCapacityTime (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) (t : ℝ≥0) : unitInterval := - capacityParameterOnPrefix p hi h0 hH T (2*(t:ℝ)) - -lemma slitCapacityTime_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) : Continuous (slitCapacityTime p hi h0 hH T) := - (capacityParameterOnPrefix_continuous p hi h0 hH T).comp - (continuous_const.mul continuous_subtype_val) - -@[simp] lemma slitCapacityTime_zero (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) : slitCapacityTime p hi h0 hH T 0=0 := by - have hh := capacityParameterOnPrefix_at_initialCapacity p hi h0 hH T - (s := 0) (show (0:unitInterval) ≤ T from T.property.1) - simpa only [slitCapacityTime,NNReal.coe_zero,mul_zero,initialCapacity_zero] using hh - -lemma slitCapacityTime_le (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) (t : ℝ≥0) : slitCapacityTime p hi h0 hH T t ≤ T := - capacityParameterOnPrefix_le p hi h0 hH T _ - -lemma slitCapacityTime_capacity (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : - initialCapacity p hi h0 hH (slitCapacityTime p hi h0 hH T t)=2*(t:ℝ) := by - apply capacityParameterOnPrefix_capacity - constructor - · positivity - · change (t:ℝ) ≤ initialCapacity p hi h0 hH T/2 at ht - linarith - -lemma slitCapacityTime_at_capacity (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {v : unitInterval} (hv : v ≤ T) : - slitCapacityTime p hi h0 hH T - ⟨initialCapacity p hi h0 hH v/2, - div_nonneg (initialCapacity_nonneg p hi h0 hH v) (by norm_num)⟩=v := by - unfold slitCapacityTime - change capacityParameterOnPrefix p hi h0 hH T (2*(initialCapacity p hi h0 hH v/2))=v - rw [mul_div_cancel₀ _ (by norm_num : (2:ℝ)≠0)] - exact capacityParameterOnPrefix_at_initialCapacity p hi h0 hH T hv - -lemma slitCapacityTime_image (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : - slitCapacityTime p hi h0 hH T '' Icc 0 t=Icc 0 (slitCapacityTime p hi h0 hH T t) := by - ext v - constructor - · rintro ⟨u,hu,rfl⟩ - refine ⟨(slitCapacityTime p hi h0 hH T u).property.1,?_⟩ - apply (initialCapacity_strictMono p hi h0 hH).le_iff_le.mp - rw [slitCapacityTime_capacity p hi h0 hH T (hu.2.trans ht), - slitCapacityTime_capacity p hi h0 hH T ht] - exact mul_le_mul_of_nonneg_left (NNReal.coe_le_coe.mpr hu.2) (by norm_num) - · intro hv - let u : ℝ≥0 := ⟨initialCapacity p hi h0 hH v/2, - div_nonneg (initialCapacity_nonneg p hi h0 hH v) (by norm_num)⟩ - refine ⟨u,⟨zero_le,?_⟩,slitCapacityTime_at_capacity p hi h0 hH T - (hv.2.trans (slitCapacityTime_le p hi h0 hH T t))⟩ - have hh := initialCapacity_monotone p hi h0 hH hv.2 - rw [slitCapacityTime_capacity p hi h0 hH T ht] at hh - change initialCapacity p hi h0 hH v/2 ≤ (t:ℝ) - linarith - -noncomputable def slitCapacityDriver (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) : Target.Driver := by - refine ⟨⟨fun t => capacityDriverOnPrefix p hi h0 hH T hT (2*(t:ℝ)), - (capacityDriverOnPrefix_continuous p hi h0 hH hT).comp - (continuous_const.mul continuous_subtype_val)⟩,?_⟩ - change initialDrivingValue p hi h0 hH (slitCapacityTime p hi h0 hH T 0) - ((slitCapacityTime_le p hi h0 hH T 0).trans_lt hT)=0 - simp only [slitCapacityTime_zero,initialDrivingValue_zero] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.InitialDrivingTip +import OAI.Probability.SLE.MomentGauge + +namespace OAI + +/-! The exact capacity-two parametrization of a finite prescribed slit. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +noncomputable def slitCapacityHorizon (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) : ℝ≥0 := + ⟨initialCapacity p hi h0 hH T/2,div_nonneg (initialCapacity_nonneg p hi h0 hH T) (by norm_num)⟩ + +noncomputable def slitCapacityTime (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) (t : ℝ≥0) : unitInterval := + capacityParameterOnPrefix p hi h0 hH T (2*(t:ℝ)) + +lemma slitCapacityTime_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) : Continuous (slitCapacityTime p hi h0 hH T) := + (capacityParameterOnPrefix_continuous p hi h0 hH T).comp + (continuous_const.mul continuous_subtype_val) + +@[simp] lemma slitCapacityTime_zero (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) : slitCapacityTime p hi h0 hH T 0=0 := by + have hh := capacityParameterOnPrefix_at_initialCapacity p hi h0 hH T + (s := 0) (show (0:unitInterval) ≤ T from T.property.1) + simpa only [slitCapacityTime,NNReal.coe_zero,mul_zero,initialCapacity_zero] using hh + +lemma slitCapacityTime_le (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) (t : ℝ≥0) : slitCapacityTime p hi h0 hH T t ≤ T := + capacityParameterOnPrefix_le p hi h0 hH T _ + +lemma slitCapacityTime_capacity (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : + initialCapacity p hi h0 hH (slitCapacityTime p hi h0 hH T t)=2*(t:ℝ) := by + apply capacityParameterOnPrefix_capacity + constructor + · positivity + · change (t:ℝ) ≤ initialCapacity p hi h0 hH T/2 at ht + linarith + +lemma slitCapacityTime_at_capacity (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {v : unitInterval} (hv : v ≤ T) : + slitCapacityTime p hi h0 hH T + ⟨initialCapacity p hi h0 hH v/2, + div_nonneg (initialCapacity_nonneg p hi h0 hH v) (by norm_num)⟩=v := by + unfold slitCapacityTime + change capacityParameterOnPrefix p hi h0 hH T (2*(initialCapacity p hi h0 hH v/2))=v + rw [mul_div_cancel₀ _ (by norm_num : (2:ℝ)≠0)] + exact capacityParameterOnPrefix_at_initialCapacity p hi h0 hH T hv + +lemma slitCapacityTime_image (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : + slitCapacityTime p hi h0 hH T '' Icc 0 t=Icc 0 (slitCapacityTime p hi h0 hH T t) := by + ext v + constructor + · rintro ⟨u,hu,rfl⟩ + refine ⟨(slitCapacityTime p hi h0 hH T u).property.1,?_⟩ + apply (initialCapacity_strictMono p hi h0 hH).le_iff_le.mp + rw [slitCapacityTime_capacity p hi h0 hH T (hu.2.trans ht), + slitCapacityTime_capacity p hi h0 hH T ht] + exact mul_le_mul_of_nonneg_left (NNReal.coe_le_coe.mpr hu.2) (by norm_num) + · intro hv + let u : ℝ≥0 := ⟨initialCapacity p hi h0 hH v/2, + div_nonneg (initialCapacity_nonneg p hi h0 hH v) (by norm_num)⟩ + refine ⟨u,⟨zero_le,?_⟩,slitCapacityTime_at_capacity p hi h0 hH T + (hv.2.trans (slitCapacityTime_le p hi h0 hH T t))⟩ + have hh := initialCapacity_monotone p hi h0 hH hv.2 + rw [slitCapacityTime_capacity p hi h0 hH T ht] at hh + change initialCapacity p hi h0 hH v/2 ≤ (t:ℝ) + linarith + +noncomputable def slitCapacityDriver (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) : Target.Driver := by + refine ⟨⟨fun t => capacityDriverOnPrefix p hi h0 hH T hT (2*(t:ℝ)), + (capacityDriverOnPrefix_continuous p hi h0 hH hT).comp + (continuous_const.mul continuous_subtype_val)⟩,?_⟩ + change initialDrivingValue p hi h0 hH (slitCapacityTime p hi h0 hH T 0) + ((slitCapacityTime_le p hi h0 hH T 0).trans_lt hT)=0 + simp only [slitCapacityTime_zero,initialDrivingValue_zero] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitPhysicalEvolution.lean b/lean/OAI/Probability/SLE/Loewner/SlitPhysicalEvolution.lean index ab36f312772d5c357e698ececa106ef7cfc2acb4..ef4fd58bf31761191b88d481b4ec117ec8702ae6 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitPhysicalEvolution.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitPhysicalEvolution.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Loewner.SlitPhysicalClock - -namespace OAI - -/-! Exact capacity-two forward and inverse maps on a finite physical prefix. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -noncomputable def slitCapacityForward (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) (t : ℝ≥0) : ℂ → ℂ := - initialForwardChart p hi h0 hH (slitCapacityTime p hi h0 hH T t) - -noncomputable def slitCapacityInverse (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) (t : ℝ≥0) : ℂ → ℂ := - initialInverseChart p hi h0 hH (slitCapacityTime p hi h0 hH T t) - -theorem slitCapacityForward_prefix_derivative (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {z : ℂ} - (hz : z∈upperHalfPlane \ range (initialSubarc p T)) - {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : - HasDerivWithinAt (fun u : ℝ => slitCapacityForward p hi h0 hH T u.toNNReal z) - (2/(slitCapacityForward p hi h0 hH T t z-(slitCapacityDriver p hi h0 hH hT t:ℂ))) - (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) (t:ℝ) := by - have ht' : 2*(t:ℝ)∈Icc 0 (initialCapacity p hi h0 hH T) := by - constructor - · positivity - · change (t:ℝ) ≤ initialCapacity p hi h0 hH T/2 at ht - linarith - have hd := capacityForward_hasDerivWithinAt_prefix p hi h0 hH hT hz ht' - have hmul : HasDerivWithinAt (fun u : ℝ => 2*u) 2 - (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) (t:ℝ) := by - simpa using ((hasDerivAt_id (t:ℝ)).const_mul 2).hasDerivWithinAt - have hmaps : MapsTo (fun u : ℝ => 2*u) - (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) - (Icc 0 (initialCapacity p hi h0 hH T)) := by - intro u hu - constructor - · linarith [hu.1] - · have hu2 : u ≤ initialCapacity p hi h0 hH T/2 := hu.2 - linarith - have hh := hd.scomp (t:ℝ) hmul hmaps - have he (u : ℝ) (hu : 0 ≤ u) : - slitCapacityForward p hi h0 hH T u.toNNReal z= - capacityForwardOnPrefix p hi h0 hH T z (2*u) := by - simp only [slitCapacityForward,slitCapacityTime,Real.coe_toNNReal u hu, - capacityForwardOnPrefix,capacityParameterOnPrefix] - have hres := hh.congr (fun u hu => he u hu.1) (he t t.coe_nonneg) - convert hres using 1 - change 2/(capacityForwardOnPrefix p hi h0 hH T z (2*(t:ℝ))- - (capacityDriverOnPrefix p hi h0 hH T hT (2*(t:ℝ)):ℂ))=2 • _ - simp only [real_smul,Complex.ofReal_ofNat,mul_one_div] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitPhysicalClock + +namespace OAI + +/-! Exact capacity-two forward and inverse maps on a finite physical prefix. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +noncomputable def slitCapacityForward (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) (t : ℝ≥0) : ℂ → ℂ := + initialForwardChart p hi h0 hH (slitCapacityTime p hi h0 hH T t) + +noncomputable def slitCapacityInverse (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) (t : ℝ≥0) : ℂ → ℂ := + initialInverseChart p hi h0 hH (slitCapacityTime p hi h0 hH T t) + +theorem slitCapacityForward_prefix_derivative (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {z : ℂ} + (hz : z∈upperHalfPlane \ range (initialSubarc p T)) + {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : + HasDerivWithinAt (fun u : ℝ => slitCapacityForward p hi h0 hH T u.toNNReal z) + (2/(slitCapacityForward p hi h0 hH T t z-(slitCapacityDriver p hi h0 hH hT t:ℂ))) + (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) (t:ℝ) := by + have ht' : 2*(t:ℝ)∈Icc 0 (initialCapacity p hi h0 hH T) := by + constructor + · positivity + · change (t:ℝ) ≤ initialCapacity p hi h0 hH T/2 at ht + linarith + have hd := capacityForward_hasDerivWithinAt_prefix p hi h0 hH hT hz ht' + have hmul : HasDerivWithinAt (fun u : ℝ => 2*u) 2 + (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) (t:ℝ) := by + simpa using ((hasDerivAt_id (t:ℝ)).const_mul 2).hasDerivWithinAt + have hmaps : MapsTo (fun u : ℝ => 2*u) + (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) + (Icc 0 (initialCapacity p hi h0 hH T)) := by + intro u hu + constructor + · linarith [hu.1] + · have hu2 : u ≤ initialCapacity p hi h0 hH T/2 := hu.2 + linarith + have hh := hd.scomp (t:ℝ) hmul hmaps + have he (u : ℝ) (hu : 0 ≤ u) : + slitCapacityForward p hi h0 hH T u.toNNReal z= + capacityForwardOnPrefix p hi h0 hH T z (2*u) := by + simp only [slitCapacityForward,slitCapacityTime,Real.coe_toNNReal u hu, + capacityForwardOnPrefix,capacityParameterOnPrefix] + have hres := hh.congr (fun u hu => he u hu.1) (he t t.coe_nonneg) + convert hres using 1 + change 2/(capacityForwardOnPrefix p hi h0 hH T z (2*(t:ℝ))- + (capacityDriverOnPrefix p hi h0 hH T hT (2*(t:ℝ)):ℂ))=2 • _ + simp only [real_smul,Complex.ofReal_ofNat,mul_one_div] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitPhysicalTrace.lean b/lean/OAI/Probability/SLE/Loewner/SlitPhysicalTrace.lean index d657a41e6240ad600f7cd99c4e92c94351973d86..3bbbaf75ce7b03dfdb2af1858c02dadef9872f0f 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitPhysicalTrace.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitPhysicalTrace.lean @@ -1,74 +1,74 @@ -import OAI.Probability.SLE.Loewner.SlitPhysicalClock -import OAI.Probability.SLE.Geometry.FiniteSimpleSlit - -namespace OAI - -/-! The capacity-two curve has exactly the prescribed initial ranges -and no additional swallowed points. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -noncomputable def slitCapacityCurve (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) (t : ℝ≥0) : ℂ := p (slitCapacityTime p hi h0 hH T t) - -lemma slitCapacityCurve_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) : Continuous (slitCapacityCurve p hi h0 hH T) := - p.continuous.comp (slitCapacityTime_continuous p hi h0 hH T) - -@[simp] lemma slitCapacityCurve_zero (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) : slitCapacityCurve p hi h0 hH T 0=0 := by - simp only [slitCapacityCurve,slitCapacityTime_zero,h0] - -lemma slitCapacityCurve_initial_range (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : - slitCapacityCurve p hi h0 hH T '' Icc 0 t= - range (initialSubarc p (slitCapacityTime p hi h0 hH T t)) := by - rw [initialSubarc_range] - change (p ∘ slitCapacityTime p hi h0 hH T) '' Icc 0 t=p '' _ - rw [image_comp,slitCapacityTime_image p hi h0 hH T ht] - -lemma slitCapacityCurve_injOn (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) : - InjOn (slitCapacityCurve p hi h0 hH T) (Icc 0 (slitCapacityHorizon p hi h0 hH T)) := by - intro s hs t ht he - have hc := congrArg (initialCapacity p hi h0 hH) (hi he) - rw [slitCapacityTime_capacity p hi h0 hH T hs.2, - slitCapacityTime_capacity p hi h0 hH T ht.2] at hc - apply NNReal.coe_injective - linarith - -lemma slitCapacityCurve_positive (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {t : ℝ≥0} (ht : t∈Ioc 0 (slitCapacityHorizon p hi h0 hH T)) : - slitCapacityCurve p hi h0 hH T t∈upperHalfPlane := by - apply hH - intro he - have hh := slitCapacityTime_capacity p hi h0 hH T ht.2 - rw [he,initialCapacity_zero] at hh - have hp : (0:ℝ) slitCapacityCurve_positive p hi h0 hH T ⟨hu.1,hu.2.trans ht⟩) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitPhysicalClock +import OAI.Probability.SLE.Geometry.FiniteSimpleSlit + +namespace OAI + +/-! The capacity-two curve has exactly the prescribed initial ranges +and no additional swallowed points. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +noncomputable def slitCapacityCurve (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) (t : ℝ≥0) : ℂ := p (slitCapacityTime p hi h0 hH T t) + +lemma slitCapacityCurve_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) : Continuous (slitCapacityCurve p hi h0 hH T) := + p.continuous.comp (slitCapacityTime_continuous p hi h0 hH T) + +@[simp] lemma slitCapacityCurve_zero (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) : slitCapacityCurve p hi h0 hH T 0=0 := by + simp only [slitCapacityCurve,slitCapacityTime_zero,h0] + +lemma slitCapacityCurve_initial_range (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {t : ℝ≥0} (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : + slitCapacityCurve p hi h0 hH T '' Icc 0 t= + range (initialSubarc p (slitCapacityTime p hi h0 hH T t)) := by + rw [initialSubarc_range] + change (p ∘ slitCapacityTime p hi h0 hH T) '' Icc 0 t=p '' _ + rw [image_comp,slitCapacityTime_image p hi h0 hH T ht] + +lemma slitCapacityCurve_injOn (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) : + InjOn (slitCapacityCurve p hi h0 hH T) (Icc 0 (slitCapacityHorizon p hi h0 hH T)) := by + intro s hs t ht he + have hc := congrArg (initialCapacity p hi h0 hH) (hi he) + rw [slitCapacityTime_capacity p hi h0 hH T hs.2, + slitCapacityTime_capacity p hi h0 hH T ht.2] at hc + apply NNReal.coe_injective + linarith + +lemma slitCapacityCurve_positive (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {t : ℝ≥0} (ht : t∈Ioc 0 (slitCapacityHorizon p hi h0 hH T)) : + slitCapacityCurve p hi h0 hH T t∈upperHalfPlane := by + apply hH + intro he + have hh := slitCapacityTime_capacity p hi h0 hH T ht.2 + rw [he,initialCapacity_zero] at hh + have hp : (0:ℝ) slitCapacityCurve_positive p hi h0 hH T ⟨hu.1,hu.2.trans ht⟩) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitRadialLimits.lean b/lean/OAI/Probability/SLE/Loewner/SlitRadialLimits.lean index 5c0b0a88ec1889af93a5bb13216311be01ba7bfe..3ea1f0a8db95557377ca8a3b1c944256aad242d7 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitRadialLimits.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitRadialLimits.lean @@ -1,60 +1,60 @@ -import OAI.Probability.SLE.Loewner.SlitExtendedTrace - -namespace OAI - -/-! Radial inverse limits for the entire extended slit, including the -joining time and every time in its constant-driver tail. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem Target.inverseFlow_radial_constant_tail (b : Driver) (A S : ℝ≥0) - (hS : 0 inverseFlow 1 b (A+S) ((b (A+S):ℂ)+(y:ℂ)*I)) - (𝓝[>] (0:ℝ)) (𝓝 (trace 1 b (A+S))) := by - have hd : ∀ t∈Icc 0 (S:ℝ), HasDerivWithinAt (driverValue (linearDriver 0)) 0 - (Icc 0 (S:ℝ)) t := fun t ht => (linearDriver_derivative 0 t ht.1).mono Icc_subset_Ici_self - have hl := tendsto_inverseFlow_smooth_trace (linearDriver 0) S (fun _ => 0) hd - (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) - have hpos := smooth_raw_trace_mem_upperHalfPlane (linearDriver 0) S hS (fun _ => 0) hd - (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) - have hq : (u:ℂ)+trace 1 (linearDriver 0) S∈upperHalfPlane := by - simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hpos - have hc := (continuousOn_inverseFlow 1 (by norm_num) b A).continuousAt - (isOpen_upperHalfPlane.mem_nhds hq) - rw [trace_constant_tail b A S hS u hb] - apply (hc.tendsto.comp (tendsto_const_nhds.add hl)).congr' - filter_upwards [self_mem_nhdsWithin] with y hy - have he := hb (A+S) (le_add_of_nonneg_right zero_le) - have hw : (u:ℂ)+(y:ℂ)*I∈upperHalfPlane := by simpa [upperHalfPlane] using hy - rw [he,inverseFlow_constant_tail b A S u hb hw] - simp only [linearDriver_apply,zero_mul,ofReal_zero,zero_add,add_sub_cancel_left,Function.comp_def] - -theorem slitCapacity_radial_limit (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) (t : ℝ≥0) : - Tendsto (fun y : ℝ => Target.inverseFlow 1 (slitCapacityDriver p hi h0 hH hT) t - ((slitCapacityDriver p hi h0 hH hT t:ℂ)+(y:ℂ)*I)) (𝓝[>] (0:ℝ)) - (𝓝 (Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t)) := by - let A := slitCapacityHorizon p hi h0 hH T - by_cases ht : t ≤ A - · rw [slitCapacity_raw_trace_eq p hi h0 hH hT ht] - have hl := initialInverse_radial_driver_limit p hi h0 hH - ((slitCapacityTime_le p hi h0 hH T t).trans_lt hT) - apply hl.congr' - filter_upwards [self_mem_nhdsWithin] with y hy - rw [slitCapacityInverse_eq_raw p hi h0 hH hT ht (by simpa [upperHalfPlane] using hy)] - rfl - · have hAt : A slitCapacityDriver_after p hi h0 hH hT hs) - simpa only [←NNReal.coe_add,add_tsub_cancel_of_le hAt.le] using hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitExtendedTrace + +namespace OAI + +/-! Radial inverse limits for the entire extended slit, including the +joining time and every time in its constant-driver tail. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem Target.inverseFlow_radial_constant_tail (b : Driver) (A S : ℝ≥0) + (hS : 0 inverseFlow 1 b (A+S) ((b (A+S):ℂ)+(y:ℂ)*I)) + (𝓝[>] (0:ℝ)) (𝓝 (trace 1 b (A+S))) := by + have hd : ∀ t∈Icc 0 (S:ℝ), HasDerivWithinAt (driverValue (linearDriver 0)) 0 + (Icc 0 (S:ℝ)) t := fun t ht => (linearDriver_derivative 0 t ht.1).mono Icc_subset_Ici_self + have hl := tendsto_inverseFlow_smooth_trace (linearDriver 0) S (fun _ => 0) hd + (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) + have hpos := smooth_raw_trace_mem_upperHalfPlane (linearDriver 0) S hS (fun _ => 0) hd + (show (0:ℝ) ≤ 0 from le_rfl) (fun _ _ => by simp) + have hq : (u:ℂ)+trace 1 (linearDriver 0) S∈upperHalfPlane := by + simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hpos + have hc := (continuousOn_inverseFlow 1 (by norm_num) b A).continuousAt + (isOpen_upperHalfPlane.mem_nhds hq) + rw [trace_constant_tail b A S hS u hb] + apply (hc.tendsto.comp (tendsto_const_nhds.add hl)).congr' + filter_upwards [self_mem_nhdsWithin] with y hy + have he := hb (A+S) (le_add_of_nonneg_right zero_le) + have hw : (u:ℂ)+(y:ℂ)*I∈upperHalfPlane := by simpa [upperHalfPlane] using hy + rw [he,inverseFlow_constant_tail b A S u hb hw] + simp only [linearDriver_apply,zero_mul,ofReal_zero,zero_add,add_sub_cancel_left,Function.comp_def] + +theorem slitCapacity_radial_limit (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) (t : ℝ≥0) : + Tendsto (fun y : ℝ => Target.inverseFlow 1 (slitCapacityDriver p hi h0 hH hT) t + ((slitCapacityDriver p hi h0 hH hT t:ℂ)+(y:ℂ)*I)) (𝓝[>] (0:ℝ)) + (𝓝 (Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t)) := by + let A := slitCapacityHorizon p hi h0 hH T + by_cases ht : t ≤ A + · rw [slitCapacity_raw_trace_eq p hi h0 hH hT ht] + have hl := initialInverse_radial_driver_limit p hi h0 hH + ((slitCapacityTime_le p hi h0 hH T t).trans_lt hT) + apply hl.congr' + filter_upwards [self_mem_nhdsWithin] with y hy + rw [slitCapacityInverse_eq_raw p hi h0 hH hT ht (by simpa [upperHalfPlane] using hy)] + rfl + · have hAt : A slitCapacityDriver_after p hi h0 hH hT hs) + simpa only [←NNReal.coe_add,add_tsub_cancel_of_le hAt.le] using hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitRawCapacity.lean b/lean/OAI/Probability/SLE/Loewner/SlitRawCapacity.lean index 558f8b909e771cd3b0d09c4c0919594b6a409df1..2e3e47a8d0aafd52b6637c5e899c3db343a0c171 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitRawCapacity.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitRawCapacity.lean @@ -1,67 +1,67 @@ -import OAI.Probability.SLE.Loewner.SlitTailDomain -import OAI.Probability.SLE.Loewner.RawForwardInfinity - -namespace OAI - -/-! Every prescribed simple arc prefix, extended by a vertical image, -has an actual global capacity-two witness with its exact raw maps. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem RegularSLEPresentation.slit_raw_capacity_witness - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) - (p : C(unitInterval,ℂ)) (hi : Function.Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT0 : 0 Target.trace 1 b s) - (fun t z => Target.flow 1 b z t) (fun t => Target.inverseFlow 1 b t) := by - dsimp only - let b := slitCapacityDriver p hi h0 hH hT - have hdom := a.slitCapacity_raw_domains hκ p hi h0 hH hT0 hT - obtain ⟨hc,hz,_,hpos,_⟩ := a.slitCapacity_raw_geometry hκ p hi h0 hH hT0 hT - refine ⟨hc,hz,?_,?_,a.raw_inverse_holomorphic hκ b,?_,?_,?_,?_,?_,?_,?_,?_⟩ - · intro t - by_cases ht : 0 Target.flow 1 b z (u.toNNReal:ℝ)) - =ᶠ[𝓝[Ici 0] (t:ℝ)] Target.flow 1 b z := by - filter_upwards [self_mem_nhdsWithin] with u hu - rw [Real.coe_toNNReal u hu] - have hh := hd.congr_of_eventuallyEq he (by simp) - simpa [Target.driverValue] using hh - · intro t - simpa only [b,hdom t] using Target.flow_hydrodynamic_normalization b t - · exact slitCapacity_radial_limit p hi h0 hH hT - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitTailDomain +import OAI.Probability.SLE.Loewner.RawForwardInfinity + +namespace OAI + +/-! Every prescribed simple arc prefix, extended by a vertical image, +has an actual global capacity-two witness with its exact raw maps. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem RegularSLEPresentation.slit_raw_capacity_witness + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) + (p : C(unitInterval,ℂ)) (hi : Function.Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT0 : 0 Target.trace 1 b s) + (fun t z => Target.flow 1 b z t) (fun t => Target.inverseFlow 1 b t) := by + dsimp only + let b := slitCapacityDriver p hi h0 hH hT + have hdom := a.slitCapacity_raw_domains hκ p hi h0 hH hT0 hT + obtain ⟨hc,hz,_,hpos,_⟩ := a.slitCapacity_raw_geometry hκ p hi h0 hH hT0 hT + refine ⟨hc,hz,?_,?_,a.raw_inverse_holomorphic hκ b,?_,?_,?_,?_,?_,?_,?_,?_⟩ + · intro t + by_cases ht : 0 Target.flow 1 b z (u.toNNReal:ℝ)) + =ᶠ[𝓝[Ici 0] (t:ℝ)] Target.flow 1 b z := by + filter_upwards [self_mem_nhdsWithin] with u hu + rw [Real.coe_toNNReal u hu] + have hh := hd.congr_of_eventuallyEq he (by simp) + simpa [Target.driverValue] using hh + · intro t + simpa only [b,hdom t] using Target.flow_hydrodynamic_normalization b t + · exact slitCapacity_radial_limit p hi h0 hH hT + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitRawForward.lean b/lean/OAI/Probability/SLE/Loewner/SlitRawForward.lean index 757b21496e8efc47c1289eaaead4329c79a4a511..8df6b93c9ee2655344695873c7eb9be12934fa4e 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitRawForward.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitRawForward.lean @@ -1,50 +1,50 @@ -import OAI.Probability.SLE.Loewner.SlitInitialEvolution -import OAI.Probability.SLE.Angle.TargetPositiveFlow - -namespace OAI - -/-! The raw positive Loewner trajectory agrees exactly with the -constructed conformal slit evolution on every surviving prefix. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem slitCapacityForward_eq_raw (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {t : ℝ≥0} - (ht : t ≤ slitCapacityHorizon p hi h0 hH T) {z : ℂ} - (hz : z∈upperHalfPlane \ range (initialSubarc p (slitCapacityTime p hi h0 hH T t))) : - Target.Surviving 1 (slitCapacityDriver p hi h0 hH hT) z t ∧ - Target.flow 1 (slitCapacityDriver p hi h0 hH hT) z t= - slitCapacityForward p hi h0 hH T t z := by - let Y := fun u : ℝ => slitCapacityForward p hi h0 hH T u.toNNReal z - have hd := fun u (hu : u∈Icc 0 (t:ℝ)) => - slitCapacityForward_derivative_upto p hi h0 hH hT ht hz hu - have hc : ContinuousOn Y (Icc 0 (t:ℝ)) := fun u hu => (hd u hu).continuousWithinAt - have hpos (u : ℝ) (hu : u∈Icc 0 (t:ℝ)) : 0<(Y u).im := by - have hu' : u.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hu.2 - have hclock : slitCapacityTime p hi h0 hH T u.toNNReal ≤ slitCapacityTime p hi h0 hH T t := by - apply (initialCapacity_strictMono p hi h0 hH).le_iff_le.mp - rw [slitCapacityTime_capacity p hi h0 hH T (hu'.trans ht), - slitCapacityTime_capacity p hi h0 hH T ht,Real.coe_toNNReal u hu.1] - linarith [hu.2] - exact initialForwardChart_im_pos p hi h0 hH _ - ⟨hz.1,fun he => hz.2 (initialSubarc_range_mono p hclock he)⟩ - obtain ⟨v,hv,hmin⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr t.coe_nonneg) - (Complex.continuous_im.comp_continuousOn hc) - have hY0 : Y 0=z := by - simp only [Y,Real.toNNReal_zero,slitCapacityForward,slitCapacityTime_zero, - initialForwardChart_zero,id_eq] - have hh := Target.flow_eq_of_positive_trajectory 1 (slitCapacityDriver p hi h0 hH hT) z t - t.coe_nonneg Y hc hY0 (Y v).im (hpos v hv) (fun u hu => hmin hu) (fun u hu => ?_) - · simpa only [Y,Real.toNNReal_coe] using hh - · have he := hasDerivWithinAt_Ici_of_Icc hu (hd u (Ico_subset_Icc_self hu)) - simpa only [Target.a,div_one,ofReal_ofNat,Target.driverValue,Y] using he - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitInitialEvolution +import OAI.Probability.SLE.Angle.TargetPositiveFlow + +namespace OAI + +/-! The raw positive Loewner trajectory agrees exactly with the +constructed conformal slit evolution on every surviving prefix. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem slitCapacityForward_eq_raw (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {t : ℝ≥0} + (ht : t ≤ slitCapacityHorizon p hi h0 hH T) {z : ℂ} + (hz : z∈upperHalfPlane \ range (initialSubarc p (slitCapacityTime p hi h0 hH T t))) : + Target.Surviving 1 (slitCapacityDriver p hi h0 hH hT) z t ∧ + Target.flow 1 (slitCapacityDriver p hi h0 hH hT) z t= + slitCapacityForward p hi h0 hH T t z := by + let Y := fun u : ℝ => slitCapacityForward p hi h0 hH T u.toNNReal z + have hd := fun u (hu : u∈Icc 0 (t:ℝ)) => + slitCapacityForward_derivative_upto p hi h0 hH hT ht hz hu + have hc : ContinuousOn Y (Icc 0 (t:ℝ)) := fun u hu => (hd u hu).continuousWithinAt + have hpos (u : ℝ) (hu : u∈Icc 0 (t:ℝ)) : 0<(Y u).im := by + have hu' : u.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hu.2 + have hclock : slitCapacityTime p hi h0 hH T u.toNNReal ≤ slitCapacityTime p hi h0 hH T t := by + apply (initialCapacity_strictMono p hi h0 hH).le_iff_le.mp + rw [slitCapacityTime_capacity p hi h0 hH T (hu'.trans ht), + slitCapacityTime_capacity p hi h0 hH T ht,Real.coe_toNNReal u hu.1] + linarith [hu.2] + exact initialForwardChart_im_pos p hi h0 hH _ + ⟨hz.1,fun he => hz.2 (initialSubarc_range_mono p hclock he)⟩ + obtain ⟨v,hv,hmin⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr t.coe_nonneg) + (Complex.continuous_im.comp_continuousOn hc) + have hY0 : Y 0=z := by + simp only [Y,Real.toNNReal_zero,slitCapacityForward,slitCapacityTime_zero, + initialForwardChart_zero,id_eq] + have hh := Target.flow_eq_of_positive_trajectory 1 (slitCapacityDriver p hi h0 hH hT) z t + t.coe_nonneg Y hc hY0 (Y v).im (hpos v hv) (fun u hu => hmin hu) (fun u hu => ?_) + · simpa only [Y,Real.toNNReal_coe] using hh + · have he := hasDerivWithinAt_Ici_of_Icc hu (hd u (Ico_subset_Icc_self hu)) + simpa only [Target.a,div_one,ofReal_ofNat,Target.driverValue,Y] using he + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitRawInverse.lean b/lean/OAI/Probability/SLE/Loewner/SlitRawInverse.lean index fe342309c6818875809f2106fbca7544d4b5ed06..94fb7fa5497acf4be57dc85aad9646d5b47c731b 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitRawInverse.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitRawInverse.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Loewner.SlitRawForward -import OAI.Probability.SLE.Loewner.TargetInverseProperties - -namespace OAI - -/-! Exact raw inverse and surviving set for the constructed finite slit. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem slitCapacityInverse_eq_raw (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {t : ℝ≥0} - (ht : t ≤ slitCapacityHorizon p hi h0 hH T) {w : ℂ} (hw : w∈upperHalfPlane) : - Target.inverseFlow 1 (slitCapacityDriver p hi h0 hH hT) t w= - slitCapacityInverse p hi h0 hH T t w := by - have hz := initialInverseChart_mem_domain p hi h0 hH (slitCapacityTime p hi h0 hH T t) hw - obtain ⟨hs,he⟩ := slitCapacityForward_eq_raw p hi h0 hH hT ht hz - apply Target.inverseFlow_eq_of_surviving 1 _ t t.coe_nonneg _ w hs - exact he.trans (initialForward_inverse p hi h0 hH _ hw) - -theorem slitCapacity_surviving_eq (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {t : ℝ≥0} - (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : - {z | Target.Surviving 1 (slitCapacityDriver p hi h0 hH hT) z t}= - upperHalfPlane \ range (initialSubarc p (slitCapacityTime p hi h0 hH T t)) := by - ext z - constructor - · intro hz - have hw := Target.surviving_flow_mem_upperHalfPlane 1 _ z t t.coe_nonneg hz - have he := Target.inverseFlow_eq_of_surviving 1 (slitCapacityDriver p hi h0 hH hT) - t t.coe_nonneg z _ hz rfl - rw [slitCapacityInverse_eq_raw p hi h0 hH hT ht hw] at he - rw [← he] - exact initialInverseChart_mem_domain p hi h0 hH _ hw - · intro hz - exact (slitCapacityForward_eq_raw p hi h0 hH hT ht hz).1 - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitRawForward +import OAI.Probability.SLE.Loewner.TargetInverseProperties + +namespace OAI + +/-! Exact raw inverse and surviving set for the constructed finite slit. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem slitCapacityInverse_eq_raw (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {t : ℝ≥0} + (ht : t ≤ slitCapacityHorizon p hi h0 hH T) {w : ℂ} (hw : w∈upperHalfPlane) : + Target.inverseFlow 1 (slitCapacityDriver p hi h0 hH hT) t w= + slitCapacityInverse p hi h0 hH T t w := by + have hz := initialInverseChart_mem_domain p hi h0 hH (slitCapacityTime p hi h0 hH T t) hw + obtain ⟨hs,he⟩ := slitCapacityForward_eq_raw p hi h0 hH hT ht hz + apply Target.inverseFlow_eq_of_surviving 1 _ t t.coe_nonneg _ w hs + exact he.trans (initialForward_inverse p hi h0 hH _ hw) + +theorem slitCapacity_surviving_eq (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {t : ℝ≥0} + (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : + {z | Target.Surviving 1 (slitCapacityDriver p hi h0 hH hT) z t}= + upperHalfPlane \ range (initialSubarc p (slitCapacityTime p hi h0 hH T t)) := by + ext z + constructor + · intro hz + have hw := Target.surviving_flow_mem_upperHalfPlane 1 _ z t t.coe_nonneg hz + have he := Target.inverseFlow_eq_of_surviving 1 (slitCapacityDriver p hi h0 hH hT) + t t.coe_nonneg z _ hz rfl + rw [slitCapacityInverse_eq_raw p hi h0 hH hT ht hw] at he + rw [← he] + exact initialInverseChart_mem_domain p hi h0 hH _ hw + · intro hz + exact (slitCapacityForward_eq_raw p hi h0 hH hT ht hz).1 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitRawTrace.lean b/lean/OAI/Probability/SLE/Loewner/SlitRawTrace.lean index cb961f78fd8f9ee473c55a043bb24ba4a99a4688..be5644a2d42dc0f4720056a07578201bc9759bc5 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitRawTrace.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitRawTrace.lean @@ -1,39 +1,39 @@ -import OAI.Probability.SLE.Loewner.SlitRawInverse -import OAI.Probability.SLE.Loewner.SlitPhysicalTrace - -namespace OAI - -/-! The exact raw trace of the continuous based driver is the prescribed -simple slit throughout its capacity-two prefix. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem slitCapacity_raw_trace_eq (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {t : ℝ≥0} - (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : - Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t= - slitCapacityCurve p hi h0 hH T t := by - let s := slitCapacityTime p hi h0 hH T t - have hs : s<1 := (slitCapacityTime_le p hi h0 hH T t).trans_lt hT - have htip := initialInverse_radial_driver_limit p hi h0 hH hs - have he : slitCapacityDriver p hi h0 hH hT t=initialDrivingValue p hi h0 hH s hs := rfl - have hr : Tendsto (fun y : ℝ => Target.inverseFlow 1 (slitCapacityDriver p hi h0 hH hT) t - ((slitCapacityDriver p hi h0 hH hT t:ℂ)+(y:ℂ)*I)) (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by - apply htip.congr' - filter_upwards [self_mem_nhdsWithin] with y hy - have hw : (slitCapacityDriver p hi h0 hH hT t:ℂ)+(y:ℂ)*I∈upperHalfPlane := by - simpa [upperHalfPlane] using hy - rw [slitCapacityInverse_eq_raw p hi h0 hH hT ht hw] - rfl - simpa only [Target.trace,one_mul,Target.driverValue,Real.toNNReal_coe,slitCapacityCurve,s] using - hr.limUnder_eq - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitRawInverse +import OAI.Probability.SLE.Loewner.SlitPhysicalTrace + +namespace OAI + +/-! The exact raw trace of the continuous based driver is the prescribed +simple slit throughout its capacity-two prefix. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem slitCapacity_raw_trace_eq (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {t : ℝ≥0} + (ht : t ≤ slitCapacityHorizon p hi h0 hH T) : + Target.trace 1 (slitCapacityDriver p hi h0 hH hT) t= + slitCapacityCurve p hi h0 hH T t := by + let s := slitCapacityTime p hi h0 hH T t + have hs : s<1 := (slitCapacityTime_le p hi h0 hH T t).trans_lt hT + have htip := initialInverse_radial_driver_limit p hi h0 hH hs + have he : slitCapacityDriver p hi h0 hH hT t=initialDrivingValue p hi h0 hH s hs := rfl + have hr : Tendsto (fun y : ℝ => Target.inverseFlow 1 (slitCapacityDriver p hi h0 hH hT) t + ((slitCapacityDriver p hi h0 hH hT t:ℂ)+(y:ℂ)*I)) (𝓝[>] (0:ℝ)) (𝓝 (p s)) := by + apply htip.congr' + filter_upwards [self_mem_nhdsWithin] with y hy + have hw : (slitCapacityDriver p hi h0 hH hT t:ℂ)+(y:ℂ)*I∈upperHalfPlane := by + simpa [upperHalfPlane] using hy + rw [slitCapacityInverse_eq_raw p hi h0 hH hT ht hw] + rfl + simpa only [Target.trace,one_mul,Target.driverValue,Real.toNNReal_coe,slitCapacityCurve,s] using + hr.limUnder_eq + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitRealBoundaryFlow.lean b/lean/OAI/Probability/SLE/Loewner/SlitRealBoundaryFlow.lean index e047d5ee21941cb5d81d6a0a6feb6dbd0e10bd03..02308d8c21d718fc1ace40e0665caf1540e9247d 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitRealBoundaryFlow.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitRealBoundaryFlow.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Geometry.ReflectedRealSeparation -import OAI.Probability.SLE.Loewner.SlitPhysicalClock - -namespace OAI - -/-! The exact capacity-clock real boundary coordinates are continuous, -real valued and uniformly separated from the driver on a finite prefix. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology NNReal - -namespace SLEExactGauge - -noncomputable def slitReflectedFlow (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) (z : ℂ) (t : ℝ) : ℂ := - initialReflectedForward p hi h0 hH (slitCapacityTime p hi h0 hH T t.toNNReal) z - -lemma slitReflectedFlow_real_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {x : ℝ} (hx : x≠0) : - Continuous (slitReflectedFlow p hi h0 hH T (x:ℂ)) := - (initialReflectedForward_real_continuous p hi h0 hH hx).comp - ((slitCapacityTime_continuous p hi h0 hH T).comp continuous_real_toNNReal) - -lemma slitReflectedFlow_real (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {x : ℝ} (hx : x≠0) (t : ℝ) : - (slitReflectedFlow p hi h0 hH T (x:ℂ) t).im=0 := - initialReflectedForward_real p hi h0 hH _ hx - -lemma slitReflectedFlow_separated (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) (t : ℝ) : - slitReflectedFlow p hi h0 hH T (x:ℂ) t- - (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)≠0 := by - apply sub_ne_zero.mpr - exact initialReflectedForward_ne_driver p hi h0 hH - ((slitCapacityTime_le p hi h0 hH T t.toNNReal).trans_lt hT) hx - -theorem slitReflectedFlow_uniform_limit (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {x : ℝ} (hx : x≠0) : - TendstoUniformly (fun y : ℝ => slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I)) - (slitReflectedFlow p hi h0 hH T (x:ℂ)) (𝓝[>] (0:ℝ)) := by - have hv : Tendsto (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (x:ℂ)) := by - have hc : Continuous (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) := by fun_prop - simpa only [ofReal_zero,zero_mul,add_zero] using - (hc.tendsto 0).mono_left nhdsWithin_le_nhds - apply Metric.tendstoUniformly_iff.mpr - intro ε hε - have he := Metric.equicontinuousAt_iff_right.mp - (initialReflectedForward_equicontinuousAt_real p hi h0 hH hx) ε hε - filter_upwards [hv.eventually he] with y hy t - exact hy (slitCapacityTime p hi h0 hH T t.toNNReal) - -lemma slitReflectedFlow_positive_gap (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) : - ∃ m : ℝ, 0 < m ∧ ∀ t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ), - m ≤ ‖slitReflectedFlow p hi h0 hH T (x:ℂ) t- - (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)‖ := by - have hc : Continuous (fun t : ℝ => ‖slitReflectedFlow p hi h0 hH T (x:ℂ) t- - (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)‖) := - ((slitReflectedFlow_real_continuous p hi h0 hH T hx).sub - (Complex.continuous_ofReal.comp - ((slitCapacityDriver p hi h0 hH hT).continuous.comp continuous_real_toNNReal))).norm - obtain ⟨t,ht,hmin⟩ := isCompact_Icc.exists_isMinOn - (nonempty_Icc.mpr (slitCapacityHorizon p hi h0 hH T).coe_nonneg) hc.continuousOn - exact ⟨_,norm_pos_iff.mpr (slitReflectedFlow_separated p hi h0 hH hT hx t), - fun u hu => hmin hu⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ReflectedRealSeparation +import OAI.Probability.SLE.Loewner.SlitPhysicalClock + +namespace OAI + +/-! The exact capacity-clock real boundary coordinates are continuous, +real valued and uniformly separated from the driver on a finite prefix. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology NNReal + +namespace SLEExactGauge + +noncomputable def slitReflectedFlow (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) (z : ℂ) (t : ℝ) : ℂ := + initialReflectedForward p hi h0 hH (slitCapacityTime p hi h0 hH T t.toNNReal) z + +lemma slitReflectedFlow_real_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {x : ℝ} (hx : x≠0) : + Continuous (slitReflectedFlow p hi h0 hH T (x:ℂ)) := + (initialReflectedForward_real_continuous p hi h0 hH hx).comp + ((slitCapacityTime_continuous p hi h0 hH T).comp continuous_real_toNNReal) + +lemma slitReflectedFlow_real (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {x : ℝ} (hx : x≠0) (t : ℝ) : + (slitReflectedFlow p hi h0 hH T (x:ℂ) t).im=0 := + initialReflectedForward_real p hi h0 hH _ hx + +lemma slitReflectedFlow_separated (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) (t : ℝ) : + slitReflectedFlow p hi h0 hH T (x:ℂ) t- + (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)≠0 := by + apply sub_ne_zero.mpr + exact initialReflectedForward_ne_driver p hi h0 hH + ((slitCapacityTime_le p hi h0 hH T t.toNNReal).trans_lt hT) hx + +theorem slitReflectedFlow_uniform_limit (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {x : ℝ} (hx : x≠0) : + TendstoUniformly (fun y : ℝ => slitReflectedFlow p hi h0 hH T ((x:ℂ)+(y:ℂ)*I)) + (slitReflectedFlow p hi h0 hH T (x:ℂ)) (𝓝[>] (0:ℝ)) := by + have hv : Tendsto (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) (𝓝[>] (0:ℝ)) (𝓝 (x:ℂ)) := by + have hc : Continuous (fun y : ℝ => (x:ℂ)+(y:ℂ)*I) := by fun_prop + simpa only [ofReal_zero,zero_mul,add_zero] using + (hc.tendsto 0).mono_left nhdsWithin_le_nhds + apply Metric.tendstoUniformly_iff.mpr + intro ε hε + have he := Metric.equicontinuousAt_iff_right.mp + (initialReflectedForward_equicontinuousAt_real p hi h0 hH hx) ε hε + filter_upwards [hv.eventually he] with y hy t + exact hy (slitCapacityTime p hi h0 hH T t.toNNReal) + +lemma slitReflectedFlow_positive_gap (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {x : ℝ} (hx : x≠0) : + ∃ m : ℝ, 0 < m ∧ ∀ t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ), + m ≤ ‖slitReflectedFlow p hi h0 hH T (x:ℂ) t- + (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)‖ := by + have hc : Continuous (fun t : ℝ => ‖slitReflectedFlow p hi h0 hH T (x:ℂ) t- + (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)‖) := + ((slitReflectedFlow_real_continuous p hi h0 hH T hx).sub + (Complex.continuous_ofReal.comp + ((slitCapacityDriver p hi h0 hH hT).continuous.comp continuous_real_toNNReal))).norm + obtain ⟨t,ht,hmin⟩ := isCompact_Icc.exists_isMinOn + (nonempty_Icc.mpr (slitCapacityHorizon p hi h0 hH T).coe_nonneg) hc.continuousOn + exact ⟨_,norm_pos_iff.mpr (slitReflectedFlow_separated p hi h0 hH hT hx t), + fun u hu => hmin hu⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitRealCharacteristics.lean b/lean/OAI/Probability/SLE/Loewner/SlitRealCharacteristics.lean index acd5cdfc8e4d4220f33f76b74c90c8a40640e890..e56e1ae5caed80fc1d9abffd44929b6193cc29db 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitRealCharacteristics.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitRealCharacteristics.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.Loewner.SlitBoundaryIntegralLimit -import OAI.Probability.SLE.Geometry.RegularSlitTube - -namespace OAI - -/-! The exact driver of a prescribed simple slit has noncolliding real -Loewner characteristics throughout every finite capacity prefix. -/ -noncomputable section -open Set Filter Function Complex _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem slit_regularRealCharacteristics (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) : - RegularRealCharacteristics (slitCapacityDriver p hi h0 hH hT) - (slitCapacityHorizon p hi h0 hH T) := by - intro x hx - let X := slitReflectedFlow p hi h0 hH T (x:ℂ) - let q := fun t : ℝ => (X t).re - let b := slitCapacityDriver p hi h0 hH hT - let A := slitCapacityHorizon p hi h0 hH T - have hX (t : ℝ) : X t=(q t:ℂ) := by - apply Complex.ext - · rfl - · simpa only [ofReal_im] using slitReflectedFlow_real p hi h0 hH T hx t - have hX0 : X 0=(x:ℂ) := by - dsimp [X,slitReflectedFlow] - simp [slitCapacityTime_zero,initialReflectedForward] - have hqgap (t : ℝ) : q t-b t.toNNReal≠0 := by - intro he - have hh := slitReflectedFlow_separated p hi h0 hH hT hx t - change X t-(b t.toNNReal:ℂ)≠0 at hh - rw [hX] at hh - apply hh - exact_mod_cast he - refine ⟨q,?_,?_,fun t _ => hqgap t⟩ - · change (X 0).re=x - rw [hX0,ofReal_re] - · intro t ht - let k := fun u : ℝ => 2/(X u-(b u.toNNReal:ℂ)) - have hk : Continuous k := slitReflectedFlow_real_kernel_continuous p hi h0 hH hT hx - have hI : HasDerivAt (fun u => (x:ℂ)+∫ v in 0..u, k v) (k t) t := by - simpa only [zero_add] using (intervalIntegral.integral_hasDerivAt_right - (hk.intervalIntegrable 0 t) hk.aestronglyMeasurable.stronglyMeasurableAtFilter - hk.continuousAt).const_add (x:ℂ) - have hXd : HasDerivWithinAt X (k t) (Icc 0 (A:ℝ)) t := by - apply hI.hasDerivWithinAt.congr_of_mem (fun u hu => ?_) ht - rw [slitReflectedFlow_boundary_integral p hi h0 hH hT hx hu] - dsimp only [X] - abel - have hqd := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hXd - change HasDerivWithinAt q (k t).re (Icc 0 (A:ℝ)) t at hqd - have he : (k t).re=2/(q t-b t.toNNReal) := by - dsimp only [k] - rw [hX] - norm_cast - rwa [he] at hqd - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitBoundaryIntegralLimit +import OAI.Probability.SLE.Geometry.RegularSlitTube + +namespace OAI + +/-! The exact driver of a prescribed simple slit has noncolliding real +Loewner characteristics throughout every finite capacity prefix. -/ +noncomputable section +open Set Filter Function Complex _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem slit_regularRealCharacteristics (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) : + RegularRealCharacteristics (slitCapacityDriver p hi h0 hH hT) + (slitCapacityHorizon p hi h0 hH T) := by + intro x hx + let X := slitReflectedFlow p hi h0 hH T (x:ℂ) + let q := fun t : ℝ => (X t).re + let b := slitCapacityDriver p hi h0 hH hT + let A := slitCapacityHorizon p hi h0 hH T + have hX (t : ℝ) : X t=(q t:ℂ) := by + apply Complex.ext + · rfl + · simpa only [ofReal_im] using slitReflectedFlow_real p hi h0 hH T hx t + have hX0 : X 0=(x:ℂ) := by + dsimp [X,slitReflectedFlow] + simp [slitCapacityTime_zero,initialReflectedForward] + have hqgap (t : ℝ) : q t-b t.toNNReal≠0 := by + intro he + have hh := slitReflectedFlow_separated p hi h0 hH hT hx t + change X t-(b t.toNNReal:ℂ)≠0 at hh + rw [hX] at hh + apply hh + exact_mod_cast he + refine ⟨q,?_,?_,fun t _ => hqgap t⟩ + · change (X 0).re=x + rw [hX0,ofReal_re] + · intro t ht + let k := fun u : ℝ => 2/(X u-(b u.toNNReal:ℂ)) + have hk : Continuous k := slitReflectedFlow_real_kernel_continuous p hi h0 hH hT hx + have hI : HasDerivAt (fun u => (x:ℂ)+∫ v in 0..u, k v) (k t) t := by + simpa only [zero_add] using (intervalIntegral.integral_hasDerivAt_right + (hk.intervalIntegrable 0 t) hk.aestronglyMeasurable.stronglyMeasurableAtFilter + hk.continuousAt).const_add (x:ℂ) + have hXd : HasDerivWithinAt X (k t) (Icc 0 (A:ℝ)) t := by + apply hI.hasDerivWithinAt.congr_of_mem (fun u hu => ?_) ht + rw [slitReflectedFlow_boundary_integral p hi h0 hH hT hx hu] + dsimp only [X] + abel + have hqd := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t hXd + change HasDerivWithinAt q (k t).re (Icc 0 (A:ℝ)) t at hqd + have he : (k t).re=2/(q t-b t.toNNReal) := by + dsimp only [k] + rw [hX] + norm_cast + rwa [he] at hqd + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitReciprocalChart.lean b/lean/OAI/Probability/SLE/Loewner/SlitReciprocalChart.lean index 1480d1f61fdb551af161d3e1c702b09ea77f3b01..f2cf8fa8af2c0d19fe24502964630905c085c74f 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitReciprocalChart.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitReciprocalChart.lean @@ -1,80 +1,80 @@ -import OAI.Probability.SLE.Loewner.SlitHydrodynamicChart -import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Analytic - -namespace OAI - -/-! In reciprocal coordinates the hydrodynamic chart is analytic across -infinity and has derivative one. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -def slitReciprocalMap (f : ℂ → ℂ) (a : ℝ) (w : ℂ) : ℂ := - w/(slitInfinityFactor f a w-deriv (slitInfinityFactor f a) 0*w) - -@[simp] lemma slitReciprocalMap_zero (f : ℂ → ℂ) (a : ℝ) : - slitReciprocalMap f a 0=0 := by simp [slitReciprocalMap] - -lemma slitReciprocalMap_analyticAt {f : ℂ → ℂ} {a : ℝ} - (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : - AnalyticAt ℂ (slitReciprocalMap f a) 0 := by - apply analyticAt_id.div - ((slitInfinityFactor_analyticAt hf ha hd).sub (analyticAt_const.mul analyticAt_id)) - change slitInfinityFactor f a 0-deriv (slitInfinityFactor f a) 0*0 ≠ 0 - simp only [mul_zero,sub_zero,slitInfinityFactor_zero ha hd,ne_eq,one_ne_zero,not_false_eq_true] - -lemma slitReciprocalMap_deriv {f : ℂ → ℂ} {a : ℝ} - (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : - deriv (slitReciprocalMap f a) 0=1 := by - have hh := ((hasDerivAt_id (0:ℂ)).div - (((slitInfinityFactor_analyticAt hf ha hd).differentiableAt.hasDerivAt).sub - ((hasDerivAt_id (0:ℂ)).const_mul (deriv (slitInfinityFactor f a) 0))) (by - change slitInfinityFactor f a 0-deriv (slitInfinityFactor f a) 0*0 ≠ 0 - simp only [mul_zero,sub_zero,slitInfinityFactor_zero ha hd,ne_eq,one_ne_zero, - not_false_eq_true])) - convert! hh.deriv using 1 - simp only [Pi.sub_apply,id_eq,mul_zero,sub_zero, - slitInfinityFactor_zero ha hd,zero_mul,one_pow,div_one,one_mul] - -lemma slitReciprocalMap_identity {f : ℂ → ℂ} (hf0 : f 0=0) (a : ℝ) - {z : ℂ} (hz : z ≠ 0) : - (slitReciprocalMap f a z⁻¹)⁻¹=slitHydrodynamicMap f a z := by - rw [slitReciprocalMap,inv_div,div_eq_mul_inv,inv_inv,sub_mul,mul_comm _ z, - slitInfinityFactor_identity hf0 a hz] - simp only [slitHydrodynamicMap] - rw [mul_assoc,inv_mul_cancel₀ hz,mul_one] - -lemma slitReciprocalMap_reflection {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) - {f : ℂ → ℂ} (hs : ∀ z ∈ D, f (conj z)=conj (f z)) (a : ℝ) : - ∀ z ∈ D, slitReciprocalMap f a (conj z)=conj (slitReciprocalMap f a z) := by - have hH := slitInfinityFactor_reflection hD hs a - have hd := derivative_real_of_reflection hD hH - intro z hz - simp only [slitReciprocalMap,hH z hz,map_div₀,map_sub,map_mul,hd] - -theorem slitReciprocalMap_exists_local_inverse {f : ℂ → ℂ} {a : ℝ} - (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : - ∃ r : ℂ → ℂ, AnalyticAt ℂ r 0 ∧ r 0=0 ∧ deriv r 0=1 ∧ - (∀ᶠ z in 𝓝 (0:ℂ), r (slitReciprocalMap f a z)=z) ∧ - (∀ᶠ z in 𝓝 (0:ℂ), slitReciprocalMap f a (r z)=z) := by - have hq := slitReciprocalMap_analyticAt hf ha hd - have hqd := slitReciprocalMap_deriv hf ha hd - have hn : deriv (slitReciprocalMap f a) 0 ≠ 0 := by rw [hqd]; exact one_ne_zero - let r := hq.hasStrictDerivAt.localInverse _ _ _ hn - have hr := hq.analyticAt_localInverse hn - have hzero := (hq.hasStrictDerivAt.eventually_left_inverse hn).self_of_nhds - have hrd := hq.hasStrictDerivAt.to_localInverse hn - refine ⟨r,?_,?_,?_,?_,?_⟩ - · simpa only [slitReciprocalMap_zero] using hr - · simpa only [slitReciprocalMap_zero] using hzero - · simpa only [slitReciprocalMap_zero,hqd,inv_one] using hrd.hasDerivAt.deriv - · exact hq.hasStrictDerivAt.eventually_left_inverse hn - · simpa only [slitReciprocalMap_zero] using hq.hasStrictDerivAt.eventually_right_inverse hn - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitHydrodynamicChart +import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Analytic + +namespace OAI + +/-! In reciprocal coordinates the hydrodynamic chart is analytic across +infinity and has derivative one. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology ComplexConjugate + +namespace SLEExactGauge + +def slitReciprocalMap (f : ℂ → ℂ) (a : ℝ) (w : ℂ) : ℂ := + w/(slitInfinityFactor f a w-deriv (slitInfinityFactor f a) 0*w) + +@[simp] lemma slitReciprocalMap_zero (f : ℂ → ℂ) (a : ℝ) : + slitReciprocalMap f a 0=0 := by simp [slitReciprocalMap] + +lemma slitReciprocalMap_analyticAt {f : ℂ → ℂ} {a : ℝ} + (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : + AnalyticAt ℂ (slitReciprocalMap f a) 0 := by + apply analyticAt_id.div + ((slitInfinityFactor_analyticAt hf ha hd).sub (analyticAt_const.mul analyticAt_id)) + change slitInfinityFactor f a 0-deriv (slitInfinityFactor f a) 0*0 ≠ 0 + simp only [mul_zero,sub_zero,slitInfinityFactor_zero ha hd,ne_eq,one_ne_zero,not_false_eq_true] + +lemma slitReciprocalMap_deriv {f : ℂ → ℂ} {a : ℝ} + (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : + deriv (slitReciprocalMap f a) 0=1 := by + have hh := ((hasDerivAt_id (0:ℂ)).div + (((slitInfinityFactor_analyticAt hf ha hd).differentiableAt.hasDerivAt).sub + ((hasDerivAt_id (0:ℂ)).const_mul (deriv (slitInfinityFactor f a) 0))) (by + change slitInfinityFactor f a 0-deriv (slitInfinityFactor f a) 0*0 ≠ 0 + simp only [mul_zero,sub_zero,slitInfinityFactor_zero ha hd,ne_eq,one_ne_zero, + not_false_eq_true])) + convert! hh.deriv using 1 + simp only [Pi.sub_apply,id_eq,mul_zero,sub_zero, + slitInfinityFactor_zero ha hd,zero_mul,one_pow,div_one,one_mul] + +lemma slitReciprocalMap_identity {f : ℂ → ℂ} (hf0 : f 0=0) (a : ℝ) + {z : ℂ} (hz : z ≠ 0) : + (slitReciprocalMap f a z⁻¹)⁻¹=slitHydrodynamicMap f a z := by + rw [slitReciprocalMap,inv_div,div_eq_mul_inv,inv_inv,sub_mul,mul_comm _ z, + slitInfinityFactor_identity hf0 a hz] + simp only [slitHydrodynamicMap] + rw [mul_assoc,inv_mul_cancel₀ hz,mul_one] + +lemma slitReciprocalMap_reflection {D : Set ℂ} (hD : D ∈ 𝓝 (0:ℂ)) + {f : ℂ → ℂ} (hs : ∀ z ∈ D, f (conj z)=conj (f z)) (a : ℝ) : + ∀ z ∈ D, slitReciprocalMap f a (conj z)=conj (slitReciprocalMap f a z) := by + have hH := slitInfinityFactor_reflection hD hs a + have hd := derivative_real_of_reflection hD hH + intro z hz + simp only [slitReciprocalMap,hH z hz,map_div₀,map_sub,map_mul,hd] + +theorem slitReciprocalMap_exists_local_inverse {f : ℂ → ℂ} {a : ℝ} + (hf : AnalyticAt ℂ f 0) (ha : a ≠ 0) (hd : deriv f 0=(a:ℂ)) : + ∃ r : ℂ → ℂ, AnalyticAt ℂ r 0 ∧ r 0=0 ∧ deriv r 0=1 ∧ + (∀ᶠ z in 𝓝 (0:ℂ), r (slitReciprocalMap f a z)=z) ∧ + (∀ᶠ z in 𝓝 (0:ℂ), slitReciprocalMap f a (r z)=z) := by + have hq := slitReciprocalMap_analyticAt hf ha hd + have hqd := slitReciprocalMap_deriv hf ha hd + have hn : deriv (slitReciprocalMap f a) 0 ≠ 0 := by rw [hqd]; exact one_ne_zero + let r := hq.hasStrictDerivAt.localInverse _ _ _ hn + have hr := hq.analyticAt_localInverse hn + have hzero := (hq.hasStrictDerivAt.eventually_left_inverse hn).self_of_nhds + have hrd := hq.hasStrictDerivAt.to_localInverse hn + refine ⟨r,?_,?_,?_,?_,?_⟩ + · simpa only [slitReciprocalMap_zero] using hr + · simpa only [slitReciprocalMap_zero] using hzero + · simpa only [slitReciprocalMap_zero,hqd,inv_one] using hrd.hasDerivAt.deriv + · exact hq.hasStrictDerivAt.eventually_left_inverse hn + · simpa only [slitReciprocalMap_zero] using hq.hasStrictDerivAt.eventually_right_inverse hn + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitReflectedInterior.lean b/lean/OAI/Probability/SLE/Loewner/SlitReflectedInterior.lean index de6ab449524b9da78e58019a79b3d2f8f5d61363..1ab01eaf5e16fbb4f7aec84147ba8c0c10051743 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitReflectedInterior.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitReflectedInterior.lean @@ -1,102 +1,102 @@ -import OAI.Probability.SLE.Loewner.SlitRealBoundaryFlow -import OAI.Probability.SLE.Loewner.SlitPhysicalEvolution - -namespace OAI - -/-! On surviving interior points the reflected coordinates obey the -exact physical-time Loewner equation and its integral identity. -/ -noncomputable section -open Set Filter Function Complex MeasureTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma slitReflectedFlow_eq_forward (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range p) (t : ℝ) : - slitReflectedFlow p hi h0 hH T z t=slitCapacityForward p hi h0 hH T t.toNNReal z := - initialReflectedForward_eq p hi h0 hH _ - ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p _ he)⟩ - -lemma slitReflectedFlow_interior_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - (T : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : - Continuous (slitReflectedFlow p hi h0 hH T z) := by - have hzD (s : unitInterval) : z∈upperHalfPlane \ range (initialSubarc p s) := - ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p s he)⟩ - have hc : Continuous (fun s => initialForwardChart p hi h0 hH s z) := by - apply SeqContinuous.continuous - intro s t hs - exact initialForwardChart_tendsto p hi h0 hH hs tendsto_const_nhds (hzD t) (fun n => hzD (s n)) - have he : slitReflectedFlow p hi h0 hH T z= - fun t => initialForwardChart p hi h0 hH (slitCapacityTime p hi h0 hH T t.toNNReal) z := - funext (slitReflectedFlow_eq_forward p hi h0 hH T hz) - rw [he] - exact hc.comp ((slitCapacityTime_continuous p hi h0 hH T).comp continuous_real_toNNReal) - -lemma slitReflectedFlow_interior_separated (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) (t : ℝ) : - slitReflectedFlow p hi h0 hH T z t-(slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)≠0 := by - have hp := initialForwardChart_mem_upper p hi h0 hH (slitCapacityTime p hi h0 hH T t.toNNReal) - ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p _ he)⟩ - rw [slitReflectedFlow_eq_forward p hi h0 hH T hz] - intro he - have hh := congrArg Complex.im he - simp only [sub_im,ofReal_im,sub_zero,zero_im] at hh - exact (show 0<(slitCapacityForward p hi h0 hH T t.toNNReal z).im from hp).ne' hh - -lemma slitReflectedFlow_interior_kernel_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : - Continuous (fun t : ℝ => 2/(slitReflectedFlow p hi h0 hH T z t- - (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))) := - continuous_const.div ((slitReflectedFlow_interior_continuous p hi h0 hH T hz).sub - (Complex.continuous_ofReal.comp - ((slitCapacityDriver p hi h0 hH hT).continuous.comp continuous_real_toNNReal))) - (slitReflectedFlow_interior_separated p hi h0 hH hT hz) - -theorem slitReflectedFlow_interior_derivative (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) - {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : - HasDerivWithinAt (slitReflectedFlow p hi h0 hH T z) - (2/(slitReflectedFlow p hi h0 hH T z t- - (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))) - (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) t := by - have hu : t.toNNReal ≤ slitCapacityHorizon p hi h0 hH T := Real.toNNReal_le_iff_le_coe.mpr ht.2 - have hh := slitCapacityForward_prefix_derivative p hi h0 hH hT - ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p T he)⟩ hu - rw [Real.coe_toNNReal t ht.1] at hh - have he : slitReflectedFlow p hi h0 hH T z= - fun u => slitCapacityForward p hi h0 hH T u.toNNReal z := - funext (slitReflectedFlow_eq_forward p hi h0 hH T hz) - simpa only [he] using hh - -theorem slitReflectedFlow_interior_integral (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) - {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : - (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T z u- - (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))= - slitReflectedFlow p hi h0 hH T z t-z := by - have hz0 : slitReflectedFlow p hi h0 hH T z 0=z := by - rw [slitReflectedFlow_eq_forward p hi h0 hH T hz] - simp only [slitCapacityForward,Real.toNNReal_zero,slitCapacityTime_zero,initialForwardChart_zero,id_eq] - have hi' : (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T z u- - (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))= - slitReflectedFlow p hi h0 hH T z t-slitReflectedFlow p hi h0 hH T z 0 := by - apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le ht.1 - (slitReflectedFlow_interior_continuous p hi h0 hH T hz).continuousOn - · intro u hu - exact (slitReflectedFlow_interior_derivative p hi h0 hH hT hz - ⟨hu.1.le,hu.2.le.trans ht.2⟩).hasDerivAt - (Icc_mem_nhds hu.1 (hu.2.trans_le ht.2)) - · exact (slitReflectedFlow_interior_kernel_continuous p hi h0 hH hT hz).intervalIntegrable 0 t - simpa only [hz0] using hi' - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitRealBoundaryFlow +import OAI.Probability.SLE.Loewner.SlitPhysicalEvolution + +namespace OAI + +/-! On surviving interior points the reflected coordinates obey the +exact physical-time Loewner equation and its integral identity. -/ +noncomputable section +open Set Filter Function Complex MeasureTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma slitReflectedFlow_eq_forward (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range p) (t : ℝ) : + slitReflectedFlow p hi h0 hH T z t=slitCapacityForward p hi h0 hH T t.toNNReal z := + initialReflectedForward_eq p hi h0 hH _ + ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p _ he)⟩ + +lemma slitReflectedFlow_interior_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + (T : unitInterval) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : + Continuous (slitReflectedFlow p hi h0 hH T z) := by + have hzD (s : unitInterval) : z∈upperHalfPlane \ range (initialSubarc p s) := + ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p s he)⟩ + have hc : Continuous (fun s => initialForwardChart p hi h0 hH s z) := by + apply SeqContinuous.continuous + intro s t hs + exact initialForwardChart_tendsto p hi h0 hH hs tendsto_const_nhds (hzD t) (fun n => hzD (s n)) + have he : slitReflectedFlow p hi h0 hH T z= + fun t => initialForwardChart p hi h0 hH (slitCapacityTime p hi h0 hH T t.toNNReal) z := + funext (slitReflectedFlow_eq_forward p hi h0 hH T hz) + rw [he] + exact hc.comp ((slitCapacityTime_continuous p hi h0 hH T).comp continuous_real_toNNReal) + +lemma slitReflectedFlow_interior_separated (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) (t : ℝ) : + slitReflectedFlow p hi h0 hH T z t-(slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ)≠0 := by + have hp := initialForwardChart_mem_upper p hi h0 hH (slitCapacityTime p hi h0 hH T t.toNNReal) + ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p _ he)⟩ + rw [slitReflectedFlow_eq_forward p hi h0 hH T hz] + intro he + have hh := congrArg Complex.im he + simp only [sub_im,ofReal_im,sub_zero,zero_im] at hh + exact (show 0<(slitCapacityForward p hi h0 hH T t.toNNReal z).im from hp).ne' hh + +lemma slitReflectedFlow_interior_kernel_continuous (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) : + Continuous (fun t : ℝ => 2/(slitReflectedFlow p hi h0 hH T z t- + (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))) := + continuous_const.div ((slitReflectedFlow_interior_continuous p hi h0 hH T hz).sub + (Complex.continuous_ofReal.comp + ((slitCapacityDriver p hi h0 hH hT).continuous.comp continuous_real_toNNReal))) + (slitReflectedFlow_interior_separated p hi h0 hH hT hz) + +theorem slitReflectedFlow_interior_derivative (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) + {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : + HasDerivWithinAt (slitReflectedFlow p hi h0 hH T z) + (2/(slitReflectedFlow p hi h0 hH T z t- + (slitCapacityDriver p hi h0 hH hT t.toNNReal:ℂ))) + (Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) t := by + have hu : t.toNNReal ≤ slitCapacityHorizon p hi h0 hH T := Real.toNNReal_le_iff_le_coe.mpr ht.2 + have hh := slitCapacityForward_prefix_derivative p hi h0 hH hT + ⟨hz.1,fun he => hz.2 (initialSubarc_range_subset p T he)⟩ hu + rw [Real.coe_toNNReal t ht.1] at hh + have he : slitReflectedFlow p hi h0 hH T z= + fun u => slitCapacityForward p hi h0 hH T u.toNNReal z := + funext (slitReflectedFlow_eq_forward p hi h0 hH T hz) + simpa only [he] using hh + +theorem slitReflectedFlow_interior_integral (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) {z : ℂ} (hz : z∈upperHalfPlane \ range p) + {t : ℝ} (ht : t∈Icc 0 (slitCapacityHorizon p hi h0 hH T:ℝ)) : + (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T z u- + (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))= + slitReflectedFlow p hi h0 hH T z t-z := by + have hz0 : slitReflectedFlow p hi h0 hH T z 0=z := by + rw [slitReflectedFlow_eq_forward p hi h0 hH T hz] + simp only [slitCapacityForward,Real.toNNReal_zero,slitCapacityTime_zero,initialForwardChart_zero,id_eq] + have hi' : (∫ u in 0..t, 2/(slitReflectedFlow p hi h0 hH T z u- + (slitCapacityDriver p hi h0 hH hT u.toNNReal:ℂ)))= + slitReflectedFlow p hi h0 hH T z t-slitReflectedFlow p hi h0 hH T z 0 := by + apply intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le ht.1 + (slitReflectedFlow_interior_continuous p hi h0 hH T hz).continuousOn + · intro u hu + exact (slitReflectedFlow_interior_derivative p hi h0 hH hT hz + ⟨hu.1.le,hu.2.le.trans ht.2⟩).hasDerivAt + (Icc_mem_nhds hu.1 (hu.2.trans_le ht.2)) + · exact (slitReflectedFlow_interior_kernel_continuous p hi h0 hH hT hz).intervalIntegrable 0 t + simpa only [hz0] using hi' + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitSegmentParameter.lean b/lean/OAI/Probability/SLE/Loewner/SlitSegmentParameter.lean index 98a325b75320398abdbccacc91e64f2b5329c880..aeba9b8a51ce88f146ac3e9ea11f7e4543fe8bc6 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitSegmentParameter.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitSegmentParameter.lean @@ -1,73 +1,73 @@ -import OAI.Probability.SLE.Loewner.InitialSlitFuture - -namespace OAI - -/-! Exact affine parameters for a segment of a prescribed simple arc. -/ -noncomputable section -open Set Function - -namespace SLEExactGauge - -def slitSegmentParameter (s t : unitInterval) (hst : s≤t) (u : unitInterval) : unitInterval := - ⟨(s:ℝ)+((t:ℝ)-(s:ℝ))*(u:ℝ),by - have hd : 0≤(t:ℝ)-(s:ℝ) := sub_nonneg.mpr hst - constructor - · exact add_nonneg s.property.1 (mul_nonneg hd u.property.1) - · have hm := mul_le_mul_of_nonneg_left u.property.2 hd - linarith [t.property.2]⟩ - -@[simp] lemma slitSegmentParameter_zero (s t : unitInterval) (hst : s≤t) : - slitSegmentParameter s t hst 0=s := by - apply Subtype.ext - change (s:ℝ)+((t:ℝ)-(s:ℝ))*0=(s:ℝ) - ring - -@[simp] lemma slitSegmentParameter_one (s t : unitInterval) (hst : s≤t) : - slitSegmentParameter s t hst 1=t := by - apply Subtype.ext - change (s:ℝ)+((t:ℝ)-(s:ℝ))*1=(t:ℝ) - ring - -lemma slitSegmentParameter_gt {s t : unitInterval} (hst : s Target.trace 1 (slitCapacityDriver p hi h0 hH hT) s) '' Icc 0 t= - slitCapacityCurve p hi h0 hH T '' Icc 0 t := by - apply image_congr - intro s hs - exact slitCapacity_raw_trace_eq p hi h0 hH hT (hs.2.trans ht) - -theorem slitCapacity_raw_complement_survives (p : C(unitInterval,ℂ)) (hi : Injective p) - (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT : T<1) (t : ℝ≥0) {z : ℂ} - (hz : z∈upperHalfPlane \ ((fun s : ℝ≥0 => - Target.trace 1 (slitCapacityDriver p hi h0 hH hT) s) '' Icc 0 t)) : - Target.Surviving 1 (slitCapacityDriver p hi h0 hH hT) z t := by - let b := slitCapacityDriver p hi h0 hH hT - let A := slitCapacityHorizon p hi h0 hH T - let u := initialDrivingValue p hi h0 hH T hT - let β := fun s : ℝ≥0 => Target.trace 1 b s - change z∈upperHalfPlane \ (β '' Icc 0 t) at hz - by_cases ht : t ≤ A - · change z∈{z | Target.Surviving 1 b z t} - rw [slitCapacity_surviving_eq p hi h0 hH hT ht, - ←slitCapacityCurve_initial_range p hi h0 hH T ht, - ←slitCapacity_raw_initial_range p hi h0 hH hT ht] - exact hz - have hAt : A hz.2 (image_mono (Icc_subset_Icc_right hAt.le) he)⟩ - have hsA : Target.Surviving 1 b z A := by - change z∈{z | Target.Surviving 1 b z A} - rw [slitCapacity_surviving_eq p hi h0 hH hT le_rfl, - ←slitCapacityCurve_initial_range p hi h0 hH T le_rfl, - ←slitCapacity_raw_initial_range p hi h0 hH hT le_rfl] - exact hpre - let w := Target.flow 1 b z A-(u:ℂ) - have hw : w∈upperHalfPlane := by - have hh := Target.surviving_flow_mem_upperHalfPlane 1 b z A A.coe_nonneg hsA - simpa only [w,upperHalfPlane,mem_ofPred_eq,sub_im,ofReal_im,sub_zero] using hh - have hshift : (u:ℂ)+w=Target.flow 1 b z A := by dsimp [w]; ring - have hb : ∀ s : ℝ≥0, A ≤ s → b s=u := - fun s hs => slitCapacityDriver_after p hi h0 hH hT hs - have hs : Target.Surviving 1 (Target.linearDriver 0) w (t-A : ℝ≥0) := by - by_contra hn - obtain ⟨v,hv,hvS,he⟩ := Target.exists_trace_time_of_not_surviving (Target.linearDriver 0) - w hw (t-A) hn (fun _ => 0) 0 - (fun s hs => Target.linearDriver_derivative 0 s hs) (fun _ _ => by simp) - have hval : β (A+v)=z := by - dsimp only [β] - rw [NNReal.coe_add,Target.trace_constant_tail b A v hv u hb,←he,hshift] - exact Target.inverseFlow_eq_of_surviving 1 b A A.coe_nonneg z _ hsA rfl - apply hz.2 - refine ⟨A+v,⟨zero_le,?_⟩,hval⟩ - calc A+v ≤ A+(t-A) := add_le_add le_rfl hvS - _ = t := add_tsub_cancel_of_le hAt.le - let q := (u:ℂ)+Target.flow 1 (Target.linearDriver 0) w (t-A : ℝ≥0) - have hq : q∈upperHalfPlane := by - have hh := Target.surviving_flow_mem_upperHalfPlane 1 (Target.linearDriver 0) w - (t-A : ℝ≥0) (t-A).coe_nonneg hs - simpa only [q,upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hh - have hF : Target.inverseFlow 1 b t q=z := by - have he := Target.inverseFlow_constant_tail b A (t-A) u hb hq - rw [←NNReal.coe_add,add_tsub_cancel_of_le hAt.le] at he - rw [he] - have hqsub : q-(u:ℂ)=Target.flow 1 (Target.linearDriver 0) w (t-A : ℝ≥0) := by - dsimp [q]; ring - rw [hqsub,Target.inverseFlow_eq_of_surviving 1 (Target.linearDriver 0) - (t-A : ℝ≥0) (t-A).coe_nonneg w _ hs rfl,hshift] - exact Target.inverseFlow_eq_of_surviving 1 b A A.coe_nonneg z _ hsA rfl - rw [←hF] - exact Target.inverseFlow_surviving 1 (by norm_num) b t q hq - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} - -theorem RegularSLEPresentation.slitCapacity_raw_domains - (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) - (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {T : unitInterval} (hT0 : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) s) t := by - let b := slitCapacityDriver p hi h0 hH hT - let β := fun s : ℝ≥0 => Target.trace 1 b s - obtain ⟨_,_,_,_,hfree⟩ := a.slitCapacity_raw_geometry hκ p hi h0 hH hT0 hT - have hsub (t : ℝ≥0) (ht : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) s) '' Icc 0 t= + slitCapacityCurve p hi h0 hH T '' Icc 0 t := by + apply image_congr + intro s hs + exact slitCapacity_raw_trace_eq p hi h0 hH hT (hs.2.trans ht) + +theorem slitCapacity_raw_complement_survives (p : C(unitInterval,ℂ)) (hi : Injective p) + (h0 : p 0=0) (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT : T<1) (t : ℝ≥0) {z : ℂ} + (hz : z∈upperHalfPlane \ ((fun s : ℝ≥0 => + Target.trace 1 (slitCapacityDriver p hi h0 hH hT) s) '' Icc 0 t)) : + Target.Surviving 1 (slitCapacityDriver p hi h0 hH hT) z t := by + let b := slitCapacityDriver p hi h0 hH hT + let A := slitCapacityHorizon p hi h0 hH T + let u := initialDrivingValue p hi h0 hH T hT + let β := fun s : ℝ≥0 => Target.trace 1 b s + change z∈upperHalfPlane \ (β '' Icc 0 t) at hz + by_cases ht : t ≤ A + · change z∈{z | Target.Surviving 1 b z t} + rw [slitCapacity_surviving_eq p hi h0 hH hT ht, + ←slitCapacityCurve_initial_range p hi h0 hH T ht, + ←slitCapacity_raw_initial_range p hi h0 hH hT ht] + exact hz + have hAt : A hz.2 (image_mono (Icc_subset_Icc_right hAt.le) he)⟩ + have hsA : Target.Surviving 1 b z A := by + change z∈{z | Target.Surviving 1 b z A} + rw [slitCapacity_surviving_eq p hi h0 hH hT le_rfl, + ←slitCapacityCurve_initial_range p hi h0 hH T le_rfl, + ←slitCapacity_raw_initial_range p hi h0 hH hT le_rfl] + exact hpre + let w := Target.flow 1 b z A-(u:ℂ) + have hw : w∈upperHalfPlane := by + have hh := Target.surviving_flow_mem_upperHalfPlane 1 b z A A.coe_nonneg hsA + simpa only [w,upperHalfPlane,mem_ofPred_eq,sub_im,ofReal_im,sub_zero] using hh + have hshift : (u:ℂ)+w=Target.flow 1 b z A := by dsimp [w]; ring + have hb : ∀ s : ℝ≥0, A ≤ s → b s=u := + fun s hs => slitCapacityDriver_after p hi h0 hH hT hs + have hs : Target.Surviving 1 (Target.linearDriver 0) w (t-A : ℝ≥0) := by + by_contra hn + obtain ⟨v,hv,hvS,he⟩ := Target.exists_trace_time_of_not_surviving (Target.linearDriver 0) + w hw (t-A) hn (fun _ => 0) 0 + (fun s hs => Target.linearDriver_derivative 0 s hs) (fun _ _ => by simp) + have hval : β (A+v)=z := by + dsimp only [β] + rw [NNReal.coe_add,Target.trace_constant_tail b A v hv u hb,←he,hshift] + exact Target.inverseFlow_eq_of_surviving 1 b A A.coe_nonneg z _ hsA rfl + apply hz.2 + refine ⟨A+v,⟨zero_le,?_⟩,hval⟩ + calc A+v ≤ A+(t-A) := add_le_add le_rfl hvS + _ = t := add_tsub_cancel_of_le hAt.le + let q := (u:ℂ)+Target.flow 1 (Target.linearDriver 0) w (t-A : ℝ≥0) + have hq : q∈upperHalfPlane := by + have hh := Target.surviving_flow_mem_upperHalfPlane 1 (Target.linearDriver 0) w + (t-A : ℝ≥0) (t-A).coe_nonneg hs + simpa only [q,upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,zero_add] using hh + have hF : Target.inverseFlow 1 b t q=z := by + have he := Target.inverseFlow_constant_tail b A (t-A) u hb hq + rw [←NNReal.coe_add,add_tsub_cancel_of_le hAt.le] at he + rw [he] + have hqsub : q-(u:ℂ)=Target.flow 1 (Target.linearDriver 0) w (t-A : ℝ≥0) := by + dsimp [q]; ring + rw [hqsub,Target.inverseFlow_eq_of_surviving 1 (Target.linearDriver 0) + (t-A : ℝ≥0) (t-A).coe_nonneg w _ hs rfl,hshift] + exact Target.inverseFlow_eq_of_surviving 1 b A A.coe_nonneg z _ hsA rfl + rw [←hF] + exact Target.inverseFlow_surviving 1 (by norm_num) b t q hq + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} + +theorem RegularSLEPresentation.slitCapacity_raw_domains + (a : RegularSLEPresentation P κ γ) (hκ : 0<κ) + (p : C(unitInterval,ℂ)) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {T : unitInterval} (hT0 : 0 Target.trace 1 (slitCapacityDriver p hi h0 hH hT) s) t := by + let b := slitCapacityDriver p hi h0 hH hT + let β := fun s : ℝ≥0 => Target.trace 1 b s + obtain ⟨_,_,_,_,hfree⟩ := a.slitCapacity_raw_geometry hκ p hi h0 hH hT0 hT + have hsub (t : ℝ≥0) (ht : 0 if s ≤ T then f s else g s) d (Ici 0) t := by - rcases lt_trichotomy t T with hlt | heq | hgt - · apply (hf hlt.le).congr_of_eventuallyEq - · filter_upwards [mem_nhdsWithin_of_mem_nhds (gt_mem_nhds hlt)] with s hs - exact ite_eq_left hs.le - · exact ite_eq_left hlt.le - · subst t - have hl := (hf le_rfl).mono (show Ici 0 ∩ Iic T ⊆ Ici 0 from inter_subset_left) - have hr := (hg le_rfl).mono (show Ici 0 ∩ Ici T ⊆ Ici T from inter_subset_right) - have hl' : HasDerivWithinAt (fun s => if s ≤ T then f s else g s) d - (Ici 0 ∩ Iic T) T := hl.congr (fun s hs => ite_eq_left hs.2) (ite_eq_left le_rfl) - have hr' : HasDerivWithinAt (fun s => if s ≤ T then f s else g s) d - (Ici 0 ∩ Ici T) T := by - apply hr.congr - · intro s hs - by_cases h : s ≤ T - · have hsT : s=T := le_antisymm h hs.2 - simpa only [hsT,ite_eq_left le_rfl] using he - · exact ite_eq_right h - · simpa only [ite_eq_left le_rfl] using he - have hu : (Ici 0 ∩ Iic T) ∪ (Ici 0 ∩ Ici T)=Ici 0 := by - ext s - simp only [mem_union,mem_inter_iff,mem_Ici,mem_Iic] - constructor - · rintro (h | h) <;> exact h.1 - · intro h - exact (le_total s T).elim (fun hs => Or.inl ⟨h,hs⟩) (fun hs => Or.inr ⟨h,hs⟩) - simpa only [hu] using hl'.union hr' - · apply ((hg hgt.le).mono_of_mem_nhdsWithin ?_).congr_of_eventuallyEq - · filter_upwards [mem_nhdsWithin_of_mem_nhds (lt_mem_nhds hgt)] with s hs - exact ite_eq_right (not_le.mpr hs) - · exact ite_eq_right (not_le.mpr hgt) - · exact mem_nhdsWithin_of_mem_nhds (Ici_mem_nhds hgt) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.LoewnerCharts + +namespace OAI + +/-! One-sided time derivatives glue at a slit concatenation time. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge + +lemma hasDerivWithinAt_time_splice + {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] + (f g : ℝ → E) (T t : ℝ) (d : E) + (he : f T=g T) + (hf : t ≤ T → HasDerivWithinAt f d (Ici 0) t) + (hg : T ≤ t → HasDerivWithinAt g d (Ici T) t) : + HasDerivWithinAt (fun s => if s ≤ T then f s else g s) d (Ici 0) t := by + rcases lt_trichotomy t T with hlt | heq | hgt + · apply (hf hlt.le).congr_of_eventuallyEq + · filter_upwards [mem_nhdsWithin_of_mem_nhds (gt_mem_nhds hlt)] with s hs + exact ite_eq_left hs.le + · exact ite_eq_left hlt.le + · subst t + have hl := (hf le_rfl).mono (show Ici 0 ∩ Iic T ⊆ Ici 0 from inter_subset_left) + have hr := (hg le_rfl).mono (show Ici 0 ∩ Ici T ⊆ Ici T from inter_subset_right) + have hl' : HasDerivWithinAt (fun s => if s ≤ T then f s else g s) d + (Ici 0 ∩ Iic T) T := hl.congr (fun s hs => ite_eq_left hs.2) (ite_eq_left le_rfl) + have hr' : HasDerivWithinAt (fun s => if s ≤ T then f s else g s) d + (Ici 0 ∩ Ici T) T := by + apply hr.congr + · intro s hs + by_cases h : s ≤ T + · have hsT : s=T := le_antisymm h hs.2 + simpa only [hsT,ite_eq_left le_rfl] using he + · exact ite_eq_right h + · simpa only [ite_eq_left le_rfl] using he + have hu : (Ici 0 ∩ Iic T) ∪ (Ici 0 ∩ Ici T)=Ici 0 := by + ext s + simp only [mem_union,mem_inter_iff,mem_Ici,mem_Iic] + constructor + · rintro (h | h) <;> exact h.1 + · intro h + exact (le_total s T).elim (fun hs => Or.inl ⟨h,hs⟩) (fun hs => Or.inr ⟨h,hs⟩) + simpa only [hu] using hl'.union hr' + · apply ((hg hgt.le).mono_of_mem_nhdsWithin ?_).congr_of_eventuallyEq + · filter_upwards [mem_nhdsWithin_of_mem_nhds (lt_mem_nhds hgt)] with s hs + exact ite_eq_right (not_le.mpr hs) + · exact ite_eq_right (not_le.mpr hgt) + · exact mem_nhdsWithin_of_mem_nhds (Ici_mem_nhds hgt) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SlitUniformForwardBound.lean b/lean/OAI/Probability/SLE/Loewner/SlitUniformForwardBound.lean index ab7f74b3a2b72f25de61821f778571f81aaefc1c..81883550f9a1b1fedb7369572459de16a9e8ce5e 100644 --- a/lean/OAI/Probability/SLE/Loewner/SlitUniformForwardBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/SlitUniformForwardBound.lean @@ -1,118 +1,118 @@ -import OAI.Probability.SLE.Loewner.ReflectedChartLocalBound -import OAI.Probability.SLE.Loewner.CapacityRadiusBound - -namespace OAI - -/-! A numerical, slit-radius bound on all near-boundary forward -coordinates. It is independent of the shape of the simple slit. -/ -noncomputable section -open Set Filter Function Complex Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -lemma SimpleSlitChart.forward_on_two_radius {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 (hp t).le) - have hdiv : ‖(P.capacity:ℂ)/z‖=P.capacity/(2*R) := by - rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos P.capacity_pos,hz] - have htri := norm_sub_le_norm_sub_add_norm_sub (P.forward z-z) ((P.capacity:ℂ)/z) 0 - simp only [sub_zero] at htri - rw [hdiv] at htri - rw [hz] at hb - have he : 10*P.capacity*R/(2*R)^2+P.capacity/(2*R)=3*P.capacity/R := by - field_simp - ring - have hcost : 10*P.capacity*R/(2*R)^2+P.capacity/(2*R)≤3*R := by - rw [he] - apply (div_le_iff₀ hR).mpr - nlinarith - have hdelta : ‖P.forward z-z‖≤3*R := by linarith - have hsum := norm_le_norm_sub_add (P.forward z) z - rw [hz] at hsum - linarith - -lemma ReflectedCapacityChart.normalizing_constant_bound {p : C(unitInterval,ℂ)} - (Q : ReflectedCapacityChart p) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 (hp t).trans (by linarith)) hnot hzn.le - have hG := Q.chart.forward_on_two_radius hi h0 hH hR hp hzH hzn - have he : deriv (slitInfinityFactor Q.diskMap Q.slope) 0= - (Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)-Q.chart.forward z := by - rw [Q.forward_eq] - simp only [slitHydrodynamicMap] - ring - rw [he] - have ht := norm_sub_le ((Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)) - (Q.chart.forward z) - nlinarith - -lemma ReflectedCapacityChart.forward_bounded_in_two_radius {p : C(unitInterval,ℂ)} - (Q : ReflectedCapacityChart p) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 (hp t).trans (by linarith)) hz hzR - have hb := Q.normalizing_constant_bound hi h0 hH hR hp - rw [Q.forward_eq] - change ‖(Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)- - deriv (slitInfinityFactor Q.diskMap Q.slope) 0‖≤_ - have ht := norm_sub_le ((Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)) - (deriv (slitInfinityFactor Q.diskMap Q.slope) 0) - nlinarith - -lemma not_reflectedArcSet_of_mem_slit_domain {p : C(unitInterval,ℂ)} (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {z : ℂ} (hz : z∈upperHalfPlane \ range p) : z∉reflectedArcSet p := by - rintro (hp|⟨w,⟨t,rfl⟩,rfl⟩) - · exact hz.2 hp - · have hn : 0≤(p t).im := by - by_cases ht : t=0 - · simp only [ht,h0,zero_im,le_refl] - · exact (hH t ht).le - have hh : 0 < -(p t).im := by - have hx := hz.1 - change 0 < (conj (p t)).im at hx - simpa only [conj_im] using hx - linarith - -theorem SimpleSlitChart.forward_uniform_near {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 (hp t).le) + have hdiv : ‖(P.capacity:ℂ)/z‖=P.capacity/(2*R) := by + rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos P.capacity_pos,hz] + have htri := norm_sub_le_norm_sub_add_norm_sub (P.forward z-z) ((P.capacity:ℂ)/z) 0 + simp only [sub_zero] at htri + rw [hdiv] at htri + rw [hz] at hb + have he : 10*P.capacity*R/(2*R)^2+P.capacity/(2*R)=3*P.capacity/R := by + field_simp + ring + have hcost : 10*P.capacity*R/(2*R)^2+P.capacity/(2*R)≤3*R := by + rw [he] + apply (div_le_iff₀ hR).mpr + nlinarith + have hdelta : ‖P.forward z-z‖≤3*R := by linarith + have hsum := norm_le_norm_sub_add (P.forward z) z + rw [hz] at hsum + linarith + +lemma ReflectedCapacityChart.normalizing_constant_bound {p : C(unitInterval,ℂ)} + (Q : ReflectedCapacityChart p) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 (hp t).trans (by linarith)) hnot hzn.le + have hG := Q.chart.forward_on_two_radius hi h0 hH hR hp hzH hzn + have he : deriv (slitInfinityFactor Q.diskMap Q.slope) 0= + (Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)-Q.chart.forward z := by + rw [Q.forward_eq] + simp only [slitHydrodynamicMap] + ring + rw [he] + have ht := norm_sub_le ((Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)) + (Q.chart.forward z) + nlinarith + +lemma ReflectedCapacityChart.forward_bounded_in_two_radius {p : C(unitInterval,ℂ)} + (Q : ReflectedCapacityChart p) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 (hp t).trans (by linarith)) hz hzR + have hb := Q.normalizing_constant_bound hi h0 hH hR hp + rw [Q.forward_eq] + change ‖(Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)- + deriv (slitInfinityFactor Q.diskMap Q.slope) 0‖≤_ + have ht := norm_sub_le ((Q.slope:ℂ)*exteriorJoukowski (slitExteriorMap Q.diskMap z)) + (deriv (slitInfinityFactor Q.diskMap Q.slope) 0) + nlinarith + +lemma not_reflectedArcSet_of_mem_slit_domain {p : C(unitInterval,ℂ)} (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {z : ℂ} (hz : z∈upperHalfPlane \ range p) : z∉reflectedArcSet p := by + rintro (hp|⟨w,⟨t,rfl⟩,rfl⟩) + · exact hz.2 hp + · have hn : 0≤(p t).im := by + by_cases ht : t=0 + · simp only [ht,h0,zero_im,le_refl] + · exact (hH t ht).le + have hh : 0 < -(p t).im := by + have hx := hz.1 + change 0 < (conj (p t)).im at hx + simpa only [conj_im] using hx + linarith + +theorem SimpleSlitChart.forward_uniform_near {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 (hp t).le) - have hquad : 4*R^2≤‖z‖^2 := by nlinarith - have hfirst : 10*P.capacity*R/‖z‖^2≤5*R/2 := by - apply (div_le_iff₀ (sq_pos_of_pos hn)).mpr - nlinarith [mul_le_mul_of_nonneg_right hc hR.le, - mul_le_mul_of_nonneg_left hquad hR.le] - have hsecond : P.capacity/‖z‖≤R/2 := by - apply (div_le_iff₀ hn).mpr - nlinarith [mul_le_mul_of_nonneg_left hz hR.le] - have hdiv : ‖(P.capacity:ℂ)/z‖=P.capacity/‖z‖ := by - rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos P.capacity_pos] - have htri := norm_sub_le_norm_sub_add_norm_sub (P.forward z-z) ((P.capacity:ℂ)/z) 0 - simp only [sub_zero] at htri - rw [hdiv] at htri - linarith - -theorem SimpleSlitChart.inverse_uniform_bound {p : C(unitInterval,ℂ)} - (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) - (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) - {R : ℝ} (hR : 0 hp _) hw - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SlitUniformForwardBound +import OAI.Probability.SLE.Loewner.InitialChartIdentities + +namespace OAI + +/-! A common radius for the physical slit gives a common bound on all +inverse charts above a bounded coordinate set. -/ +noncomputable section +open Set Filter Function Complex Metric +open scoped Topology + +namespace SLEExactGauge + +lemma SimpleSlitChart.exterior_displacement_le_radius {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 (hp t).le) + have hquad : 4*R^2≤‖z‖^2 := by nlinarith + have hfirst : 10*P.capacity*R/‖z‖^2≤5*R/2 := by + apply (div_le_iff₀ (sq_pos_of_pos hn)).mpr + nlinarith [mul_le_mul_of_nonneg_right hc hR.le, + mul_le_mul_of_nonneg_left hquad hR.le] + have hsecond : P.capacity/‖z‖≤R/2 := by + apply (div_le_iff₀ hn).mpr + nlinarith [mul_le_mul_of_nonneg_left hz hR.le] + have hdiv : ‖(P.capacity:ℂ)/z‖=P.capacity/‖z‖ := by + rw [norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos P.capacity_pos] + have htri := norm_sub_le_norm_sub_add_norm_sub (P.forward z-z) ((P.capacity:ℂ)/z) 0 + simp only [sub_zero] at htri + rw [hdiv] at htri + linarith + +theorem SimpleSlitChart.inverse_uniform_bound {p : C(unitInterval,ℂ)} + (P : SimpleSlitChart p) (hi : Injective p) (h0 : p 0=0) + (hH : ∀ t : unitInterval, t≠0 → p t∈upperHalfPlane) + {R : ℝ} (hR : 0 hp _) hw + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmallDriverRise.lean b/lean/OAI/Probability/SLE/Loewner/SmallDriverRise.lean index 322def1f3ad4e020e9d4cb47cf15e0ed288437a4..3f3182ceb58e4a40f81c536112b5d89647bb8881 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmallDriverRise.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmallDriverRise.lean @@ -1,30 +1,30 @@ -import OAI.Probability.SLE.Loewner.LoewnerSmallDriverHeight - -namespace OAI - -/-! A fixed small-driver neighborhood forces a vertical rise; the radius -does not depend on a choice of continuous capacity-two trace. -/ -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem exists_small_driver_rise (T : ℝ≥0) (H : ℝ) - (hH : 0 ≤ H) (hHT : H < 2*Real.sqrt T) : - ∃ ε : ℝ, 0 < ε ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → - (∀ t ∈ Icc 0 T, |U t| ≤ ε) → ∃ t ∈ Icc 0 T, H < (γ t).im := by - classical - by_contra! hbad - let δ : ℕ → ℝ := fun n => 1/((n:ℝ)+1) - have hδ (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity - choose U γ G F hw hU hheight using fun n => hbad (δ n) (hδ n) - exact small_drivers_not_all_height_bounded U γ G F hw T H hH hHT δ - tendsto_one_div_add_atTop_nhds_zero_nat hU hheight - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerSmallDriverHeight + +namespace OAI + +/-! A fixed small-driver neighborhood forces a vertical rise; the radius +does not depend on a choice of continuous capacity-two trace. -/ +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem exists_small_driver_rise (T : ℝ≥0) (H : ℝ) + (hH : 0 ≤ H) (hHT : H < 2*Real.sqrt T) : + ∃ ε : ℝ, 0 < ε ∧ ∀ (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ), IsCapacityTwoWitness U γ G F → + (∀ t ∈ Icc 0 T, |U t| ≤ ε) → ∃ t ∈ Icc 0 T, H < (γ t).im := by + classical + by_contra! hbad + let δ : ℕ → ℝ := fun n => 1/((n:ℝ)+1) + have hδ (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity + choose U γ G F hw hU hheight using fun n => hbad (δ n) (hδ n) + exact small_drivers_not_all_height_bounded U γ G F hw T H hH hHT δ + tendsto_one_div_add_atTop_nhds_zero_nat hU hheight + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothDomainExhaustion.lean b/lean/OAI/Probability/SLE/Loewner/SmoothDomainExhaustion.lean index a8befdeba298fc0af50baa77289df3962e967ca8..a96c011ed3e8ead6c3b874fe6cc6d066510331fe 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothDomainExhaustion.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothDomainExhaustion.lean @@ -1,84 +1,84 @@ -import OAI.Probability.SLE.Geometry.CompactSmoothExtension - -namespace OAI - -/-! Nested compactly supported smooth cutoffs exhausting an open domain. -Only their positive sets need to be nested. -/ -noncomputable section -open Set Filter Function -open scoped Topology - -namespace SLEExactGauge - -structure SmoothDomainExhaustion (U : Set (ℝ×ℝ)) (z₀ : ℝ×ℝ) where - cutoff : ℕ → ℝ×ℝ → ℝ - smooth : ∀ n, ContDiff ℝ (⊤ : ℕ∞) (cutoff n) - compact : ∀ n, HasCompactSupport (cutoff n) - supported : ∀ n, tsupport (cutoff n) ⊆ U - based : ∀ n, cutoff n z₀=1 - bounds : ∀ n z, cutoff n z∈Icc 0 1 - increasing : Monotone (fun n => support (cutoff n)) - exhausts : ∀ K, IsCompact K → K ⊆ U → ∀ᶠ n in atTop, K ⊆ support (cutoff n) - -lemma exists_smoothDomainExhaustion (U : Set (ℝ×ℝ)) (hU : IsOpen U) - (z₀ : ℝ×ℝ) (hz₀ : z₀∈U) : Nonempty (SmoothDomainExhaustion U z₀) := by - classical - let : LocallyCompactSpace U := hU.locallyCompactSpace - let K := CompactExhaustion.choice U - let x₀ : U := ⟨z₀,hz₀⟩ - obtain ⟨N,hN⟩ := K.exists_mem x₀ - let V (n : ℕ) : Set (ℝ×ℝ) := Subtype.val '' interior (K (n+N+1)) - have hVo (n) : IsOpen (V n) := hU.isOpenEmbedding_subtypeVal.isOpenMap _ isOpen_interior - have hVK (n) : closure (V n) ⊆ Subtype.val '' K (n+N+1) := by - apply closure_minimal (image_mono interior_subset) - exact ((K.isCompact _).image continuous_subtype_val).isClosed - have hVc (n) : IsCompact (closure (V n)) := - ((K.isCompact _).image continuous_subtype_val).of_isClosed_subset isClosed_closure (hVK n) - have hVU (n) : closure (V n) ⊆ U := (hVK n).trans (by - rintro z ⟨x,hx,rfl⟩ - exact x.property) - have hV0 (n) : z₀∈V n := by - exact ⟨x₀,K.subset_interior (by omega) hN,rfl⟩ - have hVm : Monotone V := by - intro n m hnm - exact image_mono (interior_mono (K.subset (by omega))) - have hχ (n : ℕ) : ∃ χ : ℝ×ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) χ ∧ - range χ ⊆ Icc 0 1 ∧ support χ=V n ∧ χ z₀=1 := by - obtain ⟨χ,hχ,hb,hs,hone⟩ := exists_contDiff_support_eq_eq_one_iff - (n := (⊤ : ℕ∞)) (hVo n) (isClosed_singleton (x := z₀)) (singleton_subset_iff.mpr (hV0 n)) - exact ⟨χ,hχ,hb,hs,(hone z₀).mp (mem_singleton z₀)⟩ - choose χ hχ hb hs h0 using hχ - refine ⟨⟨χ,hχ,?_,?_,h0,?_,?_,?_⟩⟩ - · intro n - change IsCompact (closure (support (χ n))) - rw [hs n] - exact hVc n - · intro n - simpa only [tsupport,hs n] using hVU n - · intro n z - exact hb n (mem_range_self z) - · simpa only [hs] using hVm - · intro J hJ hJU - have him : (Subtype.val : U → ℝ×ℝ) '' ((Subtype.val : U → ℝ×ℝ) ⁻¹' J)=J := by - ext z - constructor - · rintro ⟨x,hx,rfl⟩ - exact hx - · intro hz - exact ⟨⟨z,hJU hz⟩,hz,rfl⟩ - have hJ' : IsCompact ((Subtype.val : U → ℝ×ℝ) ⁻¹' J) := by - apply (Topology.IsEmbedding.subtypeVal.isCompact_iff - (s := (Subtype.val : U → ℝ×ℝ) ⁻¹' J)).mpr - rw [him] - exact hJ - obtain ⟨m,hm⟩ := K.exists_superset_of_isCompact hJ' - filter_upwards [eventually_ge_atTop (m+1)] with n hn - intro z hz - rw [hs n] - exact ⟨⟨z,hJU hz⟩,K.subset_interior (by omega) (hm hz),rfl⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompactSmoothExtension + +namespace OAI + +/-! Nested compactly supported smooth cutoffs exhausting an open domain. +Only their positive sets need to be nested. -/ +noncomputable section +open Set Filter Function +open scoped Topology + +namespace SLEExactGauge + +structure SmoothDomainExhaustion (U : Set (ℝ×ℝ)) (z₀ : ℝ×ℝ) where + cutoff : ℕ → ℝ×ℝ → ℝ + smooth : ∀ n, ContDiff ℝ (⊤ : ℕ∞) (cutoff n) + compact : ∀ n, HasCompactSupport (cutoff n) + supported : ∀ n, tsupport (cutoff n) ⊆ U + based : ∀ n, cutoff n z₀=1 + bounds : ∀ n z, cutoff n z∈Icc 0 1 + increasing : Monotone (fun n => support (cutoff n)) + exhausts : ∀ K, IsCompact K → K ⊆ U → ∀ᶠ n in atTop, K ⊆ support (cutoff n) + +lemma exists_smoothDomainExhaustion (U : Set (ℝ×ℝ)) (hU : IsOpen U) + (z₀ : ℝ×ℝ) (hz₀ : z₀∈U) : Nonempty (SmoothDomainExhaustion U z₀) := by + classical + let : LocallyCompactSpace U := hU.locallyCompactSpace + let K := CompactExhaustion.choice U + let x₀ : U := ⟨z₀,hz₀⟩ + obtain ⟨N,hN⟩ := K.exists_mem x₀ + let V (n : ℕ) : Set (ℝ×ℝ) := Subtype.val '' interior (K (n+N+1)) + have hVo (n) : IsOpen (V n) := hU.isOpenEmbedding_subtypeVal.isOpenMap _ isOpen_interior + have hVK (n) : closure (V n) ⊆ Subtype.val '' K (n+N+1) := by + apply closure_minimal (image_mono interior_subset) + exact ((K.isCompact _).image continuous_subtype_val).isClosed + have hVc (n) : IsCompact (closure (V n)) := + ((K.isCompact _).image continuous_subtype_val).of_isClosed_subset isClosed_closure (hVK n) + have hVU (n) : closure (V n) ⊆ U := (hVK n).trans (by + rintro z ⟨x,hx,rfl⟩ + exact x.property) + have hV0 (n) : z₀∈V n := by + exact ⟨x₀,K.subset_interior (by omega) hN,rfl⟩ + have hVm : Monotone V := by + intro n m hnm + exact image_mono (interior_mono (K.subset (by omega))) + have hχ (n : ℕ) : ∃ χ : ℝ×ℝ → ℝ, ContDiff ℝ (⊤ : ℕ∞) χ ∧ + range χ ⊆ Icc 0 1 ∧ support χ=V n ∧ χ z₀=1 := by + obtain ⟨χ,hχ,hb,hs,hone⟩ := exists_contDiff_support_eq_eq_one_iff + (n := (⊤ : ℕ∞)) (hVo n) (isClosed_singleton (x := z₀)) (singleton_subset_iff.mpr (hV0 n)) + exact ⟨χ,hχ,hb,hs,(hone z₀).mp (mem_singleton z₀)⟩ + choose χ hχ hb hs h0 using hχ + refine ⟨⟨χ,hχ,?_,?_,h0,?_,?_,?_⟩⟩ + · intro n + change IsCompact (closure (support (χ n))) + rw [hs n] + exact hVc n + · intro n + simpa only [tsupport,hs n] using hVU n + · intro n z + exact hb n (mem_range_self z) + · simpa only [hs] using hVm + · intro J hJ hJU + have him : (Subtype.val : U → ℝ×ℝ) '' ((Subtype.val : U → ℝ×ℝ) ⁻¹' J)=J := by + ext z + constructor + · rintro ⟨x,hx,rfl⟩ + exact hx + · intro hz + exact ⟨⟨z,hJU hz⟩,hz,rfl⟩ + have hJ' : IsCompact ((Subtype.val : U → ℝ×ℝ) ⁻¹' J) := by + apply (Topology.IsEmbedding.subtypeVal.isCompact_iff + (s := (Subtype.val : U → ℝ×ℝ) ⁻¹' J)).mpr + rw [him] + exact hJ + obtain ⟨m,hm⟩ := K.exists_superset_of_isCompact hJ' + filter_upwards [eventually_ge_atTop (m+1)] with n hn + intro z hz + rw [hs n] + exact ⟨⟨z,hJU hz⟩,K.subset_interior (by omega) (hm hz),rfl⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothDriverLipschitz.lean b/lean/OAI/Probability/SLE/Loewner/SmoothDriverLipschitz.lean index 11097bcd5e2d3e3e685703d802eadc1ac321ed31..97918c0adc69744e5311a5b48607ebc1d4335cda 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothDriverLipschitz.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothDriverLipschitz.lean @@ -1,38 +1,38 @@ -import OAI.Probability.SLE.Loewner.SmoothRawInverse - -namespace OAI - -/-! A bounded derivative on nonnegative times gives a global Lipschitz -bound for the driver's constant extension to negative times. -/ -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem lipschitz_driverValue_of_bounded_derivative (b : Driver) (b' : ℝ → ℝ) - (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : - LipschitzWith L (driverValue b) := by - have hr : LipschitzOnWith L (driverValue b) (Ici 0) := by - apply (convex_Ici (0 : ℝ)).lipschitzOnWith_of_nnnorm_hasDerivWithin_le hd - intro t ht - exact_mod_cast (show ‖b' t‖ ≤ (L : ℝ) from by simpa only [Real.norm_eq_abs] using hb t ht) - have hnn : LipschitzWith L b := by - apply LipschitzWith.of_dist_le_mul - intro t s - simpa only [driverValue,Real.toNNReal_coe,NNReal.dist_eq,Real.dist_eq] using - hr.dist_le_mul (t : ℝ) t.coe_nonneg (s : ℝ) s.coe_nonneg - apply LipschitzWith.of_dist_le_mul - intro s t - change dist (b s.toNNReal) (b t.toNNReal) ≤ (L : ℝ)*dist s t - exact (hnn.dist_le_mul _ _).trans (by - simpa only [NNReal.coe_one,one_mul] using - mul_le_mul_of_nonneg_left (Real.lipschitzWith_toNNReal.dist_le_mul s t) L.coe_nonneg) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawInverse + +namespace OAI + +/-! A bounded derivative on nonnegative times gives a global Lipschitz +bound for the driver's constant extension to negative times. -/ +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem lipschitz_driverValue_of_bounded_derivative (b : Driver) (b' : ℝ → ℝ) + (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : + LipschitzWith L (driverValue b) := by + have hr : LipschitzOnWith L (driverValue b) (Ici 0) := by + apply (convex_Ici (0 : ℝ)).lipschitzOnWith_of_nnnorm_hasDerivWithin_le hd + intro t ht + exact_mod_cast (show ‖b' t‖ ≤ (L : ℝ) from by simpa only [Real.norm_eq_abs] using hb t ht) + have hnn : LipschitzWith L b := by + apply LipschitzWith.of_dist_le_mul + intro t s + simpa only [driverValue,Real.toNNReal_coe,NNReal.dist_eq,Real.dist_eq] using + hr.dist_le_mul (t : ℝ) t.coe_nonneg (s : ℝ) s.coe_nonneg + apply LipschitzWith.of_dist_le_mul + intro s t + change dist (b s.toNNReal) (b t.toNNReal) ≤ (L : ℝ)*dist s t + exact (hnn.dist_le_mul _ _).trans (by + simpa only [NNReal.coe_one,one_mul] using + mul_le_mul_of_nonneg_left (Real.lipschitzWith_toNNReal.dist_le_mul s t) L.coe_nonneg) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawDomain.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawDomain.lean index e6dfe9a936905afeaec0f42ae52067b39f21fc0a..7396c57bd5c32261518c9bdb5a94e622d32ee2da 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawDomain.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawDomain.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Loewner.RawLifetimeTrace -import OAI.Probability.SLE.Loewner.TargetSurvivalLifetime -import OAI.Probability.SLE.Loewner.SmoothRawGeometry - -namespace OAI - -/-! For a smooth driver the raw survival domain is the geometric slit -domain. Every lost interior point is a tip at its first lifetime. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem Target.exists_trace_time_of_not_surviving - (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) (T : ℝ≥0) - (hdead : ¬Surviving 1 b z T) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : - ∃ τ : ℝ≥0, 0 < τ ∧ τ ≤ T ∧ z=trace 1 b τ := by - obtain ⟨τ,hτ,hτT,hs⟩ := exists_survival_lifetime 1 b z hz T.coe_nonneg hdead - refine ⟨⟨τ,hτ.le⟩,hτ,hτT,?_⟩ - apply eq_trace_at_lifetime b z hz hτ - (fun t ht htτ => (hs t ht).mpr htτ) - (fun h => lt_irrefl τ ((hs τ hτ.le).mp h)) - b' (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg - (fun t ht => hb t ht.1) - -theorem Target.smooth_geometric_subset_raw_surviving - (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) : - survivingDomain (fun s : ℝ≥0 => trace 1 b s) T ⊆ {z | Surviving 1 b z T} := by - intro z hz - by_contra hn - obtain ⟨τ,_hτ,hτT,he⟩ := exists_trace_time_of_not_surviving b z hz.1.1 T hn b' L hd hb - exact hz.1.2 ⟨τ,⟨zero_le,hτT⟩,he.symm⟩ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} - -theorem RegularSLEPresentation.smooth_raw_surviving_eq_geometric - (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) - (b : Target.Driver) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) (hT : 0 < T) : - {z | Target.Surviving 1 b z T} = - survivingDomain (fun s : ℝ≥0 => Target.trace 1 b s) T := - Set.Subset.antisymm (a.smooth_raw_surviving_subset_geometric hκ b b' L hd hb T hT) - (Target.smooth_geometric_subset_raw_surviving b b' L hd hb T) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawLifetimeTrace +import OAI.Probability.SLE.Loewner.TargetSurvivalLifetime +import OAI.Probability.SLE.Loewner.SmoothRawGeometry + +namespace OAI + +/-! For a smooth driver the raw survival domain is the geometric slit +domain. Every lost interior point is a tip at its first lifetime. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem Target.exists_trace_time_of_not_surviving + (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) (T : ℝ≥0) + (hdead : ¬Surviving 1 b z T) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : + ∃ τ : ℝ≥0, 0 < τ ∧ τ ≤ T ∧ z=trace 1 b τ := by + obtain ⟨τ,hτ,hτT,hs⟩ := exists_survival_lifetime 1 b z hz T.coe_nonneg hdead + refine ⟨⟨τ,hτ.le⟩,hτ,hτT,?_⟩ + apply eq_trace_at_lifetime b z hz hτ + (fun t ht htτ => (hs t ht).mpr htτ) + (fun h => lt_irrefl τ ((hs τ hτ.le).mp h)) + b' (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg + (fun t ht => hb t ht.1) + +theorem Target.smooth_geometric_subset_raw_surviving + (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) : + survivingDomain (fun s : ℝ≥0 => trace 1 b s) T ⊆ {z | Surviving 1 b z T} := by + intro z hz + by_contra hn + obtain ⟨τ,_hτ,hτT,he⟩ := exists_trace_time_of_not_surviving b z hz.1.1 T hn b' L hd hb + exact hz.1.2 ⟨τ,⟨zero_le,hτT⟩,he.symm⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} + +theorem RegularSLEPresentation.smooth_raw_surviving_eq_geometric + (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) + (b : Target.Driver) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) (hT : 0 < T) : + {z | Target.Surviving 1 b z T} = + survivingDomain (fun s : ℝ≥0 => Target.trace 1 b s) T := + Set.Subset.antisymm (a.smooth_raw_surviving_subset_geometric hκ b b' L hd hb T hT) + (Target.smooth_geometric_subset_raw_surviving b b' L hd hb T) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawFreshness.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawFreshness.lean index 3005e3d9414370d3cd28af819c19744913b2be7f..c6669477b605109c2651be717643d676991d79e7 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawFreshness.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawFreshness.lean @@ -1,99 +1,99 @@ -import OAI.Probability.SLE.Loewner.RawReverseComposition -import OAI.Probability.SLE.Loewner.RawSurvivingMonotone -import OAI.Probability.SLE.Loewner.SmoothRawReverseEstimates -import OAI.Probability.SLE.Loewner.SmoothRawTrace -import OAI.Probability.SLE.Loewner.ReverseHeightUpper - -namespace OAI - -/-! The smooth raw tip survives at every earlier time. The proof uses -compactness of an intermediate reverse image, with a uniform positive -height, and the raw inverse composition identity. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem smooth_raw_tip_survives_short_past - (b : Driver) (T a : ℝ≥0) (ha : a ≤ T) (ha0 : 0 < a) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) - (hsmall : L^2*(a : ℝ) ≤ 1) : Surviving 1 b (trace 1 b T) (T-a : ℝ≥0) := by - let y (n : ℕ) : ℝ := 1/((n : ℝ)+1) - let w (n : ℕ) : ℂ := (b T : ℂ)+(y n : ℂ)*I - have hy (n : ℕ) : 0 < y n := by dsimp [y]; positivity - have hy1 (n : ℕ) : y n ≤ 1 := by - dsimp [y] - apply (div_le_one (by positivity)).mpr - linarith [Nat.cast_nonneg (α := ℝ) n] - have hw (n : ℕ) : w n ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy n - choose q hq0 hheight hqd heq using (fun n => exists_raw_reverse_trajectory b T (w n) (hw n)) - let v (n : ℕ) : ℂ := q n a - let z (n : ℕ) : ℂ := q n T - have hqp (n : ℕ) (t : ℝ) (ht : 0 ≤ t) : 0 < (q n t).im := - (hw n).trans_le (hheight n t ht) - have hqb (n : ℕ) : |(v n).re-b (T-a)| ≤ L*a ∧ 2*(a : ℝ) ≤ (v n).im^2 := by - have hh := smooth_raw_reverse_estimates b T b' hd hL hb (q n) - (fun t ht => (hqd n t ht.1).mono Icc_subset_Ici_self) - (fun t ht => hqp n t ht.1) (by simp [hq0,w]) a ⟨a.coe_nonneg,ha⟩ hsmall - simpa only [v,driverValue,show ((T : ℝ)-(a : ℝ)).toNNReal=T-a from rfl] using hh - have hvi (n : ℕ) : (v n).im ≤ 1+4*(a : ℝ) := by - have hh := reverse_height_square_le a.coe_nonneg (q n) - (fun t => driverValue b ((T : ℝ)-t)) - (fun t ht => (hqd n t ht.1).mono Icc_subset_Ici_self) - (fun t ht => hqp n t ht.1) - rw [hq0 n] at hh - have hwi : (w n).im=y n := by simp [w] - rw [hwi] at hh - change (v n).im^2 ≤ (y n)^2+4*(a : ℝ) at hh - nlinarith [hy n,hy1 n,hqp n a a.coe_nonneg,a.coe_nonneg,sq_nonneg (4*(a : ℝ))] - let R : ℝ := L*a+|b (T-a)|+(1+4*(a : ℝ)) - have hball (n : ℕ) : v n ∈ closedBall 0 R := by - rw [mem_closedBall,dist_zero_right] - have hre : |(v n).re| ≤ L*a+|b (T-a)| := by - calc - |(v n).re| = |((v n).re-b (T-a))+b (T-a)| := by congr 1; ring - _ ≤ |(v n).re-b (T-a)|+|b (T-a)| := abs_add_le _ _ - _ ≤ L*a+|b (T-a)| := add_le_add (hqb n).1 le_rfl - have hvp : 0 < (v n).im := hqp n a a.coe_nonneg - exact (Complex.norm_le_abs_re_add_abs_im (v n)).trans - (add_le_add hre (by simpa only [abs_of_pos hvp] using hvi n)) - obtain ⟨v₀,_,φ,hφ,hlim⟩ := (isCompact_closedBall (0 : ℂ) R).tendsto_subseq hball - have him := Complex.continuous_im.continuousAt.tendsto.comp hlim - have hnonneg : 0 ≤ v₀.im := ge_of_tendsto him - (Eventually.of_forall (fun n => (hqp (φ n) a a.coe_nonneg).le)) - have hlow : 2*(a : ℝ) ≤ v₀.im^2 := ge_of_tendsto (him.pow 2) - (Eventually.of_forall (fun n => (hqb (φ n)).2)) - have hv₀ : v₀ ∈ upperHalfPlane := by - have hap : 0 < (a : ℝ) := ha0 - change 0 < v₀.im - nlinarith - have hFv (n : ℕ) : inverseFlow 1 b (T-a : ℝ≥0) (v n)=z n := by - have hh := raw_reverse_tail_inverse b T (T-a) tsub_le_self (q n) (hqd n) (hqp n) - have hdiff : (T : ℝ)-((T-a : ℝ≥0) : ℝ)=(a : ℝ) := by rw [NNReal.coe_sub ha]; ring - simpa only [hdiff,v,z] using hh - have hFc : ContinuousAt (inverseFlow 1 b (T-a : ℝ≥0)) v₀ := - (continuousOn_inverseFlow 1 (by norm_num) b (T-a)).continuousAt - (isOpen_upperHalfPlane.mem_nhds hv₀) - have hFz : Tendsto (fun n => z (φ n)) atTop (𝓝 (inverseFlow 1 b (T-a : ℝ≥0) v₀)) := by - have hh := hFc.tendsto.comp hlim - convert hh using 1 - funext n - exact (hFv (φ n)).symm - have hy0 : Tendsto y atTop (𝓝[>] (0 : ℝ)) := tendsto_nhdsWithin_iff.mpr - ⟨tendsto_one_div_add_atTop_nhds_zero_nat,Eventually.of_forall hy⟩ - have hzlim : Tendsto z atTop (𝓝 (trace 1 b T)) := by - have hh := (tendsto_inverseFlow_smooth_trace b T b' hd hL hb).comp hy0 - convert hh using 1 - funext n - exact (heq n).symm - have hfinal := tendsto_nhds_unique hFz (hzlim.comp hφ.tendsto_atTop) - exact hfinal ▸ inverseFlow_surviving 1 (by norm_num) b (T-a) v₀ hv₀ - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawReverseComposition +import OAI.Probability.SLE.Loewner.RawSurvivingMonotone +import OAI.Probability.SLE.Loewner.SmoothRawReverseEstimates +import OAI.Probability.SLE.Loewner.SmoothRawTrace +import OAI.Probability.SLE.Loewner.ReverseHeightUpper + +namespace OAI + +/-! The smooth raw tip survives at every earlier time. The proof uses +compactness of an intermediate reverse image, with a uniform positive +height, and the raw inverse composition identity. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem smooth_raw_tip_survives_short_past + (b : Driver) (T a : ℝ≥0) (ha : a ≤ T) (ha0 : 0 < a) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) + (hsmall : L^2*(a : ℝ) ≤ 1) : Surviving 1 b (trace 1 b T) (T-a : ℝ≥0) := by + let y (n : ℕ) : ℝ := 1/((n : ℝ)+1) + let w (n : ℕ) : ℂ := (b T : ℂ)+(y n : ℂ)*I + have hy (n : ℕ) : 0 < y n := by dsimp [y]; positivity + have hy1 (n : ℕ) : y n ≤ 1 := by + dsimp [y] + apply (div_le_one (by positivity)).mpr + linarith [Nat.cast_nonneg (α := ℝ) n] + have hw (n : ℕ) : w n ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy n + choose q hq0 hheight hqd heq using (fun n => exists_raw_reverse_trajectory b T (w n) (hw n)) + let v (n : ℕ) : ℂ := q n a + let z (n : ℕ) : ℂ := q n T + have hqp (n : ℕ) (t : ℝ) (ht : 0 ≤ t) : 0 < (q n t).im := + (hw n).trans_le (hheight n t ht) + have hqb (n : ℕ) : |(v n).re-b (T-a)| ≤ L*a ∧ 2*(a : ℝ) ≤ (v n).im^2 := by + have hh := smooth_raw_reverse_estimates b T b' hd hL hb (q n) + (fun t ht => (hqd n t ht.1).mono Icc_subset_Ici_self) + (fun t ht => hqp n t ht.1) (by simp [hq0,w]) a ⟨a.coe_nonneg,ha⟩ hsmall + simpa only [v,driverValue,show ((T : ℝ)-(a : ℝ)).toNNReal=T-a from rfl] using hh + have hvi (n : ℕ) : (v n).im ≤ 1+4*(a : ℝ) := by + have hh := reverse_height_square_le a.coe_nonneg (q n) + (fun t => driverValue b ((T : ℝ)-t)) + (fun t ht => (hqd n t ht.1).mono Icc_subset_Ici_self) + (fun t ht => hqp n t ht.1) + rw [hq0 n] at hh + have hwi : (w n).im=y n := by simp [w] + rw [hwi] at hh + change (v n).im^2 ≤ (y n)^2+4*(a : ℝ) at hh + nlinarith [hy n,hy1 n,hqp n a a.coe_nonneg,a.coe_nonneg,sq_nonneg (4*(a : ℝ))] + let R : ℝ := L*a+|b (T-a)|+(1+4*(a : ℝ)) + have hball (n : ℕ) : v n ∈ closedBall 0 R := by + rw [mem_closedBall,dist_zero_right] + have hre : |(v n).re| ≤ L*a+|b (T-a)| := by + calc + |(v n).re| = |((v n).re-b (T-a))+b (T-a)| := by congr 1; ring + _ ≤ |(v n).re-b (T-a)|+|b (T-a)| := abs_add_le _ _ + _ ≤ L*a+|b (T-a)| := add_le_add (hqb n).1 le_rfl + have hvp : 0 < (v n).im := hqp n a a.coe_nonneg + exact (Complex.norm_le_abs_re_add_abs_im (v n)).trans + (add_le_add hre (by simpa only [abs_of_pos hvp] using hvi n)) + obtain ⟨v₀,_,φ,hφ,hlim⟩ := (isCompact_closedBall (0 : ℂ) R).tendsto_subseq hball + have him := Complex.continuous_im.continuousAt.tendsto.comp hlim + have hnonneg : 0 ≤ v₀.im := ge_of_tendsto him + (Eventually.of_forall (fun n => (hqp (φ n) a a.coe_nonneg).le)) + have hlow : 2*(a : ℝ) ≤ v₀.im^2 := ge_of_tendsto (him.pow 2) + (Eventually.of_forall (fun n => (hqb (φ n)).2)) + have hv₀ : v₀ ∈ upperHalfPlane := by + have hap : 0 < (a : ℝ) := ha0 + change 0 < v₀.im + nlinarith + have hFv (n : ℕ) : inverseFlow 1 b (T-a : ℝ≥0) (v n)=z n := by + have hh := raw_reverse_tail_inverse b T (T-a) tsub_le_self (q n) (hqd n) (hqp n) + have hdiff : (T : ℝ)-((T-a : ℝ≥0) : ℝ)=(a : ℝ) := by rw [NNReal.coe_sub ha]; ring + simpa only [hdiff,v,z] using hh + have hFc : ContinuousAt (inverseFlow 1 b (T-a : ℝ≥0)) v₀ := + (continuousOn_inverseFlow 1 (by norm_num) b (T-a)).continuousAt + (isOpen_upperHalfPlane.mem_nhds hv₀) + have hFz : Tendsto (fun n => z (φ n)) atTop (𝓝 (inverseFlow 1 b (T-a : ℝ≥0) v₀)) := by + have hh := hFc.tendsto.comp hlim + convert hh using 1 + funext n + exact (hFv (φ n)).symm + have hy0 : Tendsto y atTop (𝓝[>] (0 : ℝ)) := tendsto_nhdsWithin_iff.mpr + ⟨tendsto_one_div_add_atTop_nhds_zero_nat,Eventually.of_forall hy⟩ + have hzlim : Tendsto z atTop (𝓝 (trace 1 b T)) := by + have hh := (tendsto_inverseFlow_smooth_trace b T b' hd hL hb).comp hy0 + convert hh using 1 + funext n + exact (heq n).symm + have hfinal := tendsto_nhds_unique hFz (hzlim.comp hφ.tendsto_atTop) + exact hfinal ▸ inverseFlow_surviving 1 (by norm_num) b (T-a) v₀ hv₀ + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawGeometry.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawGeometry.lean index d3ea15af4551f32855bfac79a8bd0e2eb66d8ec8..9a61a01c69a79c17986122389d672a4f27d1eabf 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawGeometry.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawGeometry.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Loewner.SmoothRawSimple -import OAI.Probability.SLE.Loewner.SmoothRawTraceInterior -import OAI.Probability.SLE.Geometry.FiniteSimpleSlit - -namespace OAI - -/-! The raw smooth trace has the topology of a simple slit. Its raw -surviving domain embeds in the geometric unbounded component; equality -of these domains is treated separately. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} - -theorem RegularSLEPresentation.smooth_raw_free_complement - (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) - (b : Target.Driver) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) (hT : 0 < T) : - upperHalfPlane \ ((fun s : ℝ≥0 => Target.trace 1 b s) '' Icc 0 T) ⊆ - survivingDomain (fun s : ℝ≥0 => Target.trace 1 b s) T := by - apply finite_simple_slit_free (fun s => Target.trace 1 b s) - (Target.continuous_smooth_raw_trace b b' L hd hb) - (Target.smooth_trace_zero b b' L hd hb) T hT - (a.smooth_raw_trace_injective hκ b b' L hd hb).injOn - intro t ht - exact Target.smooth_raw_trace_mem_upperHalfPlane b t ht.1 b' - (fun s hs => (hd s hs.1).mono Icc_subset_Ici_self) L.coe_nonneg (fun s hs => hb s hs.1) - -theorem RegularSLEPresentation.smooth_raw_surviving_subset_geometric - (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) - (b : Target.Driver) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) (hT : 0 < T) : - {z | Target.Surviving 1 b z T} ⊆ - survivingDomain (fun s : ℝ≥0 => Target.trace 1 b s) T := by - intro z hz - apply a.smooth_raw_free_complement hκ b b' L hd hb T hT - refine ⟨Target.surviving_mem_upperHalfPlane 1 b z T T.coe_nonneg hz,?_⟩ - rintro ⟨s,hs,he⟩ - have hp := Target.surviving_of_le_time 1 b z s.coe_nonneg hs.2 hz - change Target.trace 1 b s=z at he - rw [← he] at hp - exact (a.smooth_raw_tip_not_surviving hκ b s b' - (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg - (fun t ht => hb t ht.1)) hp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawSimple +import OAI.Probability.SLE.Loewner.SmoothRawTraceInterior +import OAI.Probability.SLE.Geometry.FiniteSimpleSlit + +namespace OAI + +/-! The raw smooth trace has the topology of a simple slit. Its raw +surviving domain embeds in the geometric unbounded component; equality +of these domains is treated separately. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} + +theorem RegularSLEPresentation.smooth_raw_free_complement + (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) + (b : Target.Driver) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) (hT : 0 < T) : + upperHalfPlane \ ((fun s : ℝ≥0 => Target.trace 1 b s) '' Icc 0 T) ⊆ + survivingDomain (fun s : ℝ≥0 => Target.trace 1 b s) T := by + apply finite_simple_slit_free (fun s => Target.trace 1 b s) + (Target.continuous_smooth_raw_trace b b' L hd hb) + (Target.smooth_trace_zero b b' L hd hb) T hT + (a.smooth_raw_trace_injective hκ b b' L hd hb).injOn + intro t ht + exact Target.smooth_raw_trace_mem_upperHalfPlane b t ht.1 b' + (fun s hs => (hd s hs.1).mono Icc_subset_Ici_self) L.coe_nonneg (fun s hs => hb s hs.1) + +theorem RegularSLEPresentation.smooth_raw_surviving_subset_geometric + (a : RegularSLEPresentation P κ γ) (hκ : 0 < κ) + (b : Target.Driver) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (Target.driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) (hT : 0 < T) : + {z | Target.Surviving 1 b z T} ⊆ + survivingDomain (fun s : ℝ≥0 => Target.trace 1 b s) T := by + intro z hz + apply a.smooth_raw_free_complement hκ b b' L hd hb T hT + refine ⟨Target.surviving_mem_upperHalfPlane 1 b z T T.coe_nonneg hz,?_⟩ + rintro ⟨s,hs,he⟩ + have hp := Target.surviving_of_le_time 1 b z s.coe_nonneg hs.2 hz + change Target.trace 1 b s=z at he + rw [← he] at hp + exact (a.smooth_raw_tip_not_surviving hκ b s b' + (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg + (fun t ht => hb t ht.1)) hp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawInverse.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawInverse.lean index 7e3acd82f5ee6a53edf64075889215d1797b11f4..b37490cb6cf0b385c0ce605919184029f7c2e5a3 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawInverse.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawInverse.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Loewner.ReverseTrajectoryExistence -import OAI.Probability.SLE.Loewner.SmoothReverseContraction -import OAI.Probability.SLE.Loewner.TargetInverseContinuity - -namespace OAI - -/-! A boundary Cauchy estimate for the exact raw inverse map, without -assuming the existence of a capacity trace. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -lemma exists_raw_reverse_trajectory (b : Driver) (T : ℝ≥0) (w : ℂ) - (hw : w ∈ upperHalfPlane) : - ∃ q : ℝ → ℂ, q 0=w ∧ - (∀ t, 0 ≤ t → w.im ≤ (q t).im) ∧ - (∀ t, 0 ≤ t → HasDerivWithinAt q - (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici 0) t) ∧ - inverseFlow 1 b T w=q T := by - have hu : Continuous (fun t : ℝ => driverValue b ((T : ℝ)-t)) := - b.continuous.comp (continuous_real_toNNReal.comp (continuous_const.sub continuous_id)) - obtain ⟨q,hq0,hheight,hqd⟩ := exists_reverse_trajectory _ hu w hw - refine ⟨q,hq0,hheight,hqd,?_⟩ - have hh := inverseFlow_eq_reverse_trajectory 1 (by norm_num) b T w - (w.im/2) (half_pos hw) (half_le_self hw.le) q hq0 (fun t ht => ?_) - · simpa only [div_one] using hh - · have he : reverseField (w.im/2) (driverValue b ((T : ℝ)-1*t)) (q t) = - -2/(q t-(driverValue b ((T : ℝ)-t) : ℂ)) := by - rw [one_mul,reverseField,reverseCenter_eq ((half_le_self hw.le).trans (hheight t ht))] - rfl - rw [he] - exact hqd t ht - -theorem inverseFlow_vertical_contraction (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) - (hsmall : L^2*(T : ℝ) ≤ 1) {y v : ℝ} (hy : 0 < y) (hv : 0 < v) : - dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) - (inverseFlow 1 b T ((b T : ℂ)+(v : ℂ)*I)) ≤ |y-v| := by - obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T - ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) - obtain ⟨r,hr0,hry,hrd,her⟩ := exists_raw_reverse_trajectory b T - ((b T : ℂ)+(v : ℂ)*I) (by simpa [upperHalfPlane] using hv) - let u (s : ℝ) := driverValue b ((T : ℝ)-s) - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := - ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 - simp only [mul_neg,mul_one] - have hh := smooth_reverse_trajectories_contract T.coe_nonneg hL q r u - (fun s => -b' ((T : ℝ)-s)) - (fun s hs => (hqd s hs.1).mono Icc_subset_Ici_self) - (fun s hs => (hrd s hs.1).mono Icc_subset_Ici_self) hu - (fun s hs => by simpa using hb _ (hreflect s hs)) - (fun s hs => hy.trans_le (by simpa using hqy s hs.1)) - (fun s hs => hv.trans_le (by simpa using hry s hs.1)) - (by simp [hq0,u,driverValue]) (by simp [hr0,u,driverValue]) hsmall - rw [heq,her] - convert hh using 1 - rw [hq0,hr0,dist_eq_norm] - have he : ((b T : ℂ)+(y : ℂ)*I)-((b T : ℂ)+(v : ℂ)*I)=((y-v : ℝ) : ℂ)*I := by - push_cast - ring - rw [he,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one] - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReverseTrajectoryExistence +import OAI.Probability.SLE.Loewner.SmoothReverseContraction +import OAI.Probability.SLE.Loewner.TargetInverseContinuity + +namespace OAI + +/-! A boundary Cauchy estimate for the exact raw inverse map, without +assuming the existence of a capacity trace. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +lemma exists_raw_reverse_trajectory (b : Driver) (T : ℝ≥0) (w : ℂ) + (hw : w ∈ upperHalfPlane) : + ∃ q : ℝ → ℂ, q 0=w ∧ + (∀ t, 0 ≤ t → w.im ≤ (q t).im) ∧ + (∀ t, 0 ≤ t → HasDerivWithinAt q + (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Ici 0) t) ∧ + inverseFlow 1 b T w=q T := by + have hu : Continuous (fun t : ℝ => driverValue b ((T : ℝ)-t)) := + b.continuous.comp (continuous_real_toNNReal.comp (continuous_const.sub continuous_id)) + obtain ⟨q,hq0,hheight,hqd⟩ := exists_reverse_trajectory _ hu w hw + refine ⟨q,hq0,hheight,hqd,?_⟩ + have hh := inverseFlow_eq_reverse_trajectory 1 (by norm_num) b T w + (w.im/2) (half_pos hw) (half_le_self hw.le) q hq0 (fun t ht => ?_) + · simpa only [div_one] using hh + · have he : reverseField (w.im/2) (driverValue b ((T : ℝ)-1*t)) (q t) = + -2/(q t-(driverValue b ((T : ℝ)-t) : ℂ)) := by + rw [one_mul,reverseField,reverseCenter_eq ((half_le_self hw.le).trans (hheight t ht))] + rfl + rw [he] + exact hqd t ht + +theorem inverseFlow_vertical_contraction (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) + (hsmall : L^2*(T : ℝ) ≤ 1) {y v : ℝ} (hy : 0 < y) (hv : 0 < v) : + dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) + (inverseFlow 1 b T ((b T : ℂ)+(v : ℂ)*I)) ≤ |y-v| := by + obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T + ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) + obtain ⟨r,hr0,hry,hrd,her⟩ := exists_raw_reverse_trajectory b T + ((b T : ℂ)+(v : ℂ)*I) (by simpa [upperHalfPlane] using hv) + let u (s : ℝ) := driverValue b ((T : ℝ)-s) + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := + ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 + simp only [mul_neg,mul_one] + have hh := smooth_reverse_trajectories_contract T.coe_nonneg hL q r u + (fun s => -b' ((T : ℝ)-s)) + (fun s hs => (hqd s hs.1).mono Icc_subset_Ici_self) + (fun s hs => (hrd s hs.1).mono Icc_subset_Ici_self) hu + (fun s hs => by simpa using hb _ (hreflect s hs)) + (fun s hs => hy.trans_le (by simpa using hqy s hs.1)) + (fun s hs => hv.trans_le (by simpa using hry s hs.1)) + (by simp [hq0,u,driverValue]) (by simp [hr0,u,driverValue]) hsmall + rw [heq,her] + convert hh using 1 + rw [hq0,hr0,dist_eq_norm] + have he : ((b T : ℂ)+(y : ℂ)*I)-((b T : ℂ)+(v : ℂ)*I)=((y-v : ℝ) : ℂ)*I := by + push_cast + ring + rw [he,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one] + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawReverseEstimates.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawReverseEstimates.lean index 566901074d868ed296ecf97b2f79736ab40896e4..b78f75b5b00e6337223995be072d31c779e17276 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawReverseEstimates.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawReverseEstimates.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Loewner.SmoothRawInverse - -namespace OAI - -/-! Intermediate reverse-time estimates for constructing the raw slit. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -lemma smooth_raw_reverse_estimates (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) - (q : ℝ → ℂ) - (hq : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q - (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Icc 0 (T : ℝ)) t) - (hp : ∀ t ∈ Icc 0 (T : ℝ), 0 < (q t).im) (hq0 : (q 0).re=b T) - (a : ℝ) (ha : a ∈ Icc 0 (T : ℝ)) (hsmall : L^2*a ≤ 1) : - |(q a).re-driverValue b ((T : ℝ)-a)| ≤ L*a ∧ 2*a ≤ (q a).im^2 := by - let u (s : ℝ) := driverValue b ((T : ℝ)-s) - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := - ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 - simp only [mul_neg,mul_one] - have hstart : (q 0).re=u 0 := by simpa only [u,sub_zero,driverValue,Real.toNNReal_coe] using hq0 - have hb' : ∀ t ∈ Icc 0 (T : ℝ), |-b' ((T : ℝ)-t)| ≤ L := - fun t ht => by simpa only [abs_neg] using hb _ (hreflect t ht) - refine ⟨(reverse_centered_coordinates hL q u _ hq hu hb' hp hstart).1 a ha,?_⟩ - have hsub : Icc 0 a ⊆ Icc 0 (T : ℝ) := Icc_subset_Icc_right ha.2 - exact reverse_loewner_height_lower hL q u _ - (fun s hs => (hq s (hsub hs)).mono hsub) - (fun s hs => (hu s (hsub hs)).mono hsub) - (fun s hs => hb' s (hsub hs)) (fun s hs => hp s (hsub hs)) - hstart hsmall a ⟨ha.1,le_rfl⟩ - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawInverse + +namespace OAI + +/-! Intermediate reverse-time estimates for constructing the raw slit. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +lemma smooth_raw_reverse_estimates (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) + (q : ℝ → ℂ) + (hq : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt q + (-2/(q t-(driverValue b ((T : ℝ)-t) : ℂ))) (Icc 0 (T : ℝ)) t) + (hp : ∀ t ∈ Icc 0 (T : ℝ), 0 < (q t).im) (hq0 : (q 0).re=b T) + (a : ℝ) (ha : a ∈ Icc 0 (T : ℝ)) (hsmall : L^2*a ≤ 1) : + |(q a).re-driverValue b ((T : ℝ)-a)| ≤ L*a ∧ 2*a ≤ (q a).im^2 := by + let u (s : ℝ) := driverValue b ((T : ℝ)-s) + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := + ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 + simp only [mul_neg,mul_one] + have hstart : (q 0).re=u 0 := by simpa only [u,sub_zero,driverValue,Real.toNNReal_coe] using hq0 + have hb' : ∀ t ∈ Icc 0 (T : ℝ), |-b' ((T : ℝ)-t)| ≤ L := + fun t ht => by simpa only [abs_neg] using hb _ (hreflect t ht) + refine ⟨(reverse_centered_coordinates hL q u _ hq hu hb' hp hstart).1 a ha,?_⟩ + have hsub : Icc 0 a ⊆ Icc 0 (T : ℝ) := Icc_subset_Icc_right ha.2 + exact reverse_loewner_height_lower hL q u _ + (fun s hs => (hq s (hsub hs)).mono hsub) + (fun s hs => (hu s (hsub hs)).mono hsub) + (fun s hs => hb' s (hsub hs)) (fun s hs => hp s (hsub hs)) + hstart hsmall a ⟨ha.1,le_rfl⟩ + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawSimple.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawSimple.lean index 8827384c4234c195a6631a9a71b9504227b21e0f..e3d2053bc11143897db2f698d18acd5228a5d1ce 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawSimple.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawSimple.lean @@ -1,68 +1,68 @@ -import OAI.Probability.SLE.Loewner.SmoothRawFreshness -import OAI.Probability.SLE.Loewner.RawSurvivingAnalytic -import OAI.Probability.SLE.Loewner.SmoothRawTraceContinuous - -namespace OAI - -/-! Every positive-time raw tip is new: it survives all earlier times -and fails to survive its own time. Hence the smooth raw trace is simple. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma Target.smooth_raw_tip_survives_past (b : Driver) (T S : ℝ≥0) (hST : S Target.trace 1 b T) := by - have hdT (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := - (hd t ht.1).mono (show Icc 0 (T : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) - have hbT (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := hb t ht.1 - intro S T he - change Target.trace 1 b S=Target.trace 1 b T at he - rcases lt_trichotomy S T with hST | heq | hTS - · have hp := Target.smooth_raw_tip_survives_past b T S hST b' (hdT T) L.coe_nonneg (hbT T) - rw [← he] at hp - exact False.elim ((a.smooth_raw_tip_not_surviving hκ b S b' (hdT S) L.coe_nonneg (hbT S)) hp) - · exact heq - · have hp := Target.smooth_raw_tip_survives_past b S T hTS b' (hdT S) L.coe_nonneg (hbT S) - rw [he] at hp - exact False.elim ((a.smooth_raw_tip_not_surviving hκ b T b' (hdT T) L.coe_nonneg (hbT T)) hp) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawFreshness +import OAI.Probability.SLE.Loewner.RawSurvivingAnalytic +import OAI.Probability.SLE.Loewner.SmoothRawTraceContinuous + +namespace OAI + +/-! Every positive-time raw tip is new: it survives all earlier times +and fails to survive its own time. Hence the smooth raw trace is simple. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma Target.smooth_raw_tip_survives_past (b : Driver) (T S : ℝ≥0) (hST : S Target.trace 1 b T) := by + have hdT (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := + (hd t ht.1).mono (show Icc 0 (T : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) + have hbT (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := hb t ht.1 + intro S T he + change Target.trace 1 b S=Target.trace 1 b T at he + rcases lt_trichotomy S T with hST | heq | hTS + · have hp := Target.smooth_raw_tip_survives_past b T S hST b' (hdT T) L.coe_nonneg (hbT T) + rw [← he] at hp + exact False.elim ((a.smooth_raw_tip_not_surviving hκ b S b' (hdT S) L.coe_nonneg (hbT S)) hp) + · exact heq + · have hp := Target.smooth_raw_tip_survives_past b S T hTS b' (hdT S) L.coe_nonneg (hbT S) + rw [he] at hp + exact False.elim ((a.smooth_raw_tip_not_surviving hκ b T b' (hdT T) L.coe_nonneg (hbT T)) hp) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawTipGlobal.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawTipGlobal.lean index 96fed73231f4aca9547c63483e24e3030627c3e4..ddcc5489beaff8e3934dc22591105e878f859529 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawTipGlobal.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawTipGlobal.lean @@ -1,97 +1,97 @@ -import OAI.Probability.SLE.Loewner.SmoothRawTipLimit -import OAI.Probability.SLE.Loewner.ReverseLongContraction - -namespace OAI - -/-! A bounded derivative gives a finite radial tip limit at every time. -The proof first gains a positive height on a short terminal interval, -then uses ordinary interior stability for the remaining reverse time. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem inverseFlow_vertical_distance_of_smooth_driver - (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) - (b' t) (Icc 0 (T : ℝ)) t) - {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) - (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) - {y v : ℝ} (hy : 0 < y) (hv : 0 < v) : - dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) - (inverseFlow 1 b T ((b T : ℂ)+(v : ℂ)*I)) ≤ - |y-v| * Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)) := by - obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T - ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) - obtain ⟨r,hr0,hry,hrd,her⟩ := exists_raw_reverse_trajectory b T - ((b T : ℂ)+(v : ℂ)*I) (by simpa [upperHalfPlane] using hv) - let u (s : ℝ) := driverValue b ((T : ℝ)-s) - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := - ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 - simp only [mul_neg,mul_one] - have hh := smooth_reverse_long_distance ha haT hL q r u - (fun s => -b' ((T : ℝ)-s)) - (fun s hs => (hqd s hs.1).mono Icc_subset_Ici_self) - (fun s hs => (hrd s hs.1).mono Icc_subset_Ici_self) hu - (fun s hs => by simpa using hb _ (hreflect s hs)) - (fun s hs => hy.trans_le (by simpa using hqy s hs.1)) - (fun s hs => hv.trans_le (by simpa using hry s hs.1)) - (by simp [hq0,u,driverValue]) (by simp [hr0,u,driverValue]) hsmall - rw [heq,her] - convert hh using 1 - rw [hq0,hr0,dist_eq_norm] - have he : ((b T : ℂ)+(y : ℂ)*I)-((b T : ℂ)+(v : ℂ)*I)=((y-v : ℝ) : ℂ)*I := by - push_cast - ring - rw [he,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one] - -theorem inverseFlow_tip_limit_of_smooth_window - (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) - (b' t) (Icc 0 (T : ℝ)) t) - {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) - (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) : - ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) - (𝓝[>] 0) (𝓝 z) := by - let C : ℝ≥0 := ⟨Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)),(Real.exp_pos _).le⟩ - refine exists_right_limit_of_complex_lipschitz - (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) C ?_ - intro y hy v hv - exact (inverseFlow_vertical_distance_of_smooth_driver b T b' hd hL ha haT hb hsmall hy hv).trans_eq - (mul_comm _ _) - -lemma exists_smooth_window (T : ℝ) (hT : 0 < T) (L : ℝ) : - ∃ a : ℝ, 0 < a ∧ a ≤ T ∧ L^2*a ≤ 1 := by - let a : ℝ := min T (1/(L^2+1)) - have ha : 0 < a := lt_min hT (by positivity) - refine ⟨a,ha,min_le_left _ _,?_⟩ - have hden : 0 < L^2+1 := by positivity - have h : a*(L^2+1) ≤ 1 := (le_div_iff₀ hden).mp (min_le_right T (1/(L^2+1))) - nlinarith [ha.le] - -theorem inverseFlow_tip_limit_of_smooth_driver - (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : - ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) - (𝓝[>] 0) (𝓝 z) := by - by_cases hT : T=0 - · subst T - exact inverseFlow_tip_limit_of_short_smooth_driver b 0 b' hd hL hb (by simp) - obtain ⟨a,ha,haT,hsmall⟩ := exists_smooth_window T - (by exact_mod_cast (pos_iff_ne_zero.mpr hT)) L - exact inverseFlow_tip_limit_of_smooth_window b T b' hd hL ha haT hb hsmall - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawTipLimit +import OAI.Probability.SLE.Loewner.ReverseLongContraction + +namespace OAI + +/-! A bounded derivative gives a finite radial tip limit at every time. +The proof first gains a positive height on a short terminal interval, +then uses ordinary interior stability for the remaining reverse time. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem inverseFlow_vertical_distance_of_smooth_driver + (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) + (b' t) (Icc 0 (T : ℝ)) t) + {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) + (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) + {y v : ℝ} (hy : 0 < y) (hv : 0 < v) : + dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) + (inverseFlow 1 b T ((b T : ℂ)+(v : ℂ)*I)) ≤ + |y-v| * Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)) := by + obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T + ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) + obtain ⟨r,hr0,hry,hrd,her⟩ := exists_raw_reverse_trajectory b T + ((b T : ℂ)+(v : ℂ)*I) (by simpa [upperHalfPlane] using hv) + let u (s : ℝ) := driverValue b ((T : ℝ)-s) + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := + ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 + simp only [mul_neg,mul_one] + have hh := smooth_reverse_long_distance ha haT hL q r u + (fun s => -b' ((T : ℝ)-s)) + (fun s hs => (hqd s hs.1).mono Icc_subset_Ici_self) + (fun s hs => (hrd s hs.1).mono Icc_subset_Ici_self) hu + (fun s hs => by simpa using hb _ (hreflect s hs)) + (fun s hs => hy.trans_le (by simpa using hqy s hs.1)) + (fun s hs => hv.trans_le (by simpa using hry s hs.1)) + (by simp [hq0,u,driverValue]) (by simp [hr0,u,driverValue]) hsmall + rw [heq,her] + convert hh using 1 + rw [hq0,hr0,dist_eq_norm] + have he : ((b T : ℂ)+(y : ℂ)*I)-((b T : ℂ)+(v : ℂ)*I)=((y-v : ℝ) : ℂ)*I := by + push_cast + ring + rw [he,norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one] + +theorem inverseFlow_tip_limit_of_smooth_window + (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) + (b' t) (Icc 0 (T : ℝ)) t) + {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) + (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) : + ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) + (𝓝[>] 0) (𝓝 z) := by + let C : ℝ≥0 := ⟨Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)),(Real.exp_pos _).le⟩ + refine exists_right_limit_of_complex_lipschitz + (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) C ?_ + intro y hy v hv + exact (inverseFlow_vertical_distance_of_smooth_driver b T b' hd hL ha haT hb hsmall hy hv).trans_eq + (mul_comm _ _) + +lemma exists_smooth_window (T : ℝ) (hT : 0 < T) (L : ℝ) : + ∃ a : ℝ, 0 < a ∧ a ≤ T ∧ L^2*a ≤ 1 := by + let a : ℝ := min T (1/(L^2+1)) + have ha : 0 < a := lt_min hT (by positivity) + refine ⟨a,ha,min_le_left _ _,?_⟩ + have hden : 0 < L^2+1 := by positivity + have h : a*(L^2+1) ≤ 1 := (le_div_iff₀ hden).mp (min_le_right T (1/(L^2+1))) + nlinarith [ha.le] + +theorem inverseFlow_tip_limit_of_smooth_driver + (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : + ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) + (𝓝[>] 0) (𝓝 z) := by + by_cases hT : T=0 + · subst T + exact inverseFlow_tip_limit_of_short_smooth_driver b 0 b' hd hL hb (by simp) + obtain ⟨a,ha,haT,hsmall⟩ := exists_smooth_window T + (by exact_mod_cast (pos_iff_ne_zero.mpr hT)) L + exact inverseFlow_tip_limit_of_smooth_window b T b' hd hL ha haT hb hsmall + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawTipLimit.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawTipLimit.lean index 094334993c7519ce8de499afae3dd2298c547c0a..905a81a758d453c11394a4a937adc4078dc2b03e 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawTipLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawTipLimit.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Loewner.SmoothRawInverse - -namespace OAI - -/-! The exact raw inverse has a finite radial tip limit on a short -smooth-driver interval. No geometric witness is assumed. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma exists_right_limit_of_complex_lipschitz (f : ℝ → ℂ) (K : ℝ≥0) - (hf : ∀ x > 0, ∀ y > 0, dist (f x) (f y) ≤ K*|x-y|) : - ∃ z : ℂ, Tendsto f (𝓝[>] (0 : ℝ)) (𝓝 z) := by - have hre : LipschitzOnWith K (fun x => (f x).re) (Ioi (0 : ℝ)) := by - apply LipschitzOnWith.of_dist_le_mul - intro x hx y hy - calc - dist (f x).re (f y).re = |(f x-f y).re| := by simp [Real.dist_eq] - _ ≤ ‖f x-f y‖ := Complex.abs_re_le_norm _ - _ ≤ K*|x-y| := by simpa only [dist_eq_norm] using hf x hx y hy - _ = K*dist x y := by simp [Real.dist_eq] - have him : LipschitzOnWith K (fun x => (f x).im) (Ioi (0 : ℝ)) := by - apply LipschitzOnWith.of_dist_le_mul - intro x hx y hy - calc - dist (f x).im (f y).im = |(f x-f y).im| := by simp [Real.dist_eq] - _ ≤ ‖f x-f y‖ := Complex.abs_im_le_norm _ - _ ≤ K*|x-y| := by simpa only [dist_eq_norm] using hf x hx y hy - _ = K*dist x y := by simp [Real.dist_eq] - obtain ⟨r,hr,her⟩ := hre.extend_real - obtain ⟨i,hi,hei⟩ := him.extend_real - let g (x : ℝ) : ℂ := (r x : ℂ)+(i x : ℂ)*I - have hg : Continuous g := - (Complex.continuous_ofReal.comp hr.continuous).add - ((Complex.continuous_ofReal.comp hi.continuous).mul continuous_const) - refine ⟨g 0,?_⟩ - have hh : Tendsto g (𝓝[>] (0 : ℝ)) (𝓝 (g 0)) := - (hg.tendsto 0).mono_left nhdsWithin_le_nhds - apply hh.congr' - filter_upwards [self_mem_nhdsWithin] with x hx - apply Complex.ext <;> simp [g,← her hx,← hei hx] - -namespace Target - -theorem inverseFlow_tip_limit_of_short_smooth_driver - (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) - (hsmall : L^2*(T : ℝ) ≤ 1) : - ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) - (𝓝[>] 0) (𝓝 z) := - exists_right_limit_of_complex_lipschitz _ 1 (fun _ hy _ hv => by - simpa only [NNReal.coe_one,one_mul] using - inverseFlow_vertical_contraction b T b' hd hL hb hsmall hy hv) - -end Target -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawInverse + +namespace OAI + +/-! The exact raw inverse has a finite radial tip limit on a short +smooth-driver interval. No geometric witness is assumed. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma exists_right_limit_of_complex_lipschitz (f : ℝ → ℂ) (K : ℝ≥0) + (hf : ∀ x > 0, ∀ y > 0, dist (f x) (f y) ≤ K*|x-y|) : + ∃ z : ℂ, Tendsto f (𝓝[>] (0 : ℝ)) (𝓝 z) := by + have hre : LipschitzOnWith K (fun x => (f x).re) (Ioi (0 : ℝ)) := by + apply LipschitzOnWith.of_dist_le_mul + intro x hx y hy + calc + dist (f x).re (f y).re = |(f x-f y).re| := by simp [Real.dist_eq] + _ ≤ ‖f x-f y‖ := Complex.abs_re_le_norm _ + _ ≤ K*|x-y| := by simpa only [dist_eq_norm] using hf x hx y hy + _ = K*dist x y := by simp [Real.dist_eq] + have him : LipschitzOnWith K (fun x => (f x).im) (Ioi (0 : ℝ)) := by + apply LipschitzOnWith.of_dist_le_mul + intro x hx y hy + calc + dist (f x).im (f y).im = |(f x-f y).im| := by simp [Real.dist_eq] + _ ≤ ‖f x-f y‖ := Complex.abs_im_le_norm _ + _ ≤ K*|x-y| := by simpa only [dist_eq_norm] using hf x hx y hy + _ = K*dist x y := by simp [Real.dist_eq] + obtain ⟨r,hr,her⟩ := hre.extend_real + obtain ⟨i,hi,hei⟩ := him.extend_real + let g (x : ℝ) : ℂ := (r x : ℂ)+(i x : ℂ)*I + have hg : Continuous g := + (Complex.continuous_ofReal.comp hr.continuous).add + ((Complex.continuous_ofReal.comp hi.continuous).mul continuous_const) + refine ⟨g 0,?_⟩ + have hh : Tendsto g (𝓝[>] (0 : ℝ)) (𝓝 (g 0)) := + (hg.tendsto 0).mono_left nhdsWithin_le_nhds + apply hh.congr' + filter_upwards [self_mem_nhdsWithin] with x hx + apply Complex.ext <;> simp [g,← her hx,← hei hx] + +namespace Target + +theorem inverseFlow_tip_limit_of_short_smooth_driver + (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) + (hsmall : L^2*(T : ℝ) ≤ 1) : + ∃ z : ℂ, Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) + (𝓝[>] 0) (𝓝 z) := + exists_right_limit_of_complex_lipschitz _ 1 (fun _ hy _ hv => by + simpa only [NNReal.coe_one,one_mul] using + inverseFlow_vertical_contraction b T b' hd hL hb hsmall hy hv) + +end Target +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawTrace.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawTrace.lean index ff89ca13c07f8202e55afe30ebd887d4f39976dd..90a2c3b2d937b8fb6e38503c2b0c8f1300adc253 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawTrace.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawTrace.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Loewner.SmoothRawTipGlobal - -namespace OAI - -/-! The prescribed raw trace is the actual radial limit for smooth -drivers. A quantitative radial error bound is retained for time continuity. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem tendsto_inverseFlow_smooth_trace (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : - Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) - (𝓝[>] 0) (𝓝 (trace 1 b T)) := by - have he := inverseFlow_tip_limit_of_smooth_driver b T b' hd hL hb - simpa only [trace,one_mul,driverValue,Real.toNNReal_coe] using tendsto_nhds_limUnder he - -theorem inverseFlow_smooth_trace_error (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) - (b' t) (Icc 0 (T : ℝ)) t) - {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) - (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) - {y : ℝ} (hy : 0 < y) : - dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) (trace 1 b T) ≤ - y*Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)) := by - have ht := tendsto_inverseFlow_smooth_trace b T b' hd hL hb - have hr : Tendsto (fun v : ℝ => |y-v| * - Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a))) (𝓝[>] 0) - (𝓝 (y*Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)))) := by - have hid : Tendsto (fun v : ℝ => v) (𝓝[>] 0) (𝓝 (0 : ℝ)) := nhdsWithin_le_nhds - have h := (hid.const_sub y).abs.mul_const - (Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a))) - simpa only [sub_zero,abs_of_pos hy] using h - apply le_of_tendsto_of_tendsto (tendsto_const_nhds.dist ht) hr - filter_upwards [self_mem_nhdsWithin] with v hv - exact inverseFlow_vertical_distance_of_smooth_driver b T b' hd hL ha haT hb hsmall hy hv - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawTipGlobal + +namespace OAI + +/-! The prescribed raw trace is the actual radial limit for smooth +drivers. A quantitative radial error bound is retained for time continuity. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem tendsto_inverseFlow_smooth_trace (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : + Tendsto (fun y : ℝ => inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) + (𝓝[>] 0) (𝓝 (trace 1 b T)) := by + have he := inverseFlow_tip_limit_of_smooth_driver b T b' hd hL hb + simpa only [trace,one_mul,driverValue,Real.toNNReal_coe] using tendsto_nhds_limUnder he + +theorem inverseFlow_smooth_trace_error (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => b s.toNNReal) + (b' t) (Icc 0 (T : ℝ)) t) + {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) + (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) + {y : ℝ} (hy : 0 < y) : + dist (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)) (trace 1 b T) ≤ + y*Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)) := by + have ht := tendsto_inverseFlow_smooth_trace b T b' hd hL hb + have hr : Tendsto (fun v : ℝ => |y-v| * + Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a))) (𝓝[>] 0) + (𝓝 (y*Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a)))) := by + have hid : Tendsto (fun v : ℝ => v) (𝓝[>] 0) (𝓝 (0 : ℝ)) := nhdsWithin_le_nhds + have h := (hid.const_sub y).abs.mul_const + (Real.exp ((4*(Real.sqrt a)⁻¹^2)*((T : ℝ)-a))) + simpa only [sub_zero,abs_of_pos hy] using h + apply le_of_tendsto_of_tendsto (tendsto_const_nhds.dist ht) hr + filter_upwards [self_mem_nhdsWithin] with v hv + exact inverseFlow_vertical_distance_of_smooth_driver b T b' hd hL ha haT hb hsmall hy hv + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceBounds.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceBounds.lean index dcd3a55ff44353362a625217d09b0949b9477c46..82e3d4626efa22ddfeeb937cf3fc2ea1f6810742 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceBounds.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceBounds.lean @@ -1,71 +1,71 @@ -import OAI.Probability.SLE.Loewner.SmoothRawTrace -import OAI.Probability.SLE.Loewner.ReverseHeightUpper - -namespace OAI - -/-! Quantitative position bounds for the raw smooth-driver trace. -These estimates include time zero and do not assume a capacity witness. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -lemma smooth_raw_inverse_real_bound (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) - {y : ℝ} (hy : 0 < y) : - |(inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)).re| ≤ L*T := by - obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T - ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) - let u (s : ℝ) := driverValue b ((T : ℝ)-s) - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := - ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 - simp only [mul_neg,mul_one] - have hh := (reverse_centered_coordinates hL q u (fun s => -b' ((T : ℝ)-s)) - (fun s hs => (hqd s hs.1).mono Icc_subset_Ici_self) hu - (fun s hs => by simpa using hb _ (hreflect s hs)) - (fun s hs => hy.trans_le (by simpa using hqy s hs.1)) - (by simp [hq0,u,driverValue])).1 T ⟨T.coe_nonneg,le_rfl⟩ - rw [heq] - simpa only [u,sub_self,driverValue,Real.toNNReal_zero,map_zero,sub_zero] using hh - -theorem smooth_raw_trace_position_bounds (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : - |(trace 1 b T).re| ≤ L*T ∧ 0 ≤ (trace 1 b T).im ∧ - (trace 1 b T).im^2 ≤ 4*T := by - have ht := tendsto_inverseFlow_smooth_trace b T b' hd hL hb - have hid : Tendsto (fun y : ℝ => y) (𝓝[>] 0) (𝓝 (0 : ℝ)) := - nhdsWithin_le_nhds - refine ⟨?_,?_,?_⟩ - · apply le_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp ht |>.abs) - filter_upwards [self_mem_nhdsWithin] with y hy - exact smooth_raw_inverse_real_bound b T b' hd hL hb hy - · apply ge_of_tendsto (Complex.continuous_im.continuousAt.tendsto.comp ht) - filter_upwards [self_mem_nhdsWithin] with y hy - exact (inverseFlow_mem_upperHalfPlane 1 (by norm_num) b T - ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy)).le - · apply le_of_tendsto_of_tendsto - ((Complex.continuous_im.continuousAt.tendsto.comp ht).pow 2) - (show Tendsto (fun y : ℝ => y^2+4*(T : ℝ)) (𝓝[>] 0) (𝓝 (4*(T : ℝ))) from - by simpa only [zero_pow (by decide : (2:ℕ) ≠ 0),zero_add] using (hid.pow 2).add_const (4*(T : ℝ))) - filter_upwards [self_mem_nhdsWithin] with y hy - simpa only [Function.comp_apply,Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re, - Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_add] using - raw_inverse_height_square_le b T ((b T : ℂ)+(y : ℂ)*I) - (by simpa [upperHalfPlane] using hy) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawTrace +import OAI.Probability.SLE.Loewner.ReverseHeightUpper + +namespace OAI + +/-! Quantitative position bounds for the raw smooth-driver trace. +These estimates include time zero and do not assume a capacity witness. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +lemma smooth_raw_inverse_real_bound (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) + {y : ℝ} (hy : 0 < y) : + |(inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)).re| ≤ L*T := by + obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T + ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) + let u (s : ℝ) := driverValue b ((T : ℝ)-s) + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := + ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 + simp only [mul_neg,mul_one] + have hh := (reverse_centered_coordinates hL q u (fun s => -b' ((T : ℝ)-s)) + (fun s hs => (hqd s hs.1).mono Icc_subset_Ici_self) hu + (fun s hs => by simpa using hb _ (hreflect s hs)) + (fun s hs => hy.trans_le (by simpa using hqy s hs.1)) + (by simp [hq0,u,driverValue])).1 T ⟨T.coe_nonneg,le_rfl⟩ + rw [heq] + simpa only [u,sub_self,driverValue,Real.toNNReal_zero,map_zero,sub_zero] using hh + +theorem smooth_raw_trace_position_bounds (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : + |(trace 1 b T).re| ≤ L*T ∧ 0 ≤ (trace 1 b T).im ∧ + (trace 1 b T).im^2 ≤ 4*T := by + have ht := tendsto_inverseFlow_smooth_trace b T b' hd hL hb + have hid : Tendsto (fun y : ℝ => y) (𝓝[>] 0) (𝓝 (0 : ℝ)) := + nhdsWithin_le_nhds + refine ⟨?_,?_,?_⟩ + · apply le_of_tendsto (Complex.continuous_re.continuousAt.tendsto.comp ht |>.abs) + filter_upwards [self_mem_nhdsWithin] with y hy + exact smooth_raw_inverse_real_bound b T b' hd hL hb hy + · apply ge_of_tendsto (Complex.continuous_im.continuousAt.tendsto.comp ht) + filter_upwards [self_mem_nhdsWithin] with y hy + exact (inverseFlow_mem_upperHalfPlane 1 (by norm_num) b T + ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy)).le + · apply le_of_tendsto_of_tendsto + ((Complex.continuous_im.continuousAt.tendsto.comp ht).pow 2) + (show Tendsto (fun y : ℝ => y^2+4*(T : ℝ)) (𝓝[>] 0) (𝓝 (4*(T : ℝ))) from + by simpa only [zero_pow (by decide : (2:ℕ) ≠ 0),zero_add] using (hid.pow 2).add_const (4*(T : ℝ))) + filter_upwards [self_mem_nhdsWithin] with y hy + simpa only [Function.comp_apply,Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re, + Complex.I_im,Complex.I_re,mul_one,mul_zero,add_zero,zero_add] using + raw_inverse_height_square_le b T ((b T : ℂ)+(y : ℂ)*I) + (by simpa [upperHalfPlane] using hy) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuity.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuity.lean index e9e887a42bd689a286428cf190961bc74c4b384e..77722d9f4815d8dbeccf2ae34b3d0317d1c12613 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuity.lean @@ -1,57 +1,57 @@ -import OAI.Probability.SLE.Loewner.SmoothRawTrace -import OAI.Probability.SLE.Loewner.RawTipTimeContinuity - -namespace OAI - -/-! Radial approximations converge uniformly away from time zero. -Consequently the raw smooth-driver tip is continuous at positive times. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -theorem smooth_trace_uniform_radial_limit (b : Driver) (b' : ℝ → ℝ) - {L : ℝ≥0} (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) - (a R : ℝ≥0) (ha : 0 < a) (hsmall : (L : ℝ)^2*(a : ℝ) ≤ 1) : - TendstoUniformlyOn - (fun n : ℕ => fun T : ℝ≥0 => inverseFlow 1 b T - ((b T : ℂ)+((1/((n : ℝ)+1) : ℝ) : ℂ)*I)) - (fun T : ℝ≥0 => trace 1 b T) atTop (Icc a R) := by - let C : ℝ := Real.exp ((4*(Real.sqrt a)⁻¹^2)*((R : ℝ)-a)) - have hzero : Tendsto (fun n : ℕ => (1/((n : ℝ)+1))*C) atTop (𝓝 0) := by - simpa only [zero_mul] using tendsto_one_div_add_atTop_nhds_zero_nat.mul_const C - rw [Metric.tendstoUniformlyOn_iff] - intro ε hε - filter_upwards [hzero.eventually (Iio_mem_nhds hε)] with n hn T hT - have hy : 0 < 1/((n : ℝ)+1) := by positivity - have hdt (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := - (hd t ht.1).mono (show Icc 0 (T : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) - have h := inverseFlow_smooth_trace_error b T b' hdt L.coe_nonneg - (show 0 < (a : ℝ) from ha) (show (a : ℝ) ≤ T from hT.1) - (fun t ht => hb t ht.1) hsmall hy - have hbound : dist (inverseFlow 1 b T - ((b T : ℂ)+((1/((n : ℝ)+1) : ℝ) : ℂ)*I)) (trace 1 b T) ≤ - (1/((n : ℝ)+1))*C := by - apply h.trans - dsimp [C] - gcongr - exact hT.2 - simpa only [dist_comm] using hbound.trans_lt hn - -theorem continuousOn_smooth_trace_Icc (b : Driver) (b' : ℝ → ℝ) - {L : ℝ≥0} (hLip : LipschitzWith L (driverValue b)) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) - (a R : ℝ≥0) (ha : 0 < a) (hsmall : (L : ℝ)^2*(a : ℝ) ≤ 1) : - ContinuousOn (fun T : ℝ≥0 => trace 1 b T) (Icc a R) := by - exact (smooth_trace_uniform_radial_limit b b' hd hb a R ha hsmall).continuousOn - (Eventually.frequently (Eventually.of_forall (fun n => (continuous_raw_radial_time b hLip - (show 0 < 1/((n : ℝ)+1) by positivity)).continuousOn))) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawTrace +import OAI.Probability.SLE.Loewner.RawTipTimeContinuity + +namespace OAI + +/-! Radial approximations converge uniformly away from time zero. +Consequently the raw smooth-driver tip is continuous at positive times. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +theorem smooth_trace_uniform_radial_limit (b : Driver) (b' : ℝ → ℝ) + {L : ℝ≥0} (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) + (a R : ℝ≥0) (ha : 0 < a) (hsmall : (L : ℝ)^2*(a : ℝ) ≤ 1) : + TendstoUniformlyOn + (fun n : ℕ => fun T : ℝ≥0 => inverseFlow 1 b T + ((b T : ℂ)+((1/((n : ℝ)+1) : ℝ) : ℂ)*I)) + (fun T : ℝ≥0 => trace 1 b T) atTop (Icc a R) := by + let C : ℝ := Real.exp ((4*(Real.sqrt a)⁻¹^2)*((R : ℝ)-a)) + have hzero : Tendsto (fun n : ℕ => (1/((n : ℝ)+1))*C) atTop (𝓝 0) := by + simpa only [zero_mul] using tendsto_one_div_add_atTop_nhds_zero_nat.mul_const C + rw [Metric.tendstoUniformlyOn_iff] + intro ε hε + filter_upwards [hzero.eventually (Iio_mem_nhds hε)] with n hn T hT + have hy : 0 < 1/((n : ℝ)+1) := by positivity + have hdt (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) := + (hd t ht.1).mono (show Icc 0 (T : ℝ) ⊆ Ici 0 from Icc_subset_Ici_self) + have h := inverseFlow_smooth_trace_error b T b' hdt L.coe_nonneg + (show 0 < (a : ℝ) from ha) (show (a : ℝ) ≤ T from hT.1) + (fun t ht => hb t ht.1) hsmall hy + have hbound : dist (inverseFlow 1 b T + ((b T : ℂ)+((1/((n : ℝ)+1) : ℝ) : ℂ)*I)) (trace 1 b T) ≤ + (1/((n : ℝ)+1))*C := by + apply h.trans + dsimp [C] + gcongr + exact hT.2 + simpa only [dist_comm] using hbound.trans_lt hn + +theorem continuousOn_smooth_trace_Icc (b : Driver) (b' : ℝ → ℝ) + {L : ℝ≥0} (hLip : LipschitzWith L (driverValue b)) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) + (a R : ℝ≥0) (ha : 0 < a) (hsmall : (L : ℝ)^2*(a : ℝ) ≤ 1) : + ContinuousOn (fun T : ℝ≥0 => trace 1 b T) (Icc a R) := by + exact (smooth_trace_uniform_radial_limit b b' hd hb a R ha hsmall).continuousOn + (Eventually.frequently (Eventually.of_forall (fun n => (continuous_raw_radial_time b hLip + (show 0 < 1/((n : ℝ)+1) by positivity)).continuousOn))) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuous.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuous.lean index 0d35437196ae987fc11638bdc6f098f968a82522..5c23ee0b471c7104d728e27c32c3e463ad27970b 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuous.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceContinuous.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Loewner.SmoothRawTraceContinuity -import OAI.Probability.SLE.Loewner.SmoothRawTraceBounds -import OAI.Probability.SLE.Loewner.SmoothDriverLipschitz - -namespace OAI - -/-! Smooth drivers with bounded derivative generate a continuous raw tip -at all nonnegative times, including the initial boundary point. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -lemma smooth_trace_zero (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : trace 1 b 0=0 := by - have hh := smooth_raw_trace_position_bounds b 0 b' - (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg (fun t ht => hb t ht.1) - simp only [NNReal.coe_zero,mul_zero] at hh - apply Complex.ext - · simpa only [Complex.zero_re] using abs_nonpos_iff.mp hh.1 - · simpa only [Complex.zero_im] using (show (trace 1 b 0).im=0 by nlinarith [sq_nonneg (trace 1 b 0).im]) - -lemma smooth_trace_norm_le (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) : - ‖trace 1 b T‖ ≤ (L : ℝ)*T+2*Real.sqrt T := by - have hh := smooth_raw_trace_position_bounds b T b' - (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg (fun t ht => hb t ht.1) - have him : (trace 1 b T).im ≤ 2*Real.sqrt T := by - nlinarith [Real.sq_sqrt T.coe_nonneg,Real.sqrt_nonneg (T : ℝ)] - calc - _ ≤ |(trace 1 b T).re|+|(trace 1 b T).im| := Complex.norm_le_abs_re_add_abs_im _ - _ ≤ (L : ℝ)*T+2*Real.sqrt T := by rw [abs_of_nonneg hh.2.1]; exact add_le_add hh.1 him - -theorem continuous_smooth_raw_trace (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) - (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) - (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : - Continuous (fun T : ℝ≥0 => trace 1 b T) := by - have hLip := lipschitz_driverValue_of_bounded_derivative b b' L hd hb - apply continuous_iff_continuousAt.mpr - intro T - by_cases hT : T=0 - · subst T - change Tendsto (fun T : ℝ≥0 => trace 1 b T) (𝓝 0) (𝓝 (trace 1 b 0)) - rw [smooth_trace_zero b b' L hd hb] - apply tendsto_zero_iff_norm_tendsto_zero.mpr - apply squeeze_zero (fun _ => norm_nonneg _) (smooth_trace_norm_le b b' L hd hb) - have hc : Continuous (fun T : ℝ≥0 => (L : ℝ)*T+2*Real.sqrt T) := by fun_prop - simpa only [NNReal.coe_zero,mul_zero,Real.sqrt_zero,add_zero] using hc.tendsto 0 - · have hTp : 0 < (T : ℝ) := by exact_mod_cast (pos_iff_ne_zero.mpr hT) - obtain ⟨ar,har,harT,hsmall⟩ := exists_smooth_window ((T : ℝ)/2) (half_pos hTp) L - let a : ℝ≥0 := ⟨ar,har.le⟩ - have ha : 0 < a := har - have haT : a < T := harT.trans_lt (half_lt_self hTp) - exact (continuousOn_smooth_trace_Icc b b' hLip hd hb a (T+1) ha hsmall).continuousAt - (Icc_mem_nhds haT (by simp)) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawTraceContinuity +import OAI.Probability.SLE.Loewner.SmoothRawTraceBounds +import OAI.Probability.SLE.Loewner.SmoothDriverLipschitz + +namespace OAI + +/-! Smooth drivers with bounded derivative generate a continuous raw tip +at all nonnegative times, including the initial boundary point. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +lemma smooth_trace_zero (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : trace 1 b 0=0 := by + have hh := smooth_raw_trace_position_bounds b 0 b' + (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg (fun t ht => hb t ht.1) + simp only [NNReal.coe_zero,mul_zero] at hh + apply Complex.ext + · simpa only [Complex.zero_re] using abs_nonpos_iff.mp hh.1 + · simpa only [Complex.zero_im] using (show (trace 1 b 0).im=0 by nlinarith [sq_nonneg (trace 1 b 0).im]) + +lemma smooth_trace_norm_le (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) (T : ℝ≥0) : + ‖trace 1 b T‖ ≤ (L : ℝ)*T+2*Real.sqrt T := by + have hh := smooth_raw_trace_position_bounds b T b' + (fun t ht => (hd t ht.1).mono Icc_subset_Ici_self) L.coe_nonneg (fun t ht => hb t ht.1) + have him : (trace 1 b T).im ≤ 2*Real.sqrt T := by + nlinarith [Real.sq_sqrt T.coe_nonneg,Real.sqrt_nonneg (T : ℝ)] + calc + _ ≤ |(trace 1 b T).re|+|(trace 1 b T).im| := Complex.norm_le_abs_re_add_abs_im _ + _ ≤ (L : ℝ)*T+2*Real.sqrt T := by rw [abs_of_nonneg hh.2.1]; exact add_le_add hh.1 him + +theorem continuous_smooth_raw_trace (b : Driver) (b' : ℝ → ℝ) (L : ℝ≥0) + (hd : ∀ t, 0 ≤ t → HasDerivWithinAt (driverValue b) (b' t) (Ici 0) t) + (hb : ∀ t, 0 ≤ t → |b' t| ≤ L) : + Continuous (fun T : ℝ≥0 => trace 1 b T) := by + have hLip := lipschitz_driverValue_of_bounded_derivative b b' L hd hb + apply continuous_iff_continuousAt.mpr + intro T + by_cases hT : T=0 + · subst T + change Tendsto (fun T : ℝ≥0 => trace 1 b T) (𝓝 0) (𝓝 (trace 1 b 0)) + rw [smooth_trace_zero b b' L hd hb] + apply tendsto_zero_iff_norm_tendsto_zero.mpr + apply squeeze_zero (fun _ => norm_nonneg _) (smooth_trace_norm_le b b' L hd hb) + have hc : Continuous (fun T : ℝ≥0 => (L : ℝ)*T+2*Real.sqrt T) := by fun_prop + simpa only [NNReal.coe_zero,mul_zero,Real.sqrt_zero,add_zero] using hc.tendsto 0 + · have hTp : 0 < (T : ℝ) := by exact_mod_cast (pos_iff_ne_zero.mpr hT) + obtain ⟨ar,har,harT,hsmall⟩ := exists_smooth_window ((T : ℝ)/2) (half_pos hTp) L + let a : ℝ≥0 := ⟨ar,har.le⟩ + have ha : 0 < a := har + have haT : a < T := harT.trans_lt (half_lt_self hTp) + exact (continuousOn_smooth_trace_Icc b b' hLip hd hb a (T+1) ha hsmall).continuousAt + (Icc_mem_nhds haT (by simp)) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceInterior.lean b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceInterior.lean index fcc428537edef56907252e681b52747f66c55f9d..af598ca21d8cc46a464ec657357d48cd91537dba 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceInterior.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothRawTraceInterior.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Loewner.SmoothRawTraceBounds -import OAI.Probability.SLE.Loewner.ReverseLongContraction - -namespace OAI - -/-! Every positive-time smooth raw tip is in the upper half-plane. -A short reverse interval gives a height bound independent of the -positive height used to approximate the tip. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -lemma smooth_raw_inverse_height_lower (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) - (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) - {y : ℝ} (hy : 0 < y) : - 2*a ≤ (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)).im^2 := by - obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T - ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) - let u (s : ℝ) := driverValue b ((T : ℝ)-s) - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := - ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 - simp only [mul_neg,mul_one] - have hQ (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt q - (-2/(q s-(u s : ℂ))) (Icc 0 (T : ℝ)) s := - (hqd s hs.1).mono Icc_subset_Ici_self - have hp (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : 0 < (q s).im := - hy.trans_le (by simpa using hqy s hs.1) - have hsub : Icc 0 a ⊆ Icc 0 (T : ℝ) := Icc_subset_Icc_right haT - have hshort := reverse_loewner_height_lower hL q u (fun s => -b' ((T : ℝ)-s)) - (fun s hs => (hQ s (hsub hs)).mono hsub) - (fun s hs => (hu s (hsub hs)).mono hsub) - (fun s hs => by simpa using hb _ (hreflect s (hsub hs))) - (fun s hs => hp s (hsub hs)) (by simp [hq0,u,driverValue]) hsmall a ⟨ha.le,le_rfl⟩ - have hmono := reverse_height_monotone q u hQ hp - have hle := hmono ⟨ha.le,haT⟩ ⟨T.coe_nonneg,le_rfl⟩ haT - rw [heq] - nlinarith [hp a ⟨ha.le,haT⟩,hp T ⟨T.coe_nonneg,le_rfl⟩] - -lemma smooth_raw_trace_height_lower (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) - (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) : - 2*a ≤ (trace 1 b T).im^2 := by - have ht := tendsto_inverseFlow_smooth_trace b T b' hd hL hb - apply ge_of_tendsto ((Complex.continuous_im.continuousAt.tendsto.comp ht).pow 2) - filter_upwards [self_mem_nhdsWithin] with y hy - exact smooth_raw_inverse_height_lower b T b' hd hL ha haT hb hsmall hy - -theorem smooth_raw_trace_mem_upperHalfPlane (b : Driver) (T : ℝ≥0) (hT : 0 < T) - (b' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) - (b' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : - trace 1 b T ∈ upperHalfPlane := by - obtain ⟨a,ha,haT,hs⟩ := exists_smooth_window T (by exact_mod_cast hT) L - have hl := smooth_raw_trace_height_lower b T b' hd hL ha haT hb hs - have hn := (smooth_raw_trace_position_bounds b T b' hd hL hb).2.1 - change 0 < (trace 1 b T).im - nlinarith - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothRawTraceBounds +import OAI.Probability.SLE.Loewner.ReverseLongContraction + +namespace OAI + +/-! Every positive-time smooth raw tip is in the upper half-plane. +A short reverse interval gives a height bound independent of the +positive height used to approximate the tip. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +lemma smooth_raw_inverse_height_lower (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) + (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) + {y : ℝ} (hy : 0 < y) : + 2*a ≤ (inverseFlow 1 b T ((b T : ℂ)+(y : ℂ)*I)).im^2 := by + obtain ⟨q,hq0,hqy,hqd,heq⟩ := exists_raw_reverse_trajectory b T + ((b T : ℂ)+(y : ℂ)*I) (by simpa [upperHalfPlane] using hy) + let u (s : ℝ) := driverValue b ((T : ℝ)-s) + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := + ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-b' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun t : ℝ => (T : ℝ)-t) (-1) (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + convert! (hd _ (hreflect s hs)).comp s hi hreflect using 1 + simp only [mul_neg,mul_one] + have hQ (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt q + (-2/(q s-(u s : ℂ))) (Icc 0 (T : ℝ)) s := + (hqd s hs.1).mono Icc_subset_Ici_self + have hp (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : 0 < (q s).im := + hy.trans_le (by simpa using hqy s hs.1) + have hsub : Icc 0 a ⊆ Icc 0 (T : ℝ) := Icc_subset_Icc_right haT + have hshort := reverse_loewner_height_lower hL q u (fun s => -b' ((T : ℝ)-s)) + (fun s hs => (hQ s (hsub hs)).mono hsub) + (fun s hs => (hu s (hsub hs)).mono hsub) + (fun s hs => by simpa using hb _ (hreflect s (hsub hs))) + (fun s hs => hp s (hsub hs)) (by simp [hq0,u,driverValue]) hsmall a ⟨ha.le,le_rfl⟩ + have hmono := reverse_height_monotone q u hQ hp + have hle := hmono ⟨ha.le,haT⟩ ⟨T.coe_nonneg,le_rfl⟩ haT + rw [heq] + nlinarith [hp a ⟨ha.le,haT⟩,hp T ⟨T.coe_nonneg,le_rfl⟩] + +lemma smooth_raw_trace_height_lower (b : Driver) (T : ℝ≥0) (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L a : ℝ} (hL : 0 ≤ L) (ha : 0 < a) (haT : a ≤ T) + (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) (hsmall : L^2*a ≤ 1) : + 2*a ≤ (trace 1 b T).im^2 := by + have ht := tendsto_inverseFlow_smooth_trace b T b' hd hL hb + apply ge_of_tendsto ((Complex.continuous_im.continuousAt.tendsto.comp ht).pow 2) + filter_upwards [self_mem_nhdsWithin] with y hy + exact smooth_raw_inverse_height_lower b T b' hd hL ha haT hb hsmall hy + +theorem smooth_raw_trace_mem_upperHalfPlane (b : Driver) (T : ℝ≥0) (hT : 0 < T) + (b' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (driverValue b) + (b' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hb : ∀ t ∈ Icc 0 (T : ℝ), |b' t| ≤ L) : + trace 1 b T ∈ upperHalfPlane := by + obtain ⟨a,ha,haT,hs⟩ := exists_smooth_window T (by exact_mod_cast hT) L + have hl := smooth_raw_trace_height_lower b T b' hd hL ha haT hb hs + have hn := (smooth_raw_trace_position_bounds b T b' hd hL hb).2.1 + change 0 < (trace 1 b T).im + nlinarith + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothReverseContraction.lean b/lean/OAI/Probability/SLE/Loewner/SmoothReverseContraction.lean index 0a9a6b63b0db92a40bdbcc3211b1767b789ba7a3..6501af1cba5d1b60bce585c99d342220aeef9b7f 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothReverseContraction.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothReverseContraction.lean @@ -1,94 +1,94 @@ -import OAI.Probability.SLE.Loewner.SmoothReverseLoewner - -namespace OAI - -/-! Short reverse trajectories from the vertical line over a smooth -driver contract. This gives a uniform Cauchy estimate at zero starting -height without assuming that a trace already exists. -/ -noncomputable section -open Set Filter Complex MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -theorem smooth_reverse_trajectories_contract {T L : ℝ} (hT : 0 ≤ T) (hL : 0 ≤ L) - (q r : ℝ → ℂ) (u u' : ℝ → ℝ) - (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Icc 0 T) t) - (hr : ∀ t ∈ Icc 0 T, HasDerivWithinAt r (-2/(r t-(u t : ℂ))) (Icc 0 T) t) - (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) - (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) - (hqp : ∀ t ∈ Icc 0 T, 0 < (q t).im) - (hrp : ∀ t ∈ Icc 0 T, 0 < (r t).im) - (hq0 : (q 0).re=u 0) (hr0 : (r 0).re=u 0) - (hsmall : L^2*T ≤ 1) : dist (q T) (r T) ≤ dist (q 0) (r 0) := by - have hqc : ContinuousOn q (Icc 0 T) := fun t ht => (hq t ht).continuousWithinAt - have hrc : ContinuousOn r (Icc 0 T) := fun t ht => (hr t ht).continuousWithinAt - have huc : ContinuousOn u (Icc 0 T) := fun t ht => (hu t ht).continuousWithinAt - have hqx := (reverse_centered_coordinates hL q u u' hq hu hb hqp hq0).1 - have hrx := (reverse_centered_coordinates hL r u u' hr hu hb hrp hr0).1 - have hqy := reverse_loewner_height_lower hL q u u' hq hu hb hqp hq0 hsmall - have hry := reverse_loewner_height_lower hL r u u' hr hu hb hrp hr0 hsmall - let c (t : ℝ) : ℂ := 2/((q t-(u t : ℂ))*(r t-(u t : ℂ))) - have hqn (t : ℝ) (ht : t ∈ Icc 0 T) : q t-(u t : ℂ) ≠ 0 := by - intro he - have hh := congrArg Complex.im he - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hh - exact (hqp t ht).ne' hh - have hrn (t : ℝ) (ht : t ∈ Icc 0 T) : r t-(u t : ℂ) ≠ 0 := by - intro he - have hh := congrArg Complex.im he - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hh - exact (hrp t ht).ne' hh - have hc : ContinuousOn c (Icc 0 T) := continuousOn_const.div - ((hqc.sub (Complex.continuous_ofReal.comp_continuousOn huc)).mul - (hrc.sub (Complex.continuous_ofReal.comp_continuousOn huc))) - (fun t ht => mul_ne_zero (hqn t ht) (hrn t ht)) - have hcn (t : ℝ) (ht : t ∈ Icc 0 T) : (c t).re ≤ 0 := by - have hprod : ((q t).re-u t)*((r t).re-u t) ≤ (L*t)^2 := by - calc - _ ≤ |((q t).re-u t)*((r t).re-u t)| := le_abs_self _ - _ = |(q t).re-u t| * |(r t).re-u t| := abs_mul _ _ - _ ≤ (L*t)*(L*t) := mul_le_mul (hqx t ht) (hrx t ht) - (abs_nonneg _) (mul_nonneg hL ht.1) - _ = _ := (pow_two _).symm - have hroot : (Real.sqrt (2*t))^2=2*t := Real.sq_sqrt (mul_nonneg (by norm_num) ht.1) - have hqroot : Real.sqrt (2*t) ≤ (q t).im := by - nlinarith [Real.sqrt_nonneg (2*t),hqp t ht,hqy t ht] - have hrroot : Real.sqrt (2*t) ≤ (r t).im := by - nlinarith [Real.sqrt_nonneg (2*t),hrp t ht,hry t ht] - have hyy : 2*t ≤ (q t).im*(r t).im := by - nlinarith [mul_le_mul hqroot hrroot (Real.sqrt_nonneg (2*t)) (hqp t ht).le] - have hsmallt := (mul_le_mul_of_nonneg_left ht.2 (sq_nonneg L)).trans hsmall - have hxx : ((q t).re-u t)*((r t).re-u t) ≤ t := by - nlinarith [mul_le_mul_of_nonneg_right hsmallt ht.1] - have hreal : ((q t-(u t : ℂ))*(r t-(u t : ℂ))).re ≤ 0 := by - simp only [Complex.mul_re,Complex.sub_re,Complex.ofReal_re, - Complex.sub_im,Complex.ofReal_im,sub_zero] - linarith [ht.1] - dsimp [c] - simp only [Complex.div_re,Complex.re_ofNat,Complex.im_ofNat,zero_mul,zero_div,add_zero] - apply div_nonpos_of_nonpos_of_nonneg - · nlinarith - · exact Complex.normSq_nonneg _ - have hd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt (fun s => q s-r s) - (c t*(q t-r t)) (Icc 0 T) t := by - convert (hq t ht).sub (hr t ht) using 1 - dsimp [c] - field_simp [hqn t ht,hrn t ht] - ring - have he := complex_linear_ode_exp T hT (fun t => q t-r t) c hc hd - have hi : (∫ t in (0 : ℝ)..T, (c t).re) ≤ 0 := by - have hh := intervalIntegral.integral_mono_on hT - ((Complex.continuous_re.comp_continuousOn hc).intervalIntegrable_of_Icc hT) - (intervalIntegrable_const (μ := volume) (c := (0 : ℝ))) hcn - simpa only [intervalIntegral.integral_zero,Function.comp_def] using hh - have hir : (∫ t in (0 : ℝ)..T, (c t).re) = (∫ t in (0 : ℝ)..T, c t).re := - Complex.reCLM.intervalIntegral_comp_comm (hc.intervalIntegrable_of_Icc hT) - rw [dist_eq_norm,dist_eq_norm,he,norm_mul,Complex.norm_exp,← hir] - exact mul_le_of_le_one_right (norm_nonneg _) (Real.exp_le_one_iff.mpr hi) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.SmoothReverseLoewner + +namespace OAI + +/-! Short reverse trajectories from the vertical line over a smooth +driver contract. This gives a uniform Cauchy estimate at zero starting +height without assuming that a trace already exists. -/ +noncomputable section +open Set Filter Complex MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +theorem smooth_reverse_trajectories_contract {T L : ℝ} (hT : 0 ≤ T) (hL : 0 ≤ L) + (q r : ℝ → ℂ) (u u' : ℝ → ℝ) + (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q (-2/(q t-(u t : ℂ))) (Icc 0 T) t) + (hr : ∀ t ∈ Icc 0 T, HasDerivWithinAt r (-2/(r t-(u t : ℂ))) (Icc 0 T) t) + (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) + (hb : ∀ t ∈ Icc 0 T, |u' t| ≤ L) + (hqp : ∀ t ∈ Icc 0 T, 0 < (q t).im) + (hrp : ∀ t ∈ Icc 0 T, 0 < (r t).im) + (hq0 : (q 0).re=u 0) (hr0 : (r 0).re=u 0) + (hsmall : L^2*T ≤ 1) : dist (q T) (r T) ≤ dist (q 0) (r 0) := by + have hqc : ContinuousOn q (Icc 0 T) := fun t ht => (hq t ht).continuousWithinAt + have hrc : ContinuousOn r (Icc 0 T) := fun t ht => (hr t ht).continuousWithinAt + have huc : ContinuousOn u (Icc 0 T) := fun t ht => (hu t ht).continuousWithinAt + have hqx := (reverse_centered_coordinates hL q u u' hq hu hb hqp hq0).1 + have hrx := (reverse_centered_coordinates hL r u u' hr hu hb hrp hr0).1 + have hqy := reverse_loewner_height_lower hL q u u' hq hu hb hqp hq0 hsmall + have hry := reverse_loewner_height_lower hL r u u' hr hu hb hrp hr0 hsmall + let c (t : ℝ) : ℂ := 2/((q t-(u t : ℂ))*(r t-(u t : ℂ))) + have hqn (t : ℝ) (ht : t ∈ Icc 0 T) : q t-(u t : ℂ) ≠ 0 := by + intro he + have hh := congrArg Complex.im he + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hh + exact (hqp t ht).ne' hh + have hrn (t : ℝ) (ht : t ∈ Icc 0 T) : r t-(u t : ℂ) ≠ 0 := by + intro he + have hh := congrArg Complex.im he + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero,Complex.zero_im] at hh + exact (hrp t ht).ne' hh + have hc : ContinuousOn c (Icc 0 T) := continuousOn_const.div + ((hqc.sub (Complex.continuous_ofReal.comp_continuousOn huc)).mul + (hrc.sub (Complex.continuous_ofReal.comp_continuousOn huc))) + (fun t ht => mul_ne_zero (hqn t ht) (hrn t ht)) + have hcn (t : ℝ) (ht : t ∈ Icc 0 T) : (c t).re ≤ 0 := by + have hprod : ((q t).re-u t)*((r t).re-u t) ≤ (L*t)^2 := by + calc + _ ≤ |((q t).re-u t)*((r t).re-u t)| := le_abs_self _ + _ = |(q t).re-u t| * |(r t).re-u t| := abs_mul _ _ + _ ≤ (L*t)*(L*t) := mul_le_mul (hqx t ht) (hrx t ht) + (abs_nonneg _) (mul_nonneg hL ht.1) + _ = _ := (pow_two _).symm + have hroot : (Real.sqrt (2*t))^2=2*t := Real.sq_sqrt (mul_nonneg (by norm_num) ht.1) + have hqroot : Real.sqrt (2*t) ≤ (q t).im := by + nlinarith [Real.sqrt_nonneg (2*t),hqp t ht,hqy t ht] + have hrroot : Real.sqrt (2*t) ≤ (r t).im := by + nlinarith [Real.sqrt_nonneg (2*t),hrp t ht,hry t ht] + have hyy : 2*t ≤ (q t).im*(r t).im := by + nlinarith [mul_le_mul hqroot hrroot (Real.sqrt_nonneg (2*t)) (hqp t ht).le] + have hsmallt := (mul_le_mul_of_nonneg_left ht.2 (sq_nonneg L)).trans hsmall + have hxx : ((q t).re-u t)*((r t).re-u t) ≤ t := by + nlinarith [mul_le_mul_of_nonneg_right hsmallt ht.1] + have hreal : ((q t-(u t : ℂ))*(r t-(u t : ℂ))).re ≤ 0 := by + simp only [Complex.mul_re,Complex.sub_re,Complex.ofReal_re, + Complex.sub_im,Complex.ofReal_im,sub_zero] + linarith [ht.1] + dsimp [c] + simp only [Complex.div_re,Complex.re_ofNat,Complex.im_ofNat,zero_mul,zero_div,add_zero] + apply div_nonpos_of_nonpos_of_nonneg + · nlinarith + · exact Complex.normSq_nonneg _ + have hd (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt (fun s => q s-r s) + (c t*(q t-r t)) (Icc 0 T) t := by + convert (hq t ht).sub (hr t ht) using 1 + dsimp [c] + field_simp [hqn t ht,hrn t ht] + ring + have he := complex_linear_ode_exp T hT (fun t => q t-r t) c hc hd + have hi : (∫ t in (0 : ℝ)..T, (c t).re) ≤ 0 := by + have hh := intervalIntegral.integral_mono_on hT + ((Complex.continuous_re.comp_continuousOn hc).intervalIntegrable_of_Icc hT) + (intervalIntegrable_const (μ := volume) (c := (0 : ℝ))) hcn + simpa only [intervalIntegral.integral_zero,Function.comp_def] using hh + have hir : (∫ t in (0 : ℝ)..T, (c t).re) = (∫ t in (0 : ℝ)..T, c t).re := + Complex.reCLM.intervalIntegral_comp_comm (hc.intervalIntegrable_of_Icc hT) + rw [dist_eq_norm,dist_eq_norm,he,norm_mul,Complex.norm_exp,← hir] + exact mul_le_of_le_one_right (norm_nonneg _) (Real.exp_le_one_iff.mpr hi) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/SmoothReverseLoewner.lean b/lean/OAI/Probability/SLE/Loewner/SmoothReverseLoewner.lean index 70e8d006d23b378913122f51074a2a979fd368af..31399d8db846578b3d663405ab49d83e945de14c 100644 --- a/lean/OAI/Probability/SLE/Loewner/SmoothReverseLoewner.lean +++ b/lean/OAI/Probability/SLE/Loewner/SmoothReverseLoewner.lean @@ -1,117 +1,117 @@ -import OAI.Probability.SLE.Loewner.ReverseSmoothBounds - -namespace OAI - -/-! Uniform height estimates for reverse Loewner trajectories driven by a -function with bounded derivative. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma reverse_centered_coordinates {T L : ℝ} (hL : 0 ≤ L) - (q : ℝ → ℂ) (u u' : ℝ → ℝ) - (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Icc 0 T) t) - (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) - (hbound : ∀ t ∈ Icc 0 T, |u' t| ≤ L) - (hpos : ∀ t ∈ Icc 0 T, 0 < (q t).im) - (hstart : (q 0).re=u 0) : - (∀ t ∈ Icc 0 T, |(q t).re-u t| ≤ L*t) ∧ - (∀ t ∈ Icc 0 T, HasDerivWithinAt (fun s => (q s).im) - (2*(q t).im/(((q t).re-u t)^2+(q t).im^2)) (Icc 0 T) t) := by - let D (t : ℝ) := ((q t).re-u t)^2+(q t).im^2 - have hD (t : ℝ) (ht : t ∈ Icc 0 T) : 0 < D t := by - have hp := hpos t ht - dsimp [D] - positivity - have hre (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt - (fun s => (q s).re-u s) (-2*((q t).re-u t)/D t-u' t) (Icc 0 T) t := by - have hd := (Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t (hq t ht)).sub (hu t ht) - change HasDerivWithinAt (fun s => (q s).re-u s) - ((-2/(q t-(u t : ℂ))).re-u' t) (Icc 0 T) t at hd - have he : (-2/(q t-(u t : ℂ))).re = -2*((q t).re-u t)/D t := by - simp [D,Complex.div_re,Complex.normSq_apply,pow_two] - rwa [he] at hd - refine ⟨damped_real_abs_barrier hL (fun s => (q s).re-u s) (fun s => -u' s) D - (sub_eq_zero.mpr hstart) (fun t ht => ?_) hD (fun t ht => by simpa using hbound t ht),?_⟩ - · convert hre t ht using 1 - · intro t ht - have hd := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hq t ht) - change HasDerivWithinAt (fun s => (q s).im) - ((-2/(q t-(u t : ℂ))).im) (Icc 0 T) t at hd - have he : (-2/(q t-(u t : ℂ))).im = - 2*(q t).im/(((q t).re-u t)^2+(q t).im^2) := by - simp [Complex.div_im,Complex.normSq_apply,pow_two] - ring - rwa [he] at hd - -theorem reverse_loewner_height_lower {T L : ℝ} (hL : 0 ≤ L) - (q : ℝ → ℂ) (u u' : ℝ → ℝ) - (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q - (-2/(q t-(u t : ℂ))) (Icc 0 T) t) - (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) - (hbound : ∀ t ∈ Icc 0 T, |u' t| ≤ L) - (hpos : ∀ t ∈ Icc 0 T, 0 < (q t).im) - (hstart : (q 0).re=u 0) (hsmall : L^2*T ≤ 1) : - ∀ t ∈ Icc 0 T, 2*t ≤ (q t).im^2 := by - obtain ⟨hx,hy⟩ := reverse_centered_coordinates hL q u u' hq hu hbound hpos hstart - exact reverse_height_square_barrier (fun s => (q s).re-u s) (fun s => (q s).im) - hpos hy hx (fun t ht => - (mul_le_mul_of_nonneg_left ht.2 (sq_nonneg L)).trans hsmall) - -theorem IsCapacityTwoWitness.inverse_tip_height_lower - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (U' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => U s.toNNReal) - (U' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hbound : ∀ t ∈ Icc 0 (T : ℝ), |U' t| ≤ L) - (hsmall : L^2*(T : ℝ) ≤ 1) {y : ℝ} (hy : 0 < y) : - 2*(T : ℝ) ≤ (F T ((U T : ℂ)+(y : ℂ)*I)).im^2 := by - let w : ℂ := (U T : ℂ)+(y : ℂ)*I - have hw : w ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy - let q (s : ℝ) := G ((T : ℝ)-s).toNNReal (F T w) - let u (s : ℝ) := U ((T : ℝ)-s).toNNReal - have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - (T : ℝ)-s ∈ Icc 0 (T : ℝ) := ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ - have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u - (-U' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by - have hi : HasDerivWithinAt (fun r : ℝ => (T : ℝ)-r) (-1) - (Icc 0 (T : ℝ)) s := by - simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt - simpa only [Function.comp_def,mul_neg,mul_one] using - (hd _ (hreflect s hs)).comp s hi hreflect - have hq0 : q 0=w := by - simp only [q,sub_zero,Real.toNNReal_coe] - exact h.2.2.2.2.2.2.2.2.1 T w hw - have hqT : q T=F T w := by - simp only [q,sub_self,Real.toNNReal_zero] - exact h.initial_value _ (h.2.2.2.2.2.2.1 T hw).1.1 - have hh := reverse_loewner_height_lower hL q u - (fun s => -U' ((T : ℝ)-s)) (fun s hs => h.reverse_path_unclipped T hw hs) hu - (fun s hs => by simpa using hbound _ (hreflect s hs)) - (fun s hs => h.imag_positive _ _ (survivingDomain_antitone γ - ((Real.toNNReal_le_iff_le_coe).mpr (hreflect s hs).2) (h.2.2.2.2.2.2.1 T hw))) - (by simp [hq0,w,u]) hsmall T ⟨T.coe_nonneg,le_rfl⟩ - simpa only [hqT,w] using hh - -theorem IsCapacityTwoWitness.smooth_trace_height_lower - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (U' : ℝ → ℝ) - (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => U s.toNNReal) - (U' t) (Icc 0 (T : ℝ)) t) - {L : ℝ} (hL : 0 ≤ L) (hbound : ∀ t ∈ Icc 0 (T : ℝ), |U' t| ≤ L) - (hsmall : L^2*(T : ℝ) ≤ 1) : - 2*(T : ℝ) ≤ (γ T).im^2 := by - have hl := ((Complex.continuous_im.continuousAt.tendsto.comp - (h.2.2.2.2.2.2.2.2.2.2.2.2 T)).pow 2) - apply ge_of_tendsto hl - filter_upwards [self_mem_nhdsWithin] with y hy - exact h.inverse_tip_height_lower T U' hd hL hbound hsmall hy - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.ReverseSmoothBounds + +namespace OAI + +/-! Uniform height estimates for reverse Loewner trajectories driven by a +function with bounded derivative. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma reverse_centered_coordinates {T L : ℝ} (hL : 0 ≤ L) + (q : ℝ → ℂ) (u u' : ℝ → ℝ) + (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Icc 0 T) t) + (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) + (hbound : ∀ t ∈ Icc 0 T, |u' t| ≤ L) + (hpos : ∀ t ∈ Icc 0 T, 0 < (q t).im) + (hstart : (q 0).re=u 0) : + (∀ t ∈ Icc 0 T, |(q t).re-u t| ≤ L*t) ∧ + (∀ t ∈ Icc 0 T, HasDerivWithinAt (fun s => (q s).im) + (2*(q t).im/(((q t).re-u t)^2+(q t).im^2)) (Icc 0 T) t) := by + let D (t : ℝ) := ((q t).re-u t)^2+(q t).im^2 + have hD (t : ℝ) (ht : t ∈ Icc 0 T) : 0 < D t := by + have hp := hpos t ht + dsimp [D] + positivity + have hre (t : ℝ) (ht : t ∈ Icc 0 T) : HasDerivWithinAt + (fun s => (q s).re-u s) (-2*((q t).re-u t)/D t-u' t) (Icc 0 T) t := by + have hd := (Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt t (hq t ht)).sub (hu t ht) + change HasDerivWithinAt (fun s => (q s).re-u s) + ((-2/(q t-(u t : ℂ))).re-u' t) (Icc 0 T) t at hd + have he : (-2/(q t-(u t : ℂ))).re = -2*((q t).re-u t)/D t := by + simp [D,Complex.div_re,Complex.normSq_apply,pow_two] + rwa [he] at hd + refine ⟨damped_real_abs_barrier hL (fun s => (q s).re-u s) (fun s => -u' s) D + (sub_eq_zero.mpr hstart) (fun t ht => ?_) hD (fun t ht => by simpa using hbound t ht),?_⟩ + · convert hre t ht using 1 + · intro t ht + have hd := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt t (hq t ht) + change HasDerivWithinAt (fun s => (q s).im) + ((-2/(q t-(u t : ℂ))).im) (Icc 0 T) t at hd + have he : (-2/(q t-(u t : ℂ))).im = + 2*(q t).im/(((q t).re-u t)^2+(q t).im^2) := by + simp [Complex.div_im,Complex.normSq_apply,pow_two] + ring + rwa [he] at hd + +theorem reverse_loewner_height_lower {T L : ℝ} (hL : 0 ≤ L) + (q : ℝ → ℂ) (u u' : ℝ → ℝ) + (hq : ∀ t ∈ Icc 0 T, HasDerivWithinAt q + (-2/(q t-(u t : ℂ))) (Icc 0 T) t) + (hu : ∀ t ∈ Icc 0 T, HasDerivWithinAt u (u' t) (Icc 0 T) t) + (hbound : ∀ t ∈ Icc 0 T, |u' t| ≤ L) + (hpos : ∀ t ∈ Icc 0 T, 0 < (q t).im) + (hstart : (q 0).re=u 0) (hsmall : L^2*T ≤ 1) : + ∀ t ∈ Icc 0 T, 2*t ≤ (q t).im^2 := by + obtain ⟨hx,hy⟩ := reverse_centered_coordinates hL q u u' hq hu hbound hpos hstart + exact reverse_height_square_barrier (fun s => (q s).re-u s) (fun s => (q s).im) + hpos hy hx (fun t ht => + (mul_le_mul_of_nonneg_left ht.2 (sq_nonneg L)).trans hsmall) + +theorem IsCapacityTwoWitness.inverse_tip_height_lower + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (U' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => U s.toNNReal) + (U' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hbound : ∀ t ∈ Icc 0 (T : ℝ), |U' t| ≤ L) + (hsmall : L^2*(T : ℝ) ≤ 1) {y : ℝ} (hy : 0 < y) : + 2*(T : ℝ) ≤ (F T ((U T : ℂ)+(y : ℂ)*I)).im^2 := by + let w : ℂ := (U T : ℂ)+(y : ℂ)*I + have hw : w ∈ upperHalfPlane := by simpa [w,upperHalfPlane] using hy + let q (s : ℝ) := G ((T : ℝ)-s).toNNReal (F T w) + let u (s : ℝ) := U ((T : ℝ)-s).toNNReal + have hreflect (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + (T : ℝ)-s ∈ Icc 0 (T : ℝ) := ⟨sub_nonneg.mpr hs.2,sub_le_self _ hs.1⟩ + have hu (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : HasDerivWithinAt u + (-U' ((T : ℝ)-s)) (Icc 0 (T : ℝ)) s := by + have hi : HasDerivWithinAt (fun r : ℝ => (T : ℝ)-r) (-1) + (Icc 0 (T : ℝ)) s := by + simpa using ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt + simpa only [Function.comp_def,mul_neg,mul_one] using + (hd _ (hreflect s hs)).comp s hi hreflect + have hq0 : q 0=w := by + simp only [q,sub_zero,Real.toNNReal_coe] + exact h.2.2.2.2.2.2.2.2.1 T w hw + have hqT : q T=F T w := by + simp only [q,sub_self,Real.toNNReal_zero] + exact h.initial_value _ (h.2.2.2.2.2.2.1 T hw).1.1 + have hh := reverse_loewner_height_lower hL q u + (fun s => -U' ((T : ℝ)-s)) (fun s hs => h.reverse_path_unclipped T hw hs) hu + (fun s hs => by simpa using hbound _ (hreflect s hs)) + (fun s hs => h.imag_positive _ _ (survivingDomain_antitone γ + ((Real.toNNReal_le_iff_le_coe).mpr (hreflect s hs).2) (h.2.2.2.2.2.2.1 T hw))) + (by simp [hq0,w,u]) hsmall T ⟨T.coe_nonneg,le_rfl⟩ + simpa only [hqT,w] using hh + +theorem IsCapacityTwoWitness.smooth_trace_height_lower + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (U' : ℝ → ℝ) + (hd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt (fun s => U s.toNNReal) + (U' t) (Icc 0 (T : ℝ)) t) + {L : ℝ} (hL : 0 ≤ L) (hbound : ∀ t ∈ Icc 0 (T : ℝ), |U' t| ≤ L) + (hsmall : L^2*(T : ℝ) ≤ 1) : + 2*(T : ℝ) ≤ (γ T).im^2 := by + have hl := ((Complex.continuous_im.continuousAt.tendsto.comp + (h.2.2.2.2.2.2.2.2.2.2.2.2 T)).pow 2) + apply ge_of_tendsto hl + filter_upwards [self_mem_nhdsWithin] with y hy + exact h.inverse_tip_height_lower T U' hd hL hbound hsmall hy + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedDriverConcatenation.lean b/lean/OAI/Probability/SLE/Loewner/StoppedDriverConcatenation.lean index bcaa31d8bcfedc81ce6c4ba5b8f7f79ed524f557..0cf231c1c297843637ead48e5a3c3d6d26bb34af 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedDriverConcatenation.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedDriverConcatenation.lean @@ -1,152 +1,152 @@ -import OAI.Probability.SLE.Loewner.RawRestartAdaptation - -namespace OAI - -/-! Reconstruct the old driver through τ+t from its stopped past at τ -and the fresh prefix through t. The reconstruction is jointly measurable -in exactly those two past sigma algebras. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -local instance concatDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance concatDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -def restartDriverPathAt (T : ℝ≥0) (p : C(ℝ≥0,ℝ)) : C(ℝ≥0,ℝ) := - ⟨fun s => p (T+s)-p T, - (p.continuous.comp (continuous_const.add continuous_id)).sub continuous_const⟩ - -def concatenateDriverPath (T : ℝ≥0) (p q : C(ℝ≥0,ℝ)) : C(ℝ≥0,ℝ) where - toFun s := if s ≤ T then p s else p T+q (s-T)-q 0 - continuous_toFun := by - change Continuous ((Iic T).piecewise p (fun s => p T+q (s-T)-q 0)) - apply Continuous.piecewise ?_ p.continuous - ((continuous_const.add (q.continuous.comp (continuous_id.sub continuous_const))).sub continuous_const) - intro s hs - have he : s=T := by simpa only [frontier_Iic,mem_singleton_iff] using hs - change p s = p T+q (s-T)-q 0 - simp only [he,tsub_self,add_sub_cancel_right] - -lemma measurable_concatenateDriverPath : - Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => - concatenateDriverPath x.1 x.2.1 x.2.2) := by - apply source_path_measurable - intro s - have hp : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.1 s) := - (continuous_eval_const s).measurable.comp (measurable_fst.comp measurable_snd) - have hpT : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.1 x.1) := - continuous_eval.measurable.comp ((measurable_fst.comp measurable_snd).prodMk measurable_fst) - have hq : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.2 (s-x.1)) := - continuous_eval.measurable.comp ((measurable_snd.comp measurable_snd).prodMk - (measurable_const.sub measurable_fst)) - have hq0 : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.2 0) := - (continuous_eval_const 0).measurable.comp (measurable_snd.comp measurable_snd) - exact hp.ite (measurableSet_le measurable_const measurable_fst) ((hpT.add hq).sub hq0) - -theorem stoppedDriverPath_concatenation (p : C(ℝ≥0,ℝ)) (T t : ℝ≥0) : - driverPathStopped (T+t) p = concatenateDriverPath T (driverPathStopped T p) - (driverPathStopped t (restartDriverPathAt T p)) := by - ext s - by_cases hs : s ≤ T - · have hs' : s ≤ T+t := hs.trans (le_add_of_nonneg_right (show 0 ≤ t from zero_le)) - simp only [driverPathStopped,concatenateDriverPath,ContinuousMap.coe_mk,ite_eq_left hs, - min_eq_left hs,min_eq_left hs'] - · have hTs : T ≤ s := (lt_of_not_ge hs).le - have he : T+min (s-T) t=min s (T+t) := by - rw [add_min,add_tsub_cancel_of_le hTs] - simp only [driverPathStopped,concatenateDriverPath,restartDriverPathAt, - ContinuousMap.coe_mk,ite_eq_right hs,min_self,zero_min,add_zero,he] - ring - -lemma restartDriverPathAt_add (p : C(ℝ≥0,ℝ)) (T t : ℝ≥0) : - restartDriverPathAt t (restartDriverPathAt T p)=restartDriverPathAt (T+t) p := by - ext s - change (p (T+(t+s))-p T)-(p (T+t)-p T)=p ((T+t)+s)-p (T+t) - rw [add_assoc] - ring - -lemma concatenate_stopped_restart (p : C(ℝ≥0,ℝ)) (t : ℝ≥0) : - concatenateDriverPath t (driverPathStopped t p) (restartDriverPathAt t p)=p := by - ext s - by_cases hs : s ≤ t - · simp only [concatenateDriverPath,driverPathStopped,ContinuousMap.coe_mk, - ite_eq_left hs,min_eq_left hs] - · have hts : t ≤ s := (lt_of_not_ge hs).le - simp only [concatenateDriverPath,driverPathStopped,restartDriverPathAt,ContinuousMap.coe_mk, - ite_eq_right hs,min_self,add_tsub_cancel_of_le hts,add_zero,sub_self,sub_zero] - ring - -lemma sourceRestartDriverPath_add - {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) - (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : - restartDriverPathAt t (sourceRestartDriverPath B hc τ ω)= - sourceRestartDriverPath B hc (fun ω => τ ω+t) ω := - restartDriverPathAt_add (sourceDriverPath B hc ω) (τ ω) t - -lemma sourceRestartDriverPath_split - {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) - (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : - sourceRestartDriverPath B hc τ ω = - concatenateDriverPath t (driverPathStopped t (sourceRestartDriverPath B hc τ ω)) - (sourceRestartDriverPath B hc (fun ω => τ ω+t) ω) := by - rw [←sourceRestartDriverPath_add] - exact (concatenate_stopped_restart _ _).symm - -lemma source_stoppedDriverPath_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) : - Measurable[hτ.measurableSpace] - (fun ω => driverPathStopped (τ ω) (sourceDriverPath B hc ω)) := by - have hprog := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc - let : MeasurableSpace Ω := hτ.measurableSpace - apply source_path_measurable - intro t - have hh := (measurable_stoppedValue hprog (hτ.min_const t)).mono - (IsStoppingTime.measurableSpace_mono (hτ.min_const t) hτ (fun _ => min_le_left _ _)) le_rfl - change Measurable[hτ.measurableSpace] - (fun ω => B ((min (τ ω : WithTop ℝ≥0) (t : WithTop ℝ≥0)).untopA) ω) at hh - change Measurable[hτ.measurableSpace] (fun ω => B (min t (τ ω)) ω) - simpa only [← WithTop.coe_min,WithTop.untopA,WithTop.untopD_coe,min_comm] using hh - -lemma source_stoppedDriverTime_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ} - (τ : Ω → ℝ≥0) (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) : - Measurable[hτ.measurableSpace] τ := by - convert hτ.measurable.untopA using 1 - funext ω - rw [WithTop.untopA_eq_untop WithTop.coe_ne_top] - rfl - -theorem measurable_past_fresh_prefix_concatenation - {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) (t : ℝ≥0) : - Measurable[hτ.measurableSpace.prod (driverPathFiltration t)] - (fun x : Ω × C(ℝ≥0,ℝ) => concatenateDriverPath (τ x.1) - (driverPathStopped (τ x.1) (sourceDriverPath B hc x.1)) (driverPathStopped t x.2)) := by - exact measurable_concatenateDriverPath.comp - (((source_stoppedDriverTime_measurable τ hτ).comp measurable_fst).prodMk - (((source_stoppedDriverPath_measurable B hm hc τ hτ).comp measurable_fst).prodMk - ((driverPathStopped_measurable_past t).comp measurable_snd))) - -theorem source_stoppedDriverPath_restart_identity - {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) - (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : - driverPathStopped (τ ω+t) (sourceDriverPath B hc ω) = - concatenateDriverPath (τ ω) (driverPathStopped (τ ω) (sourceDriverPath B hc ω)) - (driverPathStopped t (sourceRestartDriverPath B hc τ ω)) := by - exact stoppedDriverPath_concatenation (sourceDriverPath B hc ω) (τ ω) t - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawRestartAdaptation + +namespace OAI + +/-! Reconstruct the old driver through τ+t from its stopped past at τ +and the fresh prefix through t. The reconstruction is jointly measurable +in exactly those two past sigma algebras. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +local instance concatDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance concatDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +def restartDriverPathAt (T : ℝ≥0) (p : C(ℝ≥0,ℝ)) : C(ℝ≥0,ℝ) := + ⟨fun s => p (T+s)-p T, + (p.continuous.comp (continuous_const.add continuous_id)).sub continuous_const⟩ + +def concatenateDriverPath (T : ℝ≥0) (p q : C(ℝ≥0,ℝ)) : C(ℝ≥0,ℝ) where + toFun s := if s ≤ T then p s else p T+q (s-T)-q 0 + continuous_toFun := by + change Continuous ((Iic T).piecewise p (fun s => p T+q (s-T)-q 0)) + apply Continuous.piecewise ?_ p.continuous + ((continuous_const.add (q.continuous.comp (continuous_id.sub continuous_const))).sub continuous_const) + intro s hs + have he : s=T := by simpa only [frontier_Iic,mem_singleton_iff] using hs + change p s = p T+q (s-T)-q 0 + simp only [he,tsub_self,add_sub_cancel_right] + +lemma measurable_concatenateDriverPath : + Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => + concatenateDriverPath x.1 x.2.1 x.2.2) := by + apply source_path_measurable + intro s + have hp : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.1 s) := + (continuous_eval_const s).measurable.comp (measurable_fst.comp measurable_snd) + have hpT : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.1 x.1) := + continuous_eval.measurable.comp ((measurable_fst.comp measurable_snd).prodMk measurable_fst) + have hq : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.2 (s-x.1)) := + continuous_eval.measurable.comp ((measurable_snd.comp measurable_snd).prodMk + (measurable_const.sub measurable_fst)) + have hq0 : Measurable (fun x : ℝ≥0 × (C(ℝ≥0,ℝ) × C(ℝ≥0,ℝ)) => x.2.2 0) := + (continuous_eval_const 0).measurable.comp (measurable_snd.comp measurable_snd) + exact hp.ite (measurableSet_le measurable_const measurable_fst) ((hpT.add hq).sub hq0) + +theorem stoppedDriverPath_concatenation (p : C(ℝ≥0,ℝ)) (T t : ℝ≥0) : + driverPathStopped (T+t) p = concatenateDriverPath T (driverPathStopped T p) + (driverPathStopped t (restartDriverPathAt T p)) := by + ext s + by_cases hs : s ≤ T + · have hs' : s ≤ T+t := hs.trans (le_add_of_nonneg_right (show 0 ≤ t from zero_le)) + simp only [driverPathStopped,concatenateDriverPath,ContinuousMap.coe_mk,ite_eq_left hs, + min_eq_left hs,min_eq_left hs'] + · have hTs : T ≤ s := (lt_of_not_ge hs).le + have he : T+min (s-T) t=min s (T+t) := by + rw [add_min,add_tsub_cancel_of_le hTs] + simp only [driverPathStopped,concatenateDriverPath,restartDriverPathAt, + ContinuousMap.coe_mk,ite_eq_right hs,min_self,zero_min,add_zero,he] + ring + +lemma restartDriverPathAt_add (p : C(ℝ≥0,ℝ)) (T t : ℝ≥0) : + restartDriverPathAt t (restartDriverPathAt T p)=restartDriverPathAt (T+t) p := by + ext s + change (p (T+(t+s))-p T)-(p (T+t)-p T)=p ((T+t)+s)-p (T+t) + rw [add_assoc] + ring + +lemma concatenate_stopped_restart (p : C(ℝ≥0,ℝ)) (t : ℝ≥0) : + concatenateDriverPath t (driverPathStopped t p) (restartDriverPathAt t p)=p := by + ext s + by_cases hs : s ≤ t + · simp only [concatenateDriverPath,driverPathStopped,ContinuousMap.coe_mk, + ite_eq_left hs,min_eq_left hs] + · have hts : t ≤ s := (lt_of_not_ge hs).le + simp only [concatenateDriverPath,driverPathStopped,restartDriverPathAt,ContinuousMap.coe_mk, + ite_eq_right hs,min_self,add_tsub_cancel_of_le hts,add_zero,sub_self,sub_zero] + ring + +lemma sourceRestartDriverPath_add + {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) + (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : + restartDriverPathAt t (sourceRestartDriverPath B hc τ ω)= + sourceRestartDriverPath B hc (fun ω => τ ω+t) ω := + restartDriverPathAt_add (sourceDriverPath B hc ω) (τ ω) t + +lemma sourceRestartDriverPath_split + {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) + (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : + sourceRestartDriverPath B hc τ ω = + concatenateDriverPath t (driverPathStopped t (sourceRestartDriverPath B hc τ ω)) + (sourceRestartDriverPath B hc (fun ω => τ ω+t) ω) := by + rw [←sourceRestartDriverPath_add] + exact (concatenate_stopped_restart _ _).symm + +lemma source_stoppedDriverPath_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) : + Measurable[hτ.measurableSpace] + (fun ω => driverPathStopped (τ ω) (sourceDriverPath B hc ω)) := by + have hprog := (Filtration.stronglyAdapted_natural (fun t => (hm t).stronglyMeasurable)).isStronglyProgressive_of_continuous hc + let : MeasurableSpace Ω := hτ.measurableSpace + apply source_path_measurable + intro t + have hh := (measurable_stoppedValue hprog (hτ.min_const t)).mono + (IsStoppingTime.measurableSpace_mono (hτ.min_const t) hτ (fun _ => min_le_left _ _)) le_rfl + change Measurable[hτ.measurableSpace] + (fun ω => B ((min (τ ω : WithTop ℝ≥0) (t : WithTop ℝ≥0)).untopA) ω) at hh + change Measurable[hτ.measurableSpace] (fun ω => B (min t (τ ω)) ω) + simpa only [← WithTop.coe_min,WithTop.untopA,WithTop.untopD_coe,min_comm] using hh + +lemma source_stoppedDriverTime_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] {ℱ : Filtration ℝ≥0 mΩ} + (τ : Ω → ℝ≥0) (hτ : IsStoppingTime ℱ (fun ω => (τ ω : WithTop ℝ≥0))) : + Measurable[hτ.measurableSpace] τ := by + convert hτ.measurable.untopA using 1 + funext ω + rw [WithTop.untopA_eq_untop WithTop.coe_ne_top] + rfl + +theorem measurable_past_fresh_prefix_concatenation + {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) (t : ℝ≥0) : + Measurable[hτ.measurableSpace.prod (driverPathFiltration t)] + (fun x : Ω × C(ℝ≥0,ℝ) => concatenateDriverPath (τ x.1) + (driverPathStopped (τ x.1) (sourceDriverPath B hc x.1)) (driverPathStopped t x.2)) := by + exact measurable_concatenateDriverPath.comp + (((source_stoppedDriverTime_measurable τ hτ).comp measurable_fst).prodMk + (((source_stoppedDriverPath_measurable B hm hc τ hτ).comp measurable_fst).prodMk + ((driverPathStopped_measurable_past t).comp measurable_snd))) + +theorem source_stoppedDriverPath_restart_identity + {Ω : Type*} (B : ℝ≥0 → Ω → ℝ) (hc : ∀ ω, Continuous (fun t => B t ω)) + (τ : Ω → ℝ≥0) (t : ℝ≥0) (ω : Ω) : + driverPathStopped (τ ω+t) (sourceDriverPath B hc ω) = + concatenateDriverPath (τ ω) (driverPathStopped (τ ω) (sourceDriverPath B hc ω)) + (driverPathStopped t (sourceRestartDriverPath B hc τ ω)) := by + exact stoppedDriverPath_concatenation (sourceDriverPath B hc ω) (τ ω) t + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInverseExhaustion.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInverseExhaustion.lean index a40520a95ce75092159b428c796007a829315446..cff6774313f944309f2491ddbf8fa36d20bcfc27 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInverseExhaustion.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInverseExhaustion.lean @@ -1,119 +1,119 @@ -import OAI.Probability.SLE.Loewner.StoppedInversePaths - -namespace OAI - -/-! Local terminal exhaustion survives the stopped conformal weighting. -The pulled-back compact need not stay away from the real boundary: finite -terminal weighted mass is the domination used for the limit. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge - -local instance stoppedExhaustionLocallyCompact : LocallyCompactSpace upperHalfPlane := - isOpen_upperHalfPlane.locallyCompactSpace - -lemma weightedImageMass_add - (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) (μ ν : Measure upperHalfPlane) : - weightedImageMass J w (μ+ν) = weightedImageMass J w μ + weightedImageMass J w ν := by - simp only [weightedImageMass,withDensity_add_measure] - exact Measure.map_add _ _ hJ - -lemma weightedImageMass_sub - (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) {μ ν : Measure upperHalfPlane} - [SigmaFinite μ] (hle : μ ≤ ν) - [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : - weightedImageMass J w (ν-μ) = weightedImageMass J w ν-weightedImageMass J w μ := by - let := weightedImageMass_finite_on_compacts J hJ w hle - apply (Measure.add_left_inj (weightedImageMass J w μ) _ _).mp - calc - weightedImageMass J w (ν-μ)+weightedImageMass J w μ = weightedImageMass J w ν := by - rw [← weightedImageMass_add J hJ w,sigmaFinite_sub_add_cancel_of_le hle] - _ = (weightedImageMass J w ν-weightedImageMass J w μ)+weightedImageMass J w μ := - (sigmaFinite_sub_add_cancel_of_le (weightedImageMass_mono J hJ w hle)).symm - -lemma weightedImageMass_terminal_exhaustion - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] - (hNr : ∀ t, (N t).Regular) (hdom : ∀ t, N t ≤ ν) - (hlim : ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) - (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) - [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : - ∀ K : Set upperHalfPlane, IsCompact K → Tendsto - (fun t : ℝ≥0 => (weightedImageMass J w ν-weightedImageMass J w (N t)) K) - atTop (𝓝 0) := by - intro K hK - let g : upperHalfPlane → ℝ≥0∞ := (J ⁻¹' K).indicator w - have hg : Measurable g := hw.indicator (hJ hK.measurableSet) - have hfin : (∫⁻ z, g z ∂ν) ≠ ∞ := by - rw [← weightedImageMass_apply J hJ w ν K hK.measurableSet] - exact hK.measure_lt_top.ne - have he (t : ℝ≥0) : - (weightedImageMass J w ν-weightedImageMass J w (N t)) K = - ∫⁻ z, g z ∂(ν-N t) := by - let := hNr t - rw [← weightedImageMass_sub J hJ w (hdom t), - weightedImageMass_apply J hJ w (ν-N t) K hK.measurableSet] - simpa only [he] using remaining_lintegral_tendsto_zero ν N hNr hdom hlim g hg hfin - -namespace Multipoint - -local instance stoppedExhaustionDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance stoppedExhaustionDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) - (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (τ : Ω → ℝ≥0) (hc : ∀ ω, Continuous (fun t => B t ω)) (R : RawCanonicalMass) - (hm : ∀ t, Measurable (B t)) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - -include hm hτ in -theorem stoppedFreshMass_terminal_exhaustion - {P : Measure Ω} (hκ : 0 < κ) - (hdom : ∀ᵐ ω ∂P, ∀ t, R.process t (sourceRestartDriverPath B hc τ ω) ≤ - R.terminal (sourceRestartDriverPath B hc τ ω)) - (hlim : ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → Tendsto - (fun t : ℝ≥0 => (R.terminal (sourceRestartDriverPath B hc τ ω)- - R.process t (sourceRestartDriverPath B hc τ ω)) K) atTop (𝓝 0)) - (hfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) : - ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → Tendsto - (fun t : ℝ≥0 => (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω- - stoppedFreshMass B κ γ G F hw τ hc R t ω) K) atTop (𝓝 0) := by - have hJ := stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ - have hW := stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ - filter_upwards [hdom,hlim,hfin] with ω hd hl hf - let J := stoppedMassPoint B κ γ G F hw τ ω - let w := stoppedMassWeight B κ F τ ω - have hpair : @Measurable upperHalfPlane (Ω × upperHalfPlane) inferInstance - (hτ.measurableSpace.prod inferInstance) (fun z => (ω,z)) := - @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω - have hJm : Measurable J := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ hpair - have hwm : Measurable w := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW hpair - let N := fun t => R.process t (sourceRestartDriverPath B hc τ ω) - let ν := R.terminal (sourceRestartDriverPath B hc τ ω) - change ∀ K : Set upperHalfPlane, IsCompact K → Tendsto - (fun t : ℝ≥0 => (weightedImageMass J w ν-weightedImageMass J w (N t)) K) atTop (𝓝 0) - exact @weightedImageMass_terminal_exhaustion N ν - (R.terminal_radon (sourceRestartDriverPath B hc τ ω)).1 - (fun t => (R.radon t _).1) hd hl J hJm w hwm hf - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.StoppedInversePaths + +namespace OAI + +/-! Local terminal exhaustion survives the stopped conformal weighting. +The pulled-back compact need not stay away from the real boundary: finite +terminal weighted mass is the domination used for the limit. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge + +local instance stoppedExhaustionLocallyCompact : LocallyCompactSpace upperHalfPlane := + isOpen_upperHalfPlane.locallyCompactSpace + +lemma weightedImageMass_add + (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) (μ ν : Measure upperHalfPlane) : + weightedImageMass J w (μ+ν) = weightedImageMass J w μ + weightedImageMass J w ν := by + simp only [weightedImageMass,withDensity_add_measure] + exact Measure.map_add _ _ hJ + +lemma weightedImageMass_sub + (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) {μ ν : Measure upperHalfPlane} + [SigmaFinite μ] (hle : μ ≤ ν) + [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : + weightedImageMass J w (ν-μ) = weightedImageMass J w ν-weightedImageMass J w μ := by + let := weightedImageMass_finite_on_compacts J hJ w hle + apply (Measure.add_left_inj (weightedImageMass J w μ) _ _).mp + calc + weightedImageMass J w (ν-μ)+weightedImageMass J w μ = weightedImageMass J w ν := by + rw [← weightedImageMass_add J hJ w,sigmaFinite_sub_add_cancel_of_le hle] + _ = (weightedImageMass J w ν-weightedImageMass J w μ)+weightedImageMass J w μ := + (sigmaFinite_sub_add_cancel_of_le (weightedImageMass_mono J hJ w hle)).symm + +lemma weightedImageMass_terminal_exhaustion + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] + (hNr : ∀ t, (N t).Regular) (hdom : ∀ t, N t ≤ ν) + (hlim : ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) + (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) + [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : + ∀ K : Set upperHalfPlane, IsCompact K → Tendsto + (fun t : ℝ≥0 => (weightedImageMass J w ν-weightedImageMass J w (N t)) K) + atTop (𝓝 0) := by + intro K hK + let g : upperHalfPlane → ℝ≥0∞ := (J ⁻¹' K).indicator w + have hg : Measurable g := hw.indicator (hJ hK.measurableSet) + have hfin : (∫⁻ z, g z ∂ν) ≠ ∞ := by + rw [← weightedImageMass_apply J hJ w ν K hK.measurableSet] + exact hK.measure_lt_top.ne + have he (t : ℝ≥0) : + (weightedImageMass J w ν-weightedImageMass J w (N t)) K = + ∫⁻ z, g z ∂(ν-N t) := by + let := hNr t + rw [← weightedImageMass_sub J hJ w (hdom t), + weightedImageMass_apply J hJ w (ν-N t) K hK.measurableSet] + simpa only [he] using remaining_lintegral_tendsto_zero ν N hNr hdom hlim g hg hfin + +namespace Multipoint + +local instance stoppedExhaustionDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance stoppedExhaustionDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) + (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (τ : Ω → ℝ≥0) (hc : ∀ ω, Continuous (fun t => B t ω)) (R : RawCanonicalMass) + (hm : ∀ t, Measurable (B t)) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + +include hm hτ in +theorem stoppedFreshMass_terminal_exhaustion + {P : Measure Ω} (hκ : 0 < κ) + (hdom : ∀ᵐ ω ∂P, ∀ t, R.process t (sourceRestartDriverPath B hc τ ω) ≤ + R.terminal (sourceRestartDriverPath B hc τ ω)) + (hlim : ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → Tendsto + (fun t : ℝ≥0 => (R.terminal (sourceRestartDriverPath B hc τ ω)- + R.process t (sourceRestartDriverPath B hc τ ω)) K) atTop (𝓝 0)) + (hfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) : + ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → Tendsto + (fun t : ℝ≥0 => (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω- + stoppedFreshMass B κ γ G F hw τ hc R t ω) K) atTop (𝓝 0) := by + have hJ := stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ + have hW := stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ + filter_upwards [hdom,hlim,hfin] with ω hd hl hf + let J := stoppedMassPoint B κ γ G F hw τ ω + let w := stoppedMassWeight B κ F τ ω + have hpair : @Measurable upperHalfPlane (Ω × upperHalfPlane) inferInstance + (hτ.measurableSpace.prod inferInstance) (fun z => (ω,z)) := + @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω + have hJm : Measurable J := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ hpair + have hwm : Measurable w := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW hpair + let N := fun t => R.process t (sourceRestartDriverPath B hc τ ω) + let ν := R.terminal (sourceRestartDriverPath B hc τ ω) + change ∀ K : Set upperHalfPlane, IsCompact K → Tendsto + (fun t : ℝ≥0 => (weightedImageMass J w ν-weightedImageMass J w (N t)) K) atTop (𝓝 0) + exact @weightedImageMass_terminal_exhaustion N ν + (R.terminal_radon (sourceRestartDriverPath B hc τ ω)).1 + (fun t => (R.radon t _).1) hd hl J hJm w hwm hf + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInverseGreenMean.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInverseGreenMean.lean index 55c169489a6a0a1dc56a11b843a14f6364de052b..fd3b0f74370126850a6c83c85c23db871583c0dd 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInverseGreenMean.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInverseGreenMean.lean @@ -1,110 +1,110 @@ -import OAI.Probability.SLE.Mass.StoppedInverseMass -import OAI.Probability.SLE.Geometry.RestartHalfPlaneGreen -import OAI.Probability.SLE.Loewner.FreshCapacityWitness - -namespace OAI - -/-! The actual weighted fresh terminal mass has exactly the same stopped -Green kernel as the remaining original mass. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), - IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') - -include C hB hc hκ F hw τ hτ R he hFresh in -theorem CanonicalMass.stoppedFreshTerminalMass_green_test - (f : Ω × upperHalfPlane → ℝ≥0∞) - (hf : Measurable[hτ.measurableSpace.prod inferInstance] f) : - (∫⁻ ω, ∫⁻ x, f (ω,x) ∂(stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) ∂P) = - ∫⁻ ω, ∫⁻ x, f (ω,x) ∂halfPlaneGreenMeasure κ B γ G (τ ω) ω ∂P := by - rw [C.stoppedFreshTerminalMass_test B κ γ G F hw τ - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hτ hB hκ R he f hf] - apply lintegral_congr_ae - filter_upwards [hFresh] with ω hω - obtain ⟨β,G',F',hb⟩ := hω - exact restart_halfPlane_green_test (hw ω) ((hc ω).const_mul _) (τ ω) hb hκ - (fun x => f (ω,x)) - -omit [IsProbabilityMeasure P] in -lemma past_rectangle_lintegral - (μ : Ω → Measure upperHalfPlane) (E : Set Ω) (hE : MeasurableSet E) - (A : Set upperHalfPlane) (hA : MeasurableSet A) : - (∫⁻ ω, ∫⁻ z, (E ×ˢ A).indicator (fun _ => (1:ℝ≥0∞)) (ω,z) ∂μ ω ∂P) = - ∫⁻ ω in E, μ ω A ∂P := by - rw [← lintegral_indicator hE] - apply lintegral_congr - intro ω - by_cases hω : ω ∈ E - · have hsec : (fun z => (E ×ˢ A).indicator (fun _ => (1:ℝ≥0∞)) (ω,z)) = - A.indicator (fun _ => 1) := by - funext z - simp only [Set.indicator,Set.mem_prod,hω,true_and] - rw [hsec,lintegral_indicator hA,lintegral_one] - simp only [Set.indicator_of_mem hω,Measure.restrict_apply_univ] - · have hsec : (fun z => (E ×ˢ A).indicator (fun _ => (1:ℝ≥0∞)) (ω,z)) = 0 := by - funext z - simp only [Set.indicator,Set.mem_prod,hω,false_and,ite_false,Pi.zero_apply] - rw [hsec] - change (∫⁻ z, (0:ℝ≥0∞) ∂μ ω) = _ - simp only [lintegral_zero,Set.indicator_of_notMem hω] - -include C hB hc hκ F hw τ hτ R he hFresh in -theorem CanonicalMass.stoppedFreshTerminalMass_kernel - (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) : - (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) = - (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) := by - have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - have hμ := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R - have hν : Measurable (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw τ - (stopping_time_measurable_nnreal hτ)).ennreal_ofReal.lintegral_prod_right' - apply Measure.ext - intro A hA - rw [Measure.bind_apply hA hμ.aemeasurable,Measure.bind_apply hA hν.aemeasurable] - have ht := C.stoppedFreshTerminalMass_green_test hB hc hκ F hw τ hτ R he hFresh - ((E ×ˢ A).indicator (fun _ => 1)) (measurable_const.indicator (hE.prod hA)) - rw [past_rectangle_lintegral _ E (hτ.measurableSpace_le E hE) A hA, - past_rectangle_lintegral _ E (hτ.measurableSpace_le E hE) A hA] at ht - exact ht - -include C hB hc hκ F hw τ hτ R he in -theorem CanonicalMass.stoppedFreshTerminalMass_published_kernel - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) : - (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) = - (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) := by - exact C.stoppedFreshTerminalMass_kernel hB hc hκ F hw τ hτ R he - (fresh_capacity_witness_exists hB - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hpublished τ hτ) E hE - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.StoppedInverseMass +import OAI.Probability.SLE.Geometry.RestartHalfPlaneGreen +import OAI.Probability.SLE.Loewner.FreshCapacityWitness + +namespace OAI + +/-! The actual weighted fresh terminal mass has exactly the same stopped +Green kernel as the remaining original mass. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), + IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') + +include C hB hc hκ F hw τ hτ R he hFresh in +theorem CanonicalMass.stoppedFreshTerminalMass_green_test + (f : Ω × upperHalfPlane → ℝ≥0∞) + (hf : Measurable[hτ.measurableSpace.prod inferInstance] f) : + (∫⁻ ω, ∫⁻ x, f (ω,x) ∂(stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) ∂P) = + ∫⁻ ω, ∫⁻ x, f (ω,x) ∂halfPlaneGreenMeasure κ B γ G (τ ω) ω ∂P := by + rw [C.stoppedFreshTerminalMass_test B κ γ G F hw τ + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hτ hB hκ R he f hf] + apply lintegral_congr_ae + filter_upwards [hFresh] with ω hω + obtain ⟨β,G',F',hb⟩ := hω + exact restart_halfPlane_green_test (hw ω) ((hc ω).const_mul _) (τ ω) hb hκ + (fun x => f (ω,x)) + +omit [IsProbabilityMeasure P] in +lemma past_rectangle_lintegral + (μ : Ω → Measure upperHalfPlane) (E : Set Ω) (hE : MeasurableSet E) + (A : Set upperHalfPlane) (hA : MeasurableSet A) : + (∫⁻ ω, ∫⁻ z, (E ×ˢ A).indicator (fun _ => (1:ℝ≥0∞)) (ω,z) ∂μ ω ∂P) = + ∫⁻ ω in E, μ ω A ∂P := by + rw [← lintegral_indicator hE] + apply lintegral_congr + intro ω + by_cases hω : ω ∈ E + · have hsec : (fun z => (E ×ˢ A).indicator (fun _ => (1:ℝ≥0∞)) (ω,z)) = + A.indicator (fun _ => 1) := by + funext z + simp only [Set.indicator,Set.mem_prod,hω,true_and] + rw [hsec,lintegral_indicator hA,lintegral_one] + simp only [Set.indicator_of_mem hω,Measure.restrict_apply_univ] + · have hsec : (fun z => (E ×ˢ A).indicator (fun _ => (1:ℝ≥0∞)) (ω,z)) = 0 := by + funext z + simp only [Set.indicator,Set.mem_prod,hω,false_and,ite_false,Pi.zero_apply] + rw [hsec] + change (∫⁻ z, (0:ℝ≥0∞) ∂μ ω) = _ + simp only [lintegral_zero,Set.indicator_of_notMem hω] + +include C hB hc hκ F hw τ hτ R he hFresh in +theorem CanonicalMass.stoppedFreshTerminalMass_kernel + (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) : + (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) = + (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) := by + have hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + have hμ := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R + have hν : Measurable (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw τ + (stopping_time_measurable_nnreal hτ)).ennreal_ofReal.lintegral_prod_right' + apply Measure.ext + intro A hA + rw [Measure.bind_apply hA hμ.aemeasurable,Measure.bind_apply hA hν.aemeasurable] + have ht := C.stoppedFreshTerminalMass_green_test hB hc hκ F hw τ hτ R he hFresh + ((E ×ˢ A).indicator (fun _ => 1)) (measurable_const.indicator (hE.prod hA)) + rw [past_rectangle_lintegral _ E (hτ.measurableSpace_le E hE) A hA, + past_rectangle_lintegral _ E (hτ.measurableSpace_le E hE) A hA] at ht + exact ht + +include C hB hc hκ F hw τ hτ R he in +theorem CanonicalMass.stoppedFreshTerminalMass_published_kernel + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (E : Set Ω) (hE : MeasurableSet[hτ.measurableSpace] E) : + (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) = + (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) := by + exact C.stoppedFreshTerminalMass_kernel hB hc hκ F hw τ hτ R he + (fresh_capacity_witness_exists hB + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hpublished τ hτ) E hE + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInverseKernel.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInverseKernel.lean index ed80d8bfe94738ea98e05dea0772bca32d79dae7..8a3d2453217875ea4b5b68ad3bdf9f29318e18e3 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInverseKernel.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInverseKernel.lean @@ -1,137 +1,137 @@ -import OAI.Probability.SLE.Loewner.RawRestartRandomTests -import OAI.Probability.SLE.Geometry.StoppedFutureGreen -import OAI.Probability.SLE.Loewner.StoppedInversePaths - -namespace OAI - -/-! The actual inverse image of the fresh canonical mass has the original -Green remainder kernel in the full filtration after the restart. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -lemma weightedImageMass_add (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) (μ ν : Measure upperHalfPlane) : - weightedImageMass J w (μ+ν)=weightedImageMass J w μ+weightedImageMass J w ν := by - simp only [weightedImageMass,withDensity_add_measure] - exact Measure.map_add _ _ hJ - -lemma weighted_rectangle_lintegral - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (μ : Ω → Measure upperHalfPlane) - (J : Ω → upperHalfPlane → upperHalfPlane) (hJ : ∀ ω, Measurable (J ω)) - (w : Ω → upperHalfPlane → ℝ≥0∞) (hw : ∀ ω, Measurable (w ω)) - (E : Set Ω) (hE : MeasurableSet E) (A : Set upperHalfPlane) (hA : MeasurableSet A) : - (∫⁻ ω, ∫⁻ z, w ω z * (E ×ˢ A).indicator (fun _ => 1) (ω,J ω z) ∂μ ω ∂P) = - ∫⁻ ω in E, weightedImageMass (J ω) (w ω) (μ ω) A ∂P := by - have he (ω : Ω) : - (∫⁻ z, w ω z * (E ×ˢ A).indicator (fun _ => 1) (ω,J ω z) ∂μ ω) = - ∫⁻ x, (E ×ˢ A).indicator (fun _ => 1) (ω,x) - ∂weightedImageMass (J ω) (w ω) (μ ω) := by - exact (weightedImageMass_lintegral _ (hJ ω) _ (hw ω) _ _ - ((measurable_const.indicator (hE.prod hA)).comp measurable_prodMk_left)).symm - simp_rw [he] - exact past_rectangle_lintegral _ E hE A hA - -local instance stoppedKernelDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance stoppedKernelDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), - IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') - -include C hB hc hκ F hw R he τ hτ hFresh in -theorem CanonicalMass.stoppedFreshMass_green_kernel - (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : - (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) = - (P.restrict E).bind (stoppedFreshMass B κ γ G F hw τ hc R t)+ - (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R - have hNm := (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ).measurable (i := t) - have hGm : Measurable (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw - (fun ω => τ ω+t) (stopping_time_measurable_nnreal (stopping_time_add_nnreal hτ t))).ennreal_ofReal.lintegral_prod_right' - have hle : hτ.measurableSpace ≤ shiftedStoppingFiltration hτ t := - IsStoppingTime.measurableSpace_mono hτ (stopping_time_add_nnreal hτ t) - (fun ω => WithTop.coe_le_coe.mpr (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) - have hprod : hτ.measurableSpace.prod (inferInstance : MeasurableSpace upperHalfPlane) ≤ - (shiftedStoppingFiltration hτ t).prod inferInstance := - sup_le_sup (MeasurableSpace.comap_mono hle) le_rfl - have hJ := stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ - have hW := stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ - have hpair (ω : Ω) : @Measurable upperHalfPlane (Ω × upperHalfPlane) inferInstance - (hτ.measurableSpace.prod inferInstance) (fun z => (ω,z)) := - @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω - have hJsec (ω : Ω) : Measurable (stoppedMassPoint B κ γ G F hw τ ω) := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ (hpair ω) - have hWsec (ω : Ω) : Measurable (stoppedMassWeight B κ F τ ω) := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW (hpair ω) - have hcov := stopped_future_green_covariance B κ γ G F hw τ hm hc hτ hκ hFresh - have hE' : MeasurableSet E := (shiftedStoppingFiltration hτ).le t _ hE - apply Measure.ext - intro A hA - rw [Measure.bind_apply hA hTm.aemeasurable,Measure.add_apply, - Measure.bind_apply hA hNm.aemeasurable,Measure.bind_apply hA hGm.aemeasurable] - let g : Ω × upperHalfPlane → ℝ≥0∞ := fun p => stoppedMassWeight B κ F τ p.1 p.2 * - (E ×ˢ A).indicator (fun _ => 1) (p.1,stoppedMassPoint B κ γ G F hw τ p.1 p.2) - have hg : Measurable[(shiftedStoppingFiltration hτ t).prod inferInstance] g := - (hW.mono hprod le_rfl).mul - ((measurable_const.indicator (hE.prod hA)).comp - (measurable_fst.prodMk (hJ.mono hprod le_rfl))) - have hh := C.raw_restarted_random_green_test hB hc F hw R he τ hτ t g hg - change (∫⁻ ω, ∫⁻ z, stoppedMassWeight B κ F τ ω z * - (E ×ˢ A).indicator (fun _ => 1) (ω,stoppedMassPoint B κ γ G F hw τ ω z) - ∂(R.terminal (sourceRestartDriverPath B hc τ ω)) ∂P) = - (∫⁻ ω, ∫⁻ z, stoppedMassWeight B κ F τ ω z * - (E ×ˢ A).indicator (fun _ => 1) (ω,stoppedMassPoint B κ γ G F hw τ ω z) - ∂(R.process t (sourceRestartDriverPath B hc τ ω)+ - canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) ∂P) at hh - rw [weighted_rectangle_lintegral P _ _ hJsec - _ hWsec E hE' A hA, - weighted_rectangle_lintegral P _ _ hJsec - _ hWsec E hE' A hA] at hh - calc - _ = ∫⁻ ω in E, weightedImageMass (stoppedMassPoint B κ γ G F hw τ ω) - (stoppedMassWeight B κ F τ ω) - (R.process t (sourceRestartDriverPath B hc τ ω)+ - canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) A ∂P := hh - _ = ∫⁻ ω in E, stoppedFreshMass B κ γ G F hw τ hc R t ω A + - halfPlaneGreenMeasure κ B γ G (τ ω+t) ω A ∂P := by - apply lintegral_congr_ae - filter_upwards [ae_restrict_of_ae hcov] with ω hω - rw [weightedImageMass_add _ (hJsec ω),Measure.add_apply,hω t] - rfl - _ = _ := by - exact lintegral_add_left ((Measure.measurable_coe hA).comp hNm) _ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawRestartRandomTests +import OAI.Probability.SLE.Geometry.StoppedFutureGreen +import OAI.Probability.SLE.Loewner.StoppedInversePaths + +namespace OAI + +/-! The actual inverse image of the fresh canonical mass has the original +Green remainder kernel in the full filtration after the restart. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +lemma weightedImageMass_add (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) (μ ν : Measure upperHalfPlane) : + weightedImageMass J w (μ+ν)=weightedImageMass J w μ+weightedImageMass J w ν := by + simp only [weightedImageMass,withDensity_add_measure] + exact Measure.map_add _ _ hJ + +lemma weighted_rectangle_lintegral + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (μ : Ω → Measure upperHalfPlane) + (J : Ω → upperHalfPlane → upperHalfPlane) (hJ : ∀ ω, Measurable (J ω)) + (w : Ω → upperHalfPlane → ℝ≥0∞) (hw : ∀ ω, Measurable (w ω)) + (E : Set Ω) (hE : MeasurableSet E) (A : Set upperHalfPlane) (hA : MeasurableSet A) : + (∫⁻ ω, ∫⁻ z, w ω z * (E ×ˢ A).indicator (fun _ => 1) (ω,J ω z) ∂μ ω ∂P) = + ∫⁻ ω in E, weightedImageMass (J ω) (w ω) (μ ω) A ∂P := by + have he (ω : Ω) : + (∫⁻ z, w ω z * (E ×ˢ A).indicator (fun _ => 1) (ω,J ω z) ∂μ ω) = + ∫⁻ x, (E ×ˢ A).indicator (fun _ => 1) (ω,x) + ∂weightedImageMass (J ω) (w ω) (μ ω) := by + exact (weightedImageMass_lintegral _ (hJ ω) _ (hw ω) _ _ + ((measurable_const.indicator (hE.prod hA)).comp measurable_prodMk_left)).symm + simp_rw [he] + exact past_rectangle_lintegral _ E hE A hA + +local instance stoppedKernelDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance stoppedKernelDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), + IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') + +include C hB hc hκ F hw R he τ hτ hFresh in +theorem CanonicalMass.stoppedFreshMass_green_kernel + (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : + (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) = + (P.restrict E).bind (stoppedFreshMass B κ γ G F hw τ hc R t)+ + (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R + have hNm := (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ).measurable (i := t) + have hGm : Measurable (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw + (fun ω => τ ω+t) (stopping_time_measurable_nnreal (stopping_time_add_nnreal hτ t))).ennreal_ofReal.lintegral_prod_right' + have hle : hτ.measurableSpace ≤ shiftedStoppingFiltration hτ t := + IsStoppingTime.measurableSpace_mono hτ (stopping_time_add_nnreal hτ t) + (fun ω => WithTop.coe_le_coe.mpr (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) + have hprod : hτ.measurableSpace.prod (inferInstance : MeasurableSpace upperHalfPlane) ≤ + (shiftedStoppingFiltration hτ t).prod inferInstance := + sup_le_sup (MeasurableSpace.comap_mono hle) le_rfl + have hJ := stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ + have hW := stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ + have hpair (ω : Ω) : @Measurable upperHalfPlane (Ω × upperHalfPlane) inferInstance + (hτ.measurableSpace.prod inferInstance) (fun z => (ω,z)) := + @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω + have hJsec (ω : Ω) : Measurable (stoppedMassPoint B κ γ G F hw τ ω) := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ (hpair ω) + have hWsec (ω : Ω) : Measurable (stoppedMassWeight B κ F τ ω) := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW (hpair ω) + have hcov := stopped_future_green_covariance B κ γ G F hw τ hm hc hτ hκ hFresh + have hE' : MeasurableSet E := (shiftedStoppingFiltration hτ).le t _ hE + apply Measure.ext + intro A hA + rw [Measure.bind_apply hA hTm.aemeasurable,Measure.add_apply, + Measure.bind_apply hA hNm.aemeasurable,Measure.bind_apply hA hGm.aemeasurable] + let g : Ω × upperHalfPlane → ℝ≥0∞ := fun p => stoppedMassWeight B κ F τ p.1 p.2 * + (E ×ˢ A).indicator (fun _ => 1) (p.1,stoppedMassPoint B κ γ G F hw τ p.1 p.2) + have hg : Measurable[(shiftedStoppingFiltration hτ t).prod inferInstance] g := + (hW.mono hprod le_rfl).mul + ((measurable_const.indicator (hE.prod hA)).comp + (measurable_fst.prodMk (hJ.mono hprod le_rfl))) + have hh := C.raw_restarted_random_green_test hB hc F hw R he τ hτ t g hg + change (∫⁻ ω, ∫⁻ z, stoppedMassWeight B κ F τ ω z * + (E ×ˢ A).indicator (fun _ => 1) (ω,stoppedMassPoint B κ γ G F hw τ ω z) + ∂(R.terminal (sourceRestartDriverPath B hc τ ω)) ∂P) = + (∫⁻ ω, ∫⁻ z, stoppedMassWeight B κ F τ ω z * + (E ×ˢ A).indicator (fun _ => 1) (ω,stoppedMassPoint B κ γ G F hw τ ω z) + ∂(R.process t (sourceRestartDriverPath B hc τ ω)+ + canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) ∂P) at hh + rw [weighted_rectangle_lintegral P _ _ hJsec + _ hWsec E hE' A hA, + weighted_rectangle_lintegral P _ _ hJsec + _ hWsec E hE' A hA] at hh + calc + _ = ∫⁻ ω in E, weightedImageMass (stoppedMassPoint B κ γ G F hw τ ω) + (stoppedMassWeight B κ F τ ω) + (R.process t (sourceRestartDriverPath B hc τ ω)+ + canonicalGreenMass κ t (sourceRestartDriverPath B hc τ ω)) A ∂P := hh + _ = ∫⁻ ω in E, stoppedFreshMass B κ γ G F hw τ hc R t ω A + + halfPlaneGreenMeasure κ B γ G (τ ω+t) ω A ∂P := by + apply lintegral_congr_ae + filter_upwards [ae_restrict_of_ae hcov] with ω hω + rw [weightedImageMass_add _ (hJsec ω),Measure.add_apply,hω t] + rfl + _ = _ := by + exact lintegral_add_left ((Measure.measurable_coe hA).comp hNm) _ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInverseMeans.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInverseMeans.lean index 070a4cb685ba385c5e0c7f682b8653f6458ac7da..1c5e001f37543d52e7de6b40c0222be3f2e8b187 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInverseMeans.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInverseMeans.lean @@ -1,107 +1,107 @@ -import OAI.Probability.SLE.Loewner.StoppedInverseKernel -import OAI.Probability.SLE.Mass.OriginalShiftedMass -import OAI.Probability.SLE.Geometry.PositiveKernelTests - -namespace OAI - -/-! The two increment candidates have the same remaining conditional -potential and integrable positive compact tests. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), - IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') - -include C hB hc hκ F hw R he τ hτ hFresh in -lemma CanonicalMass.stoppedFreshMass_cross_kernel - (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : - (P.restrict E).bind C.rational.terminal+ - (P.restrict E).bind (stoppedFreshMass B κ γ G F hw τ hc R t) = - (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R)+ - (P.restrict E).bind (fun ω => C.process (τ ω+t) ω) := by - rw [C.original_shifted_green_kernel τ hτ t E hE, - C.stoppedFreshMass_green_kernel hB hc hκ F hw R he τ hτ hFresh t E hE] - ac_rfl - -include C hB hc hκ F hw R he τ hτ hFresh in -lemma CanonicalMass.stoppedFreshMass_mean_le (t : ℝ≥0) (E : Set Ω) : - (P.restrict E).bind (stoppedFreshMass B κ γ G F hw τ hc R t) ≤ initialGreenMeasure κ := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hNm := (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ).measurable (i := t) - have hT := C.stoppedFreshTerminalMass_kernel hB hc hκ F hw τ hτ R - (he.mono fun _ h => h.2) hFresh univ MeasurableSet.univ - have hK := C.stoppedFreshMass_green_kernel hB hc hκ F hw R he τ hτ hFresh - t univ MeasurableSet.univ - simp only [Measure.restrict_univ] at hT hK - have hTmean : P.bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) ≤ initialGreenMeasure κ := by - rw [hT] - exact C.green_at_finite_stop_mean_le τ hτ - refine (restricted_bind_le_of_ae_le P _ _ hNm hNm (ae_of_all _ (fun _ => le_rfl)) E).trans ?_ - rw [hK] at hTmean - exact (show P.bind (stoppedFreshMass B κ γ G F hw τ hc R t) ≤ - P.bind (stoppedFreshMass B κ γ G F hw τ hc R t)+ - P.bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) from Measure.le_add_right le_rfl).trans hTmean - -include C hB hc hκ F hw R he τ hτ hFresh in -lemma CanonicalMass.stoppedFreshTerminalMass_mean_le_of_fresh (E : Set Ω) : - (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) ≤ initialGreenMeasure κ := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R - have hT := C.stoppedFreshTerminalMass_kernel hB hc hκ F hw τ hτ R - (he.mono fun _ h => h.2) hFresh univ MeasurableSet.univ - simp only [Measure.restrict_univ] at hT - refine (restricted_bind_le_of_ae_le P _ _ hTm hTm (ae_of_all _ (fun _ => le_rfl)) E).trans ?_ - rw [hT] - exact C.green_at_finite_stop_mean_le τ hτ - -include C hB hc hκ F hw R he τ hτ hFresh in -lemma CanonicalMass.stoppedFreshMass_test_cross_mean - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) - (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : - (∫ ω in E, ∫ z, f z ∂C.rational.terminal ω ∂P)+ - (∫ ω in E, ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω ∂P) = - (∫ ω in E, ∫ z, f z ∂stoppedFreshTerminalMass B κ γ G F hw τ hc R ω ∂P)+ - (∫ ω in E, ∫ z, f z ∂C.process (τ ω+t) ω ∂P) := by - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R - have hNm := (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ).measurable (i := t) - have hCm := (C.original_shifted_adapted τ hτ).measurable (i := t) - have hCT := C.uniqueness_terminal_mean_le E - have hNT := C.stoppedFreshTerminalMass_mean_le_of_fresh hB hc hκ F hw R he τ hτ hFresh E - have hN := C.stoppedFreshMass_mean_le hB hc hκ F hw R he τ hτ hFresh t E - have hC := C.original_shifted_mean_le τ hτ t E - have h := congrArg (fun μ : Measure upperHalfPlane => ∫ z, f z ∂μ) - (C.stoppedFreshMass_cross_kernel hB hc hκ F hw R he τ hτ hFresh t E hE) - rw [integral_add_measure (f.integrable.mono_measure hCT) (f.integrable.mono_measure hN), - integral_add_measure (f.integrable.mono_measure hNT) (f.integrable.mono_measure hC)] at h - rw [← positive_kernel_integral_bind (P.restrict E) κ C.rational.terminal - C.rational.terminal_measurable hCT f hf, - ← positive_kernel_integral_bind (P.restrict E) κ _ hNm hN f hf, - ← positive_kernel_integral_bind (P.restrict E) κ _ hTm hNT f hf, - ← positive_kernel_integral_bind (P.restrict E) κ _ hCm hC f hf] at h - exact h - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.StoppedInverseKernel +import OAI.Probability.SLE.Mass.OriginalShiftedMass +import OAI.Probability.SLE.Geometry.PositiveKernelTests + +namespace OAI + +/-! The two increment candidates have the same remaining conditional +potential and integrable positive compact tests. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), + IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') + +include C hB hc hκ F hw R he τ hτ hFresh in +lemma CanonicalMass.stoppedFreshMass_cross_kernel + (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : + (P.restrict E).bind C.rational.terminal+ + (P.restrict E).bind (stoppedFreshMass B κ γ G F hw τ hc R t) = + (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R)+ + (P.restrict E).bind (fun ω => C.process (τ ω+t) ω) := by + rw [C.original_shifted_green_kernel τ hτ t E hE, + C.stoppedFreshMass_green_kernel hB hc hκ F hw R he τ hτ hFresh t E hE] + ac_rfl + +include C hB hc hκ F hw R he τ hτ hFresh in +lemma CanonicalMass.stoppedFreshMass_mean_le (t : ℝ≥0) (E : Set Ω) : + (P.restrict E).bind (stoppedFreshMass B κ γ G F hw τ hc R t) ≤ initialGreenMeasure κ := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hNm := (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ).measurable (i := t) + have hT := C.stoppedFreshTerminalMass_kernel hB hc hκ F hw τ hτ R + (he.mono fun _ h => h.2) hFresh univ MeasurableSet.univ + have hK := C.stoppedFreshMass_green_kernel hB hc hκ F hw R he τ hτ hFresh + t univ MeasurableSet.univ + simp only [Measure.restrict_univ] at hT hK + have hTmean : P.bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) ≤ initialGreenMeasure κ := by + rw [hT] + exact C.green_at_finite_stop_mean_le τ hτ + refine (restricted_bind_le_of_ae_le P _ _ hNm hNm (ae_of_all _ (fun _ => le_rfl)) E).trans ?_ + rw [hK] at hTmean + exact (show P.bind (stoppedFreshMass B κ γ G F hw τ hc R t) ≤ + P.bind (stoppedFreshMass B κ γ G F hw τ hc R t)+ + P.bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) from Measure.le_add_right le_rfl).trans hTmean + +include C hB hc hκ F hw R he τ hτ hFresh in +lemma CanonicalMass.stoppedFreshTerminalMass_mean_le_of_fresh (E : Set Ω) : + (P.restrict E).bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) ≤ initialGreenMeasure κ := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R + have hT := C.stoppedFreshTerminalMass_kernel hB hc hκ F hw τ hτ R + (he.mono fun _ h => h.2) hFresh univ MeasurableSet.univ + simp only [Measure.restrict_univ] at hT + refine (restricted_bind_le_of_ae_le P _ _ hTm hTm (ae_of_all _ (fun _ => le_rfl)) E).trans ?_ + rw [hT] + exact C.green_at_finite_stop_mean_le τ hτ + +include C hB hc hκ F hw R he τ hτ hFresh in +lemma CanonicalMass.stoppedFreshMass_test_cross_mean + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) + (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : + (∫ ω in E, ∫ z, f z ∂C.rational.terminal ω ∂P)+ + (∫ ω in E, ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω ∂P) = + (∫ ω in E, ∫ z, f z ∂stoppedFreshTerminalMass B κ γ G F hw τ hc R ω ∂P)+ + (∫ ω in E, ∫ z, f z ∂C.process (τ ω+t) ω ∂P) := by + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R + have hNm := (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ).measurable (i := t) + have hCm := (C.original_shifted_adapted τ hτ).measurable (i := t) + have hCT := C.uniqueness_terminal_mean_le E + have hNT := C.stoppedFreshTerminalMass_mean_le_of_fresh hB hc hκ F hw R he τ hτ hFresh E + have hN := C.stoppedFreshMass_mean_le hB hc hκ F hw R he τ hτ hFresh t E + have hC := C.original_shifted_mean_le τ hτ t E + have h := congrArg (fun μ : Measure upperHalfPlane => ∫ z, f z ∂μ) + (C.stoppedFreshMass_cross_kernel hB hc hκ F hw R he τ hτ hFresh t E hE) + rw [integral_add_measure (f.integrable.mono_measure hCT) (f.integrable.mono_measure hN), + integral_add_measure (f.integrable.mono_measure hNT) (f.integrable.mono_measure hC)] at h + rw [← positive_kernel_integral_bind (P.restrict E) κ C.rational.terminal + C.rational.terminal_measurable hCT f hf, + ← positive_kernel_integral_bind (P.restrict E) κ _ hNm hN f hf, + ← positive_kernel_integral_bind (P.restrict E) κ _ hTm hNT f hf, + ← positive_kernel_integral_bind (P.restrict E) κ _ hCm hC f hf] at h + exact h + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInversePaths.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInversePaths.lean index 70de6ecbcaab6542c7b6126977badef1f9c52609..660d1a6eb3d679ac4e3c8e833e51cd7e62e320d2 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInversePaths.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInversePaths.lean @@ -1,199 +1,199 @@ -import OAI.Probability.SLE.Loewner.RawRestartCanonical -import OAI.Probability.SLE.Loewner.StoppedInverseTerminalBound -import OAI.Probability.SLE.Mass.DominatedMassTestExhaustion - -namespace OAI - -/-! The finite fresh mass transported by the actual stopped inverse chart. -The same raw functional is used for all finite stopping times. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge - -local instance stoppedInversePathsLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma weightedImageMass_mono - (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) {μ ν : Measure upperHalfPlane} (hle : μ ≤ ν) : - weightedImageMass J w μ ≤ weightedImageMass J w ν := by - apply Measure.le_iff.mpr - intro A hA - rw [weightedImageMass_apply J hJ w μ A hA,weightedImageMass_apply J hJ w ν A hA] - exact lintegral_mono' hle le_rfl - -lemma weightedImageMass_zero (J : upperHalfPlane → upperHalfPlane) - (w : upperHalfPlane → ℝ≥0∞) : weightedImageMass J w 0=0 := by - simp only [weightedImageMass,withDensity_zero_left,Measure.map_zero] - -lemma weightedImageMass_finite_on_compacts - (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) {μ ν : Measure upperHalfPlane} - (hle : μ ≤ ν) [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : - IsFiniteMeasureOnCompacts (weightedImageMass J w μ) := - ⟨fun K hK => lt_of_le_of_lt (weightedImageMass_mono J hJ w hle K) hK.measure_lt_top⟩ - -lemma weightedImageMass_positive_test_continuous - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] - (hdom : ∀ t, N t ≤ ν) - (hc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) - (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) - [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : - Continuous (fun t => ∫ z, f z ∂(weightedImageMass J w (N t))) := by - let g := fun z => w z * ENNReal.ofReal (f (J z)) - have hg : Measurable g := hw.mul (f.continuous.measurable.comp hJ).ennreal_ofReal - have hν : (∫⁻ z, g z ∂ν) ≠ ∞ := by - rw [← weightedImageMass_lintegral J hJ w hw ν (fun z => ENNReal.ofReal (f z)) - f.continuous.measurable.ennreal_ofReal] - rw [← ofReal_integral_eq_lintegral_ofReal f.integrable (ae_of_all _ hf)] - exact ENNReal.ofReal_ne_top - have hgc := continuous_lintegral_of_radon_domination N ν hdom hc g hg hν - have hfinite (t : ℝ≥0) : (∫⁻ z, g z ∂N t) ≠ ∞ := - ne_top_of_le_ne_top hν (lintegral_mono' (hdom t) le_rfl) - have hreal : Continuous (fun t => (∫⁻ z, g z ∂N t).toReal) := by - apply continuous_iff_continuousAt.mpr - intro t - have ht : ContinuousAt (fun u : ℝ≥0 => ∫⁻ z, g z ∂N u) t := hgc.continuousAt - exact ContinuousAt.comp (f := fun u : ℝ≥0 => ∫⁻ z, g z ∂N u) (x := t) - (ENNReal.continuousAt_toReal (hfinite t)) ht - have he (t : ℝ≥0) : (∫ z, f z ∂(weightedImageMass J w (N t))) = (∫⁻ z, g z ∂N t).toReal := by - rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable, - weightedImageMass_lintegral J hJ w hw (N t) (fun z => ENNReal.ofReal (f z)) - f.continuous.measurable.ennreal_ofReal] - simpa only [he] using hreal - - -lemma positive_test_integral_measurable - {Ω : Type*} [MeasurableSpace Ω] (N : Ω → Measure upperHalfPlane) (hN : Measurable N) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : - Measurable (fun ω => ∫ z, f z ∂N ω) := by - have h := ENNReal.measurable_toReal.comp - ((Measure.measurable_lintegral f.continuous.measurable.ennreal_ofReal).comp hN) - have he (ω : Ω) : (∫ z, f z ∂N ω) = (∫⁻ z, ENNReal.ofReal (f z) ∂N ω).toReal := - integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable - convert h using 1 - funext ω - exact he ω - -structure PositiveMassPaths (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) : Prop where - zero : N 0=0 - monotone : Monotone N - dominated : ∀ t, N t ≤ ν - radon : ∀ t, (N t).Regular ∧ IsFiniteMeasureOnCompacts (N t) - continuous_tests : ∀ f : C_c(upperHalfPlane,ℝ), (∀ z, 0 ≤ f z) → - Continuous (fun t => ∫ z, f z ∂N t) - -lemma weightedImageMass_positive_paths - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] - (hz : N 0=0) (hm : Monotone N) (hdom : ∀ t, N t ≤ ν) - (hc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) - (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) - [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : - PositiveMassPaths (fun t => weightedImageMass J w (N t)) (weightedImageMass J w ν) := by - refine ⟨?_,?_,?_,?_,?_⟩ - · rw [hz,weightedImageMass_zero] - · intro s t hst - exact weightedImageMass_mono J hJ w (hm hst) - · intro t - exact weightedImageMass_mono J hJ w (hdom t) - · intro t - let := weightedImageMass_finite_on_compacts J hJ w (hdom t) - exact ⟨inferInstance,inferInstance⟩ - · intro f hf - exact weightedImageMass_positive_test_continuous N ν hdom hc J hJ w hw f hf - -namespace Multipoint - -local instance stoppedInversePathsDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance stoppedInversePathsDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) - (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (τ : Ω → ℝ≥0) (hc : ∀ ω, Continuous (fun t => B t ω)) (R : RawCanonicalMass) - -def stoppedFreshMass (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := - weightedImageMass (stoppedMassPoint B κ γ G F hw τ ω) (stoppedMassWeight B κ F τ ω) - (R.process t (sourceRestartDriverPath B hc τ ω)) - -variable (hm : ∀ t, Measurable (B t)) - (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun ω => (τ ω : WithTop ℝ≥0))) - -include hm hτ in -theorem stoppedFreshMass_adapted (hκ : 0 < κ) : - Adapted (shiftedStoppingFiltration hτ) (stoppedFreshMass B κ γ G F hw τ hc R) := by - have hraw := (sourceRestart_raw_process_progressive B hm hc τ hτ R.process R.progressive).adapted - intro t - have hle : hτ.measurableSpace ≤ shiftedStoppingFiltration hτ t := - IsStoppingTime.measurableSpace_mono hτ (stopping_time_add_nnreal hτ t) - (fun ω => WithTop.coe_le_coe.mpr (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) - have hprod : hτ.measurableSpace.prod (inferInstance : MeasurableSpace upperHalfPlane) ≤ - (shiftedStoppingFiltration hτ t).prod inferInstance := - sup_le_sup (MeasurableSpace.comap_mono hle) le_rfl - exact @weightedImageMass_measurable Ω upperHalfPlane (shiftedStoppingFiltration hτ t) inferInstance - (fun ω => R.process t (sourceRestartDriverPath B hc τ ω)) (hraw t) - (fun ω => (R.radon t _).2) - (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) - ((stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ).mono hprod le_rfl) - (fun p => stoppedMassWeight B κ F τ p.1 p.2) - ((stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ).mono hprod le_rfl) - -include hm hτ in -theorem stoppedFreshMass_paths - {P : Measure Ω} (hκ : 0 < κ) - (hpaths : ∀ᵐ ω ∂P, - R.process 0 (sourceRestartDriverPath B hc τ ω)=0 ∧ - Monotone (fun t => R.process t (sourceRestartDriverPath B hc τ ω)) ∧ - (∀ f : C_c(upperHalfPlane,ℝ), Continuous - (fun t => ∫ z, f z ∂(R.process t (sourceRestartDriverPath B hc τ ω)))) ∧ - ∀ t, R.process t (sourceRestartDriverPath B hc τ ω) ≤ - R.terminal (sourceRestartDriverPath B hc τ ω)) - (hfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) : - ∀ᵐ ω ∂P, PositiveMassPaths - (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) - (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) := by - have hJ := stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ - have hW := stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ - filter_upwards [hpaths,hfin] with ω hp hf - let J := stoppedMassPoint B κ γ G F hw τ ω - let w := stoppedMassWeight B κ F τ ω - have hpair : @Measurable upperHalfPlane (Ω × upperHalfPlane) inferInstance - (hτ.measurableSpace.prod inferInstance) (fun z => (ω,z)) := - @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω - have hJm : Measurable J := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ hpair - have hwm : Measurable w := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW hpair - let N := fun t => R.process t (sourceRestartDriverPath B hc τ ω) - let ν := R.terminal (sourceRestartDriverPath B hc τ ω) - change PositiveMassPaths (fun t => weightedImageMass J w (N t)) (weightedImageMass J w ν) - exact @weightedImageMass_positive_paths N ν - (R.terminal_radon (sourceRestartDriverPath B hc τ ω)).1 - hp.1 hp.2.1 hp.2.2.2 hp.2.2.1 J hJm w hwm hf - -include hm hτ in -theorem stoppedFreshMass_positive_test_adapted (hκ : 0 < κ) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : - Adapted (shiftedStoppingFiltration hτ) - (fun t ω => ∫ z, f z ∂(stoppedFreshMass B κ γ G F hw τ hc R t ω)) := by - intro t - exact @positive_test_integral_measurable Ω (shiftedStoppingFiltration hτ t) _ - (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ t) f hf - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.RawRestartCanonical +import OAI.Probability.SLE.Loewner.StoppedInverseTerminalBound +import OAI.Probability.SLE.Mass.DominatedMassTestExhaustion + +namespace OAI + +/-! The finite fresh mass transported by the actual stopped inverse chart. +The same raw functional is used for all finite stopping times. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge + +local instance stoppedInversePathsLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma weightedImageMass_mono + (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) {μ ν : Measure upperHalfPlane} (hle : μ ≤ ν) : + weightedImageMass J w μ ≤ weightedImageMass J w ν := by + apply Measure.le_iff.mpr + intro A hA + rw [weightedImageMass_apply J hJ w μ A hA,weightedImageMass_apply J hJ w ν A hA] + exact lintegral_mono' hle le_rfl + +lemma weightedImageMass_zero (J : upperHalfPlane → upperHalfPlane) + (w : upperHalfPlane → ℝ≥0∞) : weightedImageMass J w 0=0 := by + simp only [weightedImageMass,withDensity_zero_left,Measure.map_zero] + +lemma weightedImageMass_finite_on_compacts + (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) {μ ν : Measure upperHalfPlane} + (hle : μ ≤ ν) [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : + IsFiniteMeasureOnCompacts (weightedImageMass J w μ) := + ⟨fun K hK => lt_of_le_of_lt (weightedImageMass_mono J hJ w hle K) hK.measure_lt_top⟩ + +lemma weightedImageMass_positive_test_continuous + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] + (hdom : ∀ t, N t ≤ ν) + (hc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) + (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) + [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : + Continuous (fun t => ∫ z, f z ∂(weightedImageMass J w (N t))) := by + let g := fun z => w z * ENNReal.ofReal (f (J z)) + have hg : Measurable g := hw.mul (f.continuous.measurable.comp hJ).ennreal_ofReal + have hν : (∫⁻ z, g z ∂ν) ≠ ∞ := by + rw [← weightedImageMass_lintegral J hJ w hw ν (fun z => ENNReal.ofReal (f z)) + f.continuous.measurable.ennreal_ofReal] + rw [← ofReal_integral_eq_lintegral_ofReal f.integrable (ae_of_all _ hf)] + exact ENNReal.ofReal_ne_top + have hgc := continuous_lintegral_of_radon_domination N ν hdom hc g hg hν + have hfinite (t : ℝ≥0) : (∫⁻ z, g z ∂N t) ≠ ∞ := + ne_top_of_le_ne_top hν (lintegral_mono' (hdom t) le_rfl) + have hreal : Continuous (fun t => (∫⁻ z, g z ∂N t).toReal) := by + apply continuous_iff_continuousAt.mpr + intro t + have ht : ContinuousAt (fun u : ℝ≥0 => ∫⁻ z, g z ∂N u) t := hgc.continuousAt + exact ContinuousAt.comp (f := fun u : ℝ≥0 => ∫⁻ z, g z ∂N u) (x := t) + (ENNReal.continuousAt_toReal (hfinite t)) ht + have he (t : ℝ≥0) : (∫ z, f z ∂(weightedImageMass J w (N t))) = (∫⁻ z, g z ∂N t).toReal := by + rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable, + weightedImageMass_lintegral J hJ w hw (N t) (fun z => ENNReal.ofReal (f z)) + f.continuous.measurable.ennreal_ofReal] + simpa only [he] using hreal + + +lemma positive_test_integral_measurable + {Ω : Type*} [MeasurableSpace Ω] (N : Ω → Measure upperHalfPlane) (hN : Measurable N) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : + Measurable (fun ω => ∫ z, f z ∂N ω) := by + have h := ENNReal.measurable_toReal.comp + ((Measure.measurable_lintegral f.continuous.measurable.ennreal_ofReal).comp hN) + have he (ω : Ω) : (∫ z, f z ∂N ω) = (∫⁻ z, ENNReal.ofReal (f z) ∂N ω).toReal := + integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable + convert h using 1 + funext ω + exact he ω + +structure PositiveMassPaths (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) : Prop where + zero : N 0=0 + monotone : Monotone N + dominated : ∀ t, N t ≤ ν + radon : ∀ t, (N t).Regular ∧ IsFiniteMeasureOnCompacts (N t) + continuous_tests : ∀ f : C_c(upperHalfPlane,ℝ), (∀ z, 0 ≤ f z) → + Continuous (fun t => ∫ z, f z ∂N t) + +lemma weightedImageMass_positive_paths + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] + (hz : N 0=0) (hm : Monotone N) (hdom : ∀ t, N t ≤ ν) + (hc : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) + (J : upperHalfPlane → upperHalfPlane) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) + [IsFiniteMeasureOnCompacts (weightedImageMass J w ν)] : + PositiveMassPaths (fun t => weightedImageMass J w (N t)) (weightedImageMass J w ν) := by + refine ⟨?_,?_,?_,?_,?_⟩ + · rw [hz,weightedImageMass_zero] + · intro s t hst + exact weightedImageMass_mono J hJ w (hm hst) + · intro t + exact weightedImageMass_mono J hJ w (hdom t) + · intro t + let := weightedImageMass_finite_on_compacts J hJ w (hdom t) + exact ⟨inferInstance,inferInstance⟩ + · intro f hf + exact weightedImageMass_positive_test_continuous N ν hdom hc J hJ w hw f hf + +namespace Multipoint + +local instance stoppedInversePathsDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance stoppedInversePathsDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) + (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (τ : Ω → ℝ≥0) (hc : ∀ ω, Continuous (fun t => B t ω)) (R : RawCanonicalMass) + +def stoppedFreshMass (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := + weightedImageMass (stoppedMassPoint B κ γ G F hw τ ω) (stoppedMassWeight B κ F τ ω) + (R.process t (sourceRestartDriverPath B hc τ ω)) + +variable (hm : ∀ t, Measurable (B t)) + (hτ : IsStoppingTime (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun ω => (τ ω : WithTop ℝ≥0))) + +include hm hτ in +theorem stoppedFreshMass_adapted (hκ : 0 < κ) : + Adapted (shiftedStoppingFiltration hτ) (stoppedFreshMass B κ γ G F hw τ hc R) := by + have hraw := (sourceRestart_raw_process_progressive B hm hc τ hτ R.process R.progressive).adapted + intro t + have hle : hτ.measurableSpace ≤ shiftedStoppingFiltration hτ t := + IsStoppingTime.measurableSpace_mono hτ (stopping_time_add_nnreal hτ t) + (fun ω => WithTop.coe_le_coe.mpr (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) + have hprod : hτ.measurableSpace.prod (inferInstance : MeasurableSpace upperHalfPlane) ≤ + (shiftedStoppingFiltration hτ t).prod inferInstance := + sup_le_sup (MeasurableSpace.comap_mono hle) le_rfl + exact @weightedImageMass_measurable Ω upperHalfPlane (shiftedStoppingFiltration hτ t) inferInstance + (fun ω => R.process t (sourceRestartDriverPath B hc τ ω)) (hraw t) + (fun ω => (R.radon t _).2) + (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) + ((stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ).mono hprod le_rfl) + (fun p => stoppedMassWeight B κ F τ p.1 p.2) + ((stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ).mono hprod le_rfl) + +include hm hτ in +theorem stoppedFreshMass_paths + {P : Measure Ω} (hκ : 0 < κ) + (hpaths : ∀ᵐ ω ∂P, + R.process 0 (sourceRestartDriverPath B hc τ ω)=0 ∧ + Monotone (fun t => R.process t (sourceRestartDriverPath B hc τ ω)) ∧ + (∀ f : C_c(upperHalfPlane,ℝ), Continuous + (fun t => ∫ z, f z ∂(R.process t (sourceRestartDriverPath B hc τ ω)))) ∧ + ∀ t, R.process t (sourceRestartDriverPath B hc τ ω) ≤ + R.terminal (sourceRestartDriverPath B hc τ ω)) + (hfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) : + ∀ᵐ ω ∂P, PositiveMassPaths + (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) + (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) := by + have hJ := stoppedMassPoint_measurable B κ γ G F hw τ hm hc hτ + have hW := stoppedMassWeight_measurable B κ γ G F hw τ hm hc hτ hκ + filter_upwards [hpaths,hfin] with ω hp hf + let J := stoppedMassPoint B κ γ G F hw τ ω + let w := stoppedMassWeight B κ F τ ω + have hpair : @Measurable upperHalfPlane (Ω × upperHalfPlane) inferInstance + (hτ.measurableSpace.prod inferInstance) (fun z => (ω,z)) := + @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω + have hJm : Measurable J := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ hpair + have hwm : Measurable w := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW hpair + let N := fun t => R.process t (sourceRestartDriverPath B hc τ ω) + let ν := R.terminal (sourceRestartDriverPath B hc τ ω) + change PositiveMassPaths (fun t => weightedImageMass J w (N t)) (weightedImageMass J w ν) + exact @weightedImageMass_positive_paths N ν + (R.terminal_radon (sourceRestartDriverPath B hc τ ω)).1 + hp.1 hp.2.1 hp.2.2.2 hp.2.2.1 J hJm w hwm hf + +include hm hτ in +theorem stoppedFreshMass_positive_test_adapted (hκ : 0 < κ) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : + Adapted (shiftedStoppingFiltration hτ) + (fun t ω => ∫ z, f z ∂(stoppedFreshMass B κ γ G F hw τ hc R t ω)) := by + intro t + exact @positive_test_integral_measurable Ω (shiftedStoppingFiltration hτ t) _ + (stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ t) f hf + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalBound.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalBound.lean index 8fd746d8be4fcfec4e8683b24920c008d5de90e7..c4dd72a2910d2b2963d607eb2509426a55b87492 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalBound.lean @@ -1,92 +1,92 @@ -import OAI.Probability.SLE.Loewner.StoppedInverseGreenMean -import OAI.Probability.SLE.Mass.GreenMeasureSupport - -namespace OAI - -/-! The transported fresh terminal mass is locally integrable. The bound -comes from the original canonical Green kernel at the stopping time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace -local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma halfPlaneStoppingGreenMeasure_finite_time - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (τ : Ω → ℝ≥0) (ω : Ω) : - halfPlaneStoppingGreenMeasure κ B γ G (fun ω => (τ ω:ℝ≥0∞)) ω = - halfPlaneGreenMeasure κ B γ G (τ ω) ω := by - simp only [halfPlaneStoppingGreenMeasure,halfPlaneGreenMeasure,stoppingGreen, - ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] - -theorem CanonicalMass.green_at_finite_stop_mean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) : - P.bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) ≤ initialGreenMeasure κ := by - have hk := C.kernel (fun ω => (τ ω:WithTop ℝ≥0)) hτ univ MeasurableSet.univ - simp only [Measure.restrict_univ,C.rational.mean] at hk - rw [hk] - apply Measure.le_iff.mpr - intro A _ - simp only [Measure.add_apply] - exact le_add_left le_rfl - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun t => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - -include C hB hc hκ F hw τ hτ R he hpublished in -theorem CanonicalMass.stoppedFreshTerminalMass_mean_le : - P.bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) ≤ initialGreenMeasure κ := by - have hk := C.stoppedFreshTerminalMass_published_kernel hB hc hκ F hw τ hτ R he - hpublished univ MeasurableSet.univ - simp only [Measure.restrict_univ] at hk - rw [hk] - exact C.green_at_finite_stop_mean_le τ hτ - -include C hB hc hκ F hw τ hτ R he hpublished in -theorem CanonicalMass.stoppedFreshTerminalMass_ae_finite_on_compacts : - ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) := by - let D := localTestFamily upperHalfPlane - let M := stoppedFreshTerminalMass B κ γ G F hw τ hc R - have hM : Measurable M := stoppedFreshTerminalMass_measurable B κ γ G F hw τ - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hτ hκ R - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hbound := C.stoppedFreshTerminalMass_mean_le hB hc hκ F hw τ hτ R he hpublished - have hfin (n : ℕ) : ∀ᵐ ω ∂P, M ω (D.compactSet n) < ∞ := by - apply ae_lt_top ((Measure.measurable_coe (D.isCompact n).measurableSet).comp hM) - change (∫⁻ ω, M ω (D.compactSet n) ∂P) ≠ ∞ - rw [← Measure.bind_apply (D.isCompact n).measurableSet hM.aemeasurable] - exact (lt_of_le_of_lt (hbound (D.compactSet n)) (D.isCompact n).measure_lt_top).ne - filter_upwards [ae_all_iff.mpr hfin] with ω hω - constructor - intro K hK - obtain ⟨n,hn⟩ := D.exhaustive K hK - exact lt_of_le_of_lt (measure_mono hn) (hω n) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.StoppedInverseGreenMean +import OAI.Probability.SLE.Mass.GreenMeasureSupport + +namespace OAI + +/-! The transported fresh terminal mass is locally integrable. The bound +comes from the original canonical Green kernel at the stopping time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace +local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma halfPlaneStoppingGreenMeasure_finite_time + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (τ : Ω → ℝ≥0) (ω : Ω) : + halfPlaneStoppingGreenMeasure κ B γ G (fun ω => (τ ω:ℝ≥0∞)) ω = + halfPlaneGreenMeasure κ B γ G (τ ω) ω := by + simp only [halfPlaneStoppingGreenMeasure,halfPlaneGreenMeasure,stoppingGreen, + ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] + +theorem CanonicalMass.green_at_finite_stop_mean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) : + P.bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω) ω) ≤ initialGreenMeasure κ := by + have hk := C.kernel (fun ω => (τ ω:WithTop ℝ≥0)) hτ univ MeasurableSet.univ + simp only [Measure.restrict_univ,C.rational.mean] at hk + rw [hk] + apply Measure.le_iff.mpr + intro A _ + simp only [Measure.add_apply] + exact le_add_left le_rfl + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun t => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B t)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + +include C hB hc hκ F hw τ hτ R he hpublished in +theorem CanonicalMass.stoppedFreshTerminalMass_mean_le : + P.bind (stoppedFreshTerminalMass B κ γ G F hw τ hc R) ≤ initialGreenMeasure κ := by + have hk := C.stoppedFreshTerminalMass_published_kernel hB hc hκ F hw τ hτ R he + hpublished univ MeasurableSet.univ + simp only [Measure.restrict_univ] at hk + rw [hk] + exact C.green_at_finite_stop_mean_le τ hτ + +include C hB hc hκ F hw τ hτ R he hpublished in +theorem CanonicalMass.stoppedFreshTerminalMass_ae_finite_on_compacts : + ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) := by + let D := localTestFamily upperHalfPlane + let M := stoppedFreshTerminalMass B κ γ G F hw τ hc R + have hM : Measurable M := stoppedFreshTerminalMass_measurable B κ γ G F hw τ + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hc hτ hκ R + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hbound := C.stoppedFreshTerminalMass_mean_le hB hc hκ F hw τ hτ R he hpublished + have hfin (n : ℕ) : ∀ᵐ ω ∂P, M ω (D.compactSet n) < ∞ := by + apply ae_lt_top ((Measure.measurable_coe (D.isCompact n).measurableSet).comp hM) + change (∫⁻ ω, M ω (D.compactSet n) ∂P) ≠ ∞ + rw [← Measure.bind_apply (D.isCompact n).measurableSet hM.aemeasurable] + exact (lt_of_le_of_lt (hbound (D.compactSet n)) (D.isCompact n).measure_lt_top).ne + filter_upwards [ae_all_iff.mpr hfin] with ω hω + constructor + intro K hK + obtain ⟨n,hn⟩ := D.exhaustive K hK + exact lt_of_le_of_lt (measure_mono hn) (hω n) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalUniqueness.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalUniqueness.lean index 3a7780792c94f3a6a84c5853a75d96e626deb96c..3d823e6e02839612a570f6bad90828a6cef11b63 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalUniqueness.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInverseTerminalUniqueness.lean @@ -1,98 +1,98 @@ -import OAI.Probability.SLE.Loewner.StoppedInverseTestUniqueness -import OAI.Probability.SLE.Loewner.StoppedInverseExhaustion - -namespace OAI - -/-! The finite-time restart identity extends to the locally exhausted -terminal mass using the same raw fresh functional. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance stoppedTerminalUniqueLocallyCompact : LocallyCompactSpace upperHalfPlane := - isOpen_upperHalfPlane.locallyCompactSpace - -lemma halfPlane_nested_remainder (a b c : Measure upperHalfPlane) - [a.Regular] [b.Regular] (hab : a ≤ b) (hbc : b ≤ c) : - (c-a)-(b-a)=c-b := by - let := halfPlane_sub_finite_on_compacts b a - apply (Measure.add_left_inj (b-a) _ _).mp - have hle : b-a ≤ c-a := sub_le_sub_right_of_le (hab.trans hbc) hbc - rw [sigmaFinite_sub_add_cancel_of_le hle] - apply (Measure.add_left_inj a _ _).mp - calc - (c-a)+a = c := sigmaFinite_sub_add_cancel_of_le (hab.trans hbc) - _ = (c-b)+b := (sigmaFinite_sub_add_cancel_of_le hbc).symm - _ = ((c-b)+(b-a))+a := by rw [add_assoc,sigmaFinite_sub_add_cancel_of_le hab] - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), - IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') - - -include C hB hc hκ F hw R he τ hτ hFresh in -theorem CanonicalMass.stoppedFreshTerminalMass_eq_remainder - (hpaths : ∀ᵐ ω ∂P, PositiveMassPaths - (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) - (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) - (hfin : ∀ᵐ ω ∂P, - IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) - (hex : ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → Tendsto - (fun t : ℝ≥0 => (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω- - stoppedFreshMass B κ γ G F hw τ hc R t ω) K) atTop (𝓝 0)) : - ∀ᵐ ω ∂P, C.rational.terminal ω-C.process (τ ω) ω = - stoppedFreshTerminalMass B κ γ G F hw τ hc R ω := by - filter_upwards [C.paths,C.exhaustion,hpaths,hfin,hex, - C.stoppedFreshMass_indistinguishable hB hc hκ F hw R he τ hτ hFresh hpaths] - with ω hp hx hq hf heq hid - let := (C.rational.terminal_radon ω).1 - let := (C.radon (τ ω) ω).1 - let := halfPlane_sub_finite_on_compacts (C.rational.terminal ω) (C.process (τ ω) ω) - let := hf - apply radon_terminal_eq_of_common_exhaustion - (C.rational.terminal ω-C.process (τ ω) ω) - (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) - (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) - · intro t - rw [← hid t] - exact sub_le_sub_right_of_le (hp.dominated (τ ω)) (hp.dominated (τ ω+t)) - · exact hq.dominated - · intro K hK - have htime : Tendsto (fun t : ℝ≥0 => τ ω+t) atTop atTop := by - apply tendsto_atTop.mpr - intro b - filter_upwards [eventually_ge_atTop b] with t ht - exact ht.trans (le_add_of_nonneg_left (show 0 ≤ τ ω from zero_le)) - have hlimit := (hx K hK).comp htime - have hrem (t : ℝ≥0) : - (C.rational.terminal ω-C.process (τ ω) ω)- - stoppedFreshMass B κ γ G F hw τ hc R t ω = - C.rational.terminal ω-C.process (τ ω+t) ω := by - rw [← hid t] - let := (C.radon (τ ω+t) ω).1 - exact halfPlane_nested_remainder _ _ _ - (hp.monotone (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) (hp.dominated _) - simpa only [hrem,Function.comp_def] using hlimit - · exact heq - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.StoppedInverseTestUniqueness +import OAI.Probability.SLE.Loewner.StoppedInverseExhaustion + +namespace OAI + +/-! The finite-time restart identity extends to the locally exhausted +terminal mass using the same raw fresh functional. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance stoppedTerminalUniqueLocallyCompact : LocallyCompactSpace upperHalfPlane := + isOpen_upperHalfPlane.locallyCompactSpace + +lemma halfPlane_nested_remainder (a b c : Measure upperHalfPlane) + [a.Regular] [b.Regular] (hab : a ≤ b) (hbc : b ≤ c) : + (c-a)-(b-a)=c-b := by + let := halfPlane_sub_finite_on_compacts b a + apply (Measure.add_left_inj (b-a) _ _).mp + have hle : b-a ≤ c-a := sub_le_sub_right_of_le (hab.trans hbc) hbc + rw [sigmaFinite_sub_add_cancel_of_le hle] + apply (Measure.add_left_inj a _ _).mp + calc + (c-a)+a = c := sigmaFinite_sub_add_cancel_of_le (hab.trans hbc) + _ = (c-b)+b := (sigmaFinite_sub_add_cancel_of_le hbc).symm + _ = ((c-b)+(b-a))+a := by rw [add_assoc,sigmaFinite_sub_add_cancel_of_le hab] + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), + IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') + + +include C hB hc hκ F hw R he τ hτ hFresh in +theorem CanonicalMass.stoppedFreshTerminalMass_eq_remainder + (hpaths : ∀ᵐ ω ∂P, PositiveMassPaths + (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) + (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) + (hfin : ∀ᵐ ω ∂P, + IsFiniteMeasureOnCompacts (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) + (hex : ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → Tendsto + (fun t : ℝ≥0 => (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω- + stoppedFreshMass B κ γ G F hw τ hc R t ω) K) atTop (𝓝 0)) : + ∀ᵐ ω ∂P, C.rational.terminal ω-C.process (τ ω) ω = + stoppedFreshTerminalMass B κ γ G F hw τ hc R ω := by + filter_upwards [C.paths,C.exhaustion,hpaths,hfin,hex, + C.stoppedFreshMass_indistinguishable hB hc hκ F hw R he τ hτ hFresh hpaths] + with ω hp hx hq hf heq hid + let := (C.rational.terminal_radon ω).1 + let := (C.radon (τ ω) ω).1 + let := halfPlane_sub_finite_on_compacts (C.rational.terminal ω) (C.process (τ ω) ω) + let := hf + apply radon_terminal_eq_of_common_exhaustion + (C.rational.terminal ω-C.process (τ ω) ω) + (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) + (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) + · intro t + rw [← hid t] + exact sub_le_sub_right_of_le (hp.dominated (τ ω)) (hp.dominated (τ ω+t)) + · exact hq.dominated + · intro K hK + have htime : Tendsto (fun t : ℝ≥0 => τ ω+t) atTop atTop := by + apply tendsto_atTop.mpr + intro b + filter_upwards [eventually_ge_atTop b] with t ht + exact ht.trans (le_add_of_nonneg_left (show 0 ≤ τ ω from zero_le)) + have hlimit := (hx K hK).comp htime + have hrem (t : ℝ≥0) : + (C.rational.terminal ω-C.process (τ ω) ω)- + stoppedFreshMass B κ γ G F hw τ hc R t ω = + C.rational.terminal ω-C.process (τ ω+t) ω := by + rw [← hid t] + let := (C.radon (τ ω+t) ω).1 + exact halfPlane_nested_remainder _ _ _ + (hp.monotone (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) (hp.dominated _) + simpa only [hrem,Function.comp_def] using hlimit + · exact heq + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/StoppedInverseTestUniqueness.lean b/lean/OAI/Probability/SLE/Loewner/StoppedInverseTestUniqueness.lean index b5f6fe0024ae8b5367a537a4e25f4c23ebb543c5..b02e2041ed0431dab570bb05e534eed1884b5279 100644 --- a/lean/OAI/Probability/SLE/Loewner/StoppedInverseTestUniqueness.lean +++ b/lean/OAI/Probability/SLE/Loewner/StoppedInverseTestUniqueness.lean @@ -1,146 +1,146 @@ -import OAI.Probability.SLE.Loewner.StoppedInverseMeans -import OAI.Probability.SLE.Mass.RadonProcessAEUniqueness - -namespace OAI - -/-! The original increment and the transported fresh mass have identical -positive compact tests simultaneously at every finite future time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance stoppedTestUniqueLocallyCompact : LocallyCompactSpace upperHalfPlane := - isOpen_upperHalfPlane.locallyCompactSpace - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), - IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') - - -include C hB hc hκ F hw R he τ hτ hFresh in -lemma CanonicalMass.stoppedFreshMass_test_indistinguishable - (hpaths : ∀ᵐ ω ∂P, PositiveMassPaths - (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) - (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : - ∀ᵐ ω ∂P, ∀ t, - (∫ z, f z ∂C.process (τ ω+t) ω)-(∫ z, f z ∂C.process (τ ω) ω) = - ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hCm := C.original_shifted_adapted τ hτ - have hNm := stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ - have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R - have hCI (t : ℝ≥0) : Integrable (fun ω => ∫ z, f z ∂C.process (τ ω+t) ω) P := by - apply positive_kernel_test_integrable P κ _ (hCm.measurable (i := t)) _ f hf - simpa only [Measure.restrict_univ] using C.original_shifted_mean_le τ hτ t univ - have hC0I : Integrable (fun ω => ∫ z, f z ∂C.process (τ ω) ω) P := by - simpa only [add_zero] using hCI 0 - have hNI (t : ℝ≥0) : Integrable - (fun ω => ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω) P := by - apply positive_kernel_test_integrable P κ _ (hNm.measurable (i := t)) _ f hf - simpa only [Measure.restrict_univ] using - C.stoppedFreshMass_mean_le hB hc hκ F hw R he τ hτ hFresh t univ - have hTI : Integrable - (fun ω => ∫ z, f z ∂stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) P := by - apply positive_kernel_test_integrable P κ _ hTm _ f hf - simpa only [Measure.restrict_univ] using - C.stoppedFreshTerminalMass_mean_le_of_fresh hB hc hκ F hw R he τ hτ hFresh univ - have hCA : StronglyAdapted (shiftedStoppingFiltration hτ) - (fun t ω => ∫ z, f z ∂C.process (τ ω+t) ω) := by - intro t - exact (@positive_test_integral_measurable Ω (shiftedStoppingFiltration hτ t) - _ (hCm t) f hf).stronglyMeasurable - have hC0A : StronglyAdapted (shiftedStoppingFiltration hτ) - (fun _ ω => ∫ z, f z ∂C.process (τ ω) ω) := by - intro t - have h := (hCA 0).mono ((shiftedStoppingFiltration hτ).mono (show 0 ≤ t from zero_le)) - simpa only [add_zero] using h - have hNA : StronglyAdapted (shiftedStoppingFiltration hτ) - (fun t ω => ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω) := by - intro t - exact (@positive_test_integral_measurable Ω (shiftedStoppingFiltration hτ t) - _ (hNm t) f hf).stronglyMeasurable - have hAI (t : ℝ≥0) : Integrable (fun ω => - (∫ z, f z ∂C.process (τ ω+t) ω)-(∫ z, f z ∂C.process (τ ω) ω)) P := - (hCI t).sub hC0I - have hZI : Integrable (fun ω => - (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂C.process (τ ω) ω)) P := - (C.uniqueness_terminal_test_integrable f).sub hC0I - apply increasing_mass_unique_of_terminal_mean - (fun t ω => (∫ z, f z ∂C.process (τ ω+t) ω)-(∫ z, f z ∂C.process (τ ω) ω)) - (fun t ω => ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω) - (fun ω => (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂C.process (τ ω) ω)- - (∫ z, f z ∂stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) - (hZI.sub hTI) - (hCA.sub hC0A) hNA hAI hNI - · intro t E hE - have h := C.stoppedFreshMass_test_cross_mean hB hc hκ F hw R he τ hτ hFresh f hf t E hE - rw [integral_sub (hAI t).integrableOn (hNI t).integrableOn, - integral_sub (hCI t).integrableOn hC0I.integrableOn, - integral_sub hZI.integrableOn hTI.integrableOn, - integral_sub (C.uniqueness_terminal_test_integrable f).integrableOn hC0I.integrableOn] - linarith - · filter_upwards [C.paths] with ω hp - intro s t hst - apply sub_le_sub_right - let := (C.radon (τ ω+t) ω).1 - exact integral_mono_measure (hp.monotone (add_le_add le_rfl hst)) (ae_of_all _ hf) f.integrable - · filter_upwards [hpaths] with ω hp - intro s t hst - let := (hp.radon t).1 - exact integral_mono_measure (hp.monotone hst) (ae_of_all _ hf) f.integrable - · exact ae_of_all _ (fun ω => by simp only [add_zero,sub_self,Pi.zero_apply]) - · filter_upwards [hpaths] with ω hp - simp only [hp.zero,integral_zero_measure,Pi.zero_apply] - · filter_upwards [C.paths,hpaths] with ω hp hq - exact (((hp.continuous_tests f).comp (continuous_const.add continuous_id)).sub - continuous_const).sub (hq.continuous_tests f hf) - -include C hB hc hκ F hw R he τ hτ hFresh in -/-- Equality of the actual measure increments on one event for every finite -future time, with the unmodified raw fresh functional. -/ -theorem CanonicalMass.stoppedFreshMass_indistinguishable - (hpaths : ∀ᵐ ω ∂P, PositiveMassPaths - (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) - (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) : - ∀ᵐ ω ∂P, ∀ t, - C.process (τ ω+t) ω-C.process (τ ω) ω = - stoppedFreshMass B κ γ G F hw τ hc R t ω := by - apply random_radon_process_eq_of_positive_tests_ae P (localTestFamily upperHalfPlane) - · filter_upwards [] with ω - intro t - let := (C.radon (τ ω+t) ω).2 - let := halfPlane_sub_finite_on_compacts (C.process (τ ω+t) ω) (C.process (τ ω) ω) - infer_instance - · exact hpaths.mono (fun _ h => fun t => (h.radon t).1) - · intro f hf - filter_upwards [C.paths, - C.stoppedFreshMass_test_indistinguishable hB hc hκ F hw R he τ hτ hFresh hpaths f hf] - with ω hp ht - intro t - let := (C.radon (τ ω+t) ω).1 - let := (C.radon (τ ω) ω).1 - rw [integral_halfPlane_sub _ _ (hp.monotone (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) f] - exact ht t - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.StoppedInverseMeans +import OAI.Probability.SLE.Mass.RadonProcessAEUniqueness + +namespace OAI + +/-! The original increment and the transported fresh mass have identical +positive compact tests simultaneously at every finite future time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance stoppedTestUniqueLocallyCompact : LocallyCompactSpace upperHalfPlane := + isOpen_upperHalfPlane.locallyCompactSpace + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (hFresh : ∀ᵐ ω ∂P, ∃ (β : ℝ≥0 → ℂ) (G' F' : ℝ≥0 → ℂ → ℂ), + IsCapacityTwoWitness (fun t => Real.sqrt κ*B (τ ω+t) ω-Real.sqrt κ*B (τ ω) ω) β G' F') + + +include C hB hc hκ F hw R he τ hτ hFresh in +lemma CanonicalMass.stoppedFreshMass_test_indistinguishable + (hpaths : ∀ᵐ ω ∂P, PositiveMassPaths + (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) + (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : + ∀ᵐ ω ∂P, ∀ t, + (∫ z, f z ∂C.process (τ ω+t) ω)-(∫ z, f z ∂C.process (τ ω) ω) = + ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hCm := C.original_shifted_adapted τ hτ + have hNm := stoppedFreshMass_adapted B κ γ G F hw τ hc R hm hτ hκ + have hTm := stoppedFreshTerminalMass_measurable B κ γ G F hw τ hm hc hτ hκ R + have hCI (t : ℝ≥0) : Integrable (fun ω => ∫ z, f z ∂C.process (τ ω+t) ω) P := by + apply positive_kernel_test_integrable P κ _ (hCm.measurable (i := t)) _ f hf + simpa only [Measure.restrict_univ] using C.original_shifted_mean_le τ hτ t univ + have hC0I : Integrable (fun ω => ∫ z, f z ∂C.process (τ ω) ω) P := by + simpa only [add_zero] using hCI 0 + have hNI (t : ℝ≥0) : Integrable + (fun ω => ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω) P := by + apply positive_kernel_test_integrable P κ _ (hNm.measurable (i := t)) _ f hf + simpa only [Measure.restrict_univ] using + C.stoppedFreshMass_mean_le hB hc hκ F hw R he τ hτ hFresh t univ + have hTI : Integrable + (fun ω => ∫ z, f z ∂stoppedFreshTerminalMass B κ γ G F hw τ hc R ω) P := by + apply positive_kernel_test_integrable P κ _ hTm _ f hf + simpa only [Measure.restrict_univ] using + C.stoppedFreshTerminalMass_mean_le_of_fresh hB hc hκ F hw R he τ hτ hFresh univ + have hCA : StronglyAdapted (shiftedStoppingFiltration hτ) + (fun t ω => ∫ z, f z ∂C.process (τ ω+t) ω) := by + intro t + exact (@positive_test_integral_measurable Ω (shiftedStoppingFiltration hτ t) + _ (hCm t) f hf).stronglyMeasurable + have hC0A : StronglyAdapted (shiftedStoppingFiltration hτ) + (fun _ ω => ∫ z, f z ∂C.process (τ ω) ω) := by + intro t + have h := (hCA 0).mono ((shiftedStoppingFiltration hτ).mono (show 0 ≤ t from zero_le)) + simpa only [add_zero] using h + have hNA : StronglyAdapted (shiftedStoppingFiltration hτ) + (fun t ω => ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω) := by + intro t + exact (@positive_test_integral_measurable Ω (shiftedStoppingFiltration hτ t) + _ (hNm t) f hf).stronglyMeasurable + have hAI (t : ℝ≥0) : Integrable (fun ω => + (∫ z, f z ∂C.process (τ ω+t) ω)-(∫ z, f z ∂C.process (τ ω) ω)) P := + (hCI t).sub hC0I + have hZI : Integrable (fun ω => + (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂C.process (τ ω) ω)) P := + (C.uniqueness_terminal_test_integrable f).sub hC0I + apply increasing_mass_unique_of_terminal_mean + (fun t ω => (∫ z, f z ∂C.process (τ ω+t) ω)-(∫ z, f z ∂C.process (τ ω) ω)) + (fun t ω => ∫ z, f z ∂stoppedFreshMass B κ γ G F hw τ hc R t ω) + (fun ω => (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂C.process (τ ω) ω)- + (∫ z, f z ∂stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) + (hZI.sub hTI) + (hCA.sub hC0A) hNA hAI hNI + · intro t E hE + have h := C.stoppedFreshMass_test_cross_mean hB hc hκ F hw R he τ hτ hFresh f hf t E hE + rw [integral_sub (hAI t).integrableOn (hNI t).integrableOn, + integral_sub (hCI t).integrableOn hC0I.integrableOn, + integral_sub hZI.integrableOn hTI.integrableOn, + integral_sub (C.uniqueness_terminal_test_integrable f).integrableOn hC0I.integrableOn] + linarith + · filter_upwards [C.paths] with ω hp + intro s t hst + apply sub_le_sub_right + let := (C.radon (τ ω+t) ω).1 + exact integral_mono_measure (hp.monotone (add_le_add le_rfl hst)) (ae_of_all _ hf) f.integrable + · filter_upwards [hpaths] with ω hp + intro s t hst + let := (hp.radon t).1 + exact integral_mono_measure (hp.monotone hst) (ae_of_all _ hf) f.integrable + · exact ae_of_all _ (fun ω => by simp only [add_zero,sub_self,Pi.zero_apply]) + · filter_upwards [hpaths] with ω hp + simp only [hp.zero,integral_zero_measure,Pi.zero_apply] + · filter_upwards [C.paths,hpaths] with ω hp hq + exact (((hp.continuous_tests f).comp (continuous_const.add continuous_id)).sub + continuous_const).sub (hq.continuous_tests f hf) + +include C hB hc hκ F hw R he τ hτ hFresh in +/-- Equality of the actual measure increments on one event for every finite +future time, with the unmodified raw fresh functional. -/ +theorem CanonicalMass.stoppedFreshMass_indistinguishable + (hpaths : ∀ᵐ ω ∂P, PositiveMassPaths + (fun t => stoppedFreshMass B κ γ G F hw τ hc R t ω) + (stoppedFreshTerminalMass B κ γ G F hw τ hc R ω)) : + ∀ᵐ ω ∂P, ∀ t, + C.process (τ ω+t) ω-C.process (τ ω) ω = + stoppedFreshMass B κ γ G F hw τ hc R t ω := by + apply random_radon_process_eq_of_positive_tests_ae P (localTestFamily upperHalfPlane) + · filter_upwards [] with ω + intro t + let := (C.radon (τ ω+t) ω).2 + let := halfPlane_sub_finite_on_compacts (C.process (τ ω+t) ω) (C.process (τ ω) ω) + infer_instance + · exact hpaths.mono (fun _ h => fun t => (h.radon t).1) + · intro f hf + filter_upwards [C.paths, + C.stoppedFreshMass_test_indistinguishable hB hc hκ F hw R he τ hτ hFresh hpaths f hf] + with ω hp ht + intro t + let := (C.radon (τ ω+t) ω).1 + let := (C.radon (τ ω) ω).1 + rw [integral_halfPlane_sub _ _ (hp.monotone (le_add_of_nonneg_right (show 0 ≤ t from zero_le))) f] + exact ht t + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetBoundaryLimit.lean b/lean/OAI/Probability/SLE/Loewner/TargetBoundaryLimit.lean index bb5fd76056ad8e25b3a53c42c0c34047fc512021..53cd66932d64244a8e68fca6374111e6946e2ed1 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetBoundaryLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetBoundaryLimit.lean @@ -1,103 +1,103 @@ -import OAI.Probability.SLE.Loewner.TargetInverseContinuity -import OAI.Probability.SLE.Loewner.TargetTraceCandidate -import OAI.Probability.SLE.Geometry.RightLimitMeasurability - -namespace OAI - -/-! The complete vertical boundary limit of the raw inverse flow has a -measurable event, without a trace-generation assumption. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -def rawTipAt (κ : ℝ) (T : ℝ≥0) (y : ℝ) (b : Driver) : ℂ := - inverseFlow κ b (κ.toNNReal*T) - ((b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I) - -lemma joint_measurable_rawTipAt (κ : ℝ) (y : ℝ) : - Measurable (fun q : ℝ≥0 × Driver => rawTipAt κ q.1 y q.2) := by - have ht : Measurable (fun q : ℝ≥0 × Driver => κ.toNNReal*q.1) := - measurable_const.mul measurable_fst - have hw : Measurable (fun q : ℝ≥0 × Driver => - (q.2 (κ.toNNReal*q.1):ℂ)+(y:ℂ)*Complex.I) := - ((continuous_eval.measurable.comp (measurable_snd.prodMk ht)).complex_ofReal).add_const _ - exact @Measurable.comp (ℝ≥0 × Driver) (ℝ≥0 × (Driver × ℂ)) ℂ - inferInstance inferInstance inferInstance - (fun q => inverseFlow κ q.2.1 q.1 q.2.2) - (fun q => (κ.toNNReal*q.1,(q.2,(q.2 (κ.toNNReal*q.1):ℂ)+(y:ℂ)*Complex.I))) - (measurable_inverseFlow κ) (ht.prodMk (measurable_snd.prodMk hw)) - -lemma measurable_rawTipAt (κ : ℝ) (T : ℝ≥0) (y : ℝ) : - Measurable (rawTipAt κ T y) := by - have hw : Measurable (fun b : Driver => - (b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I) := - (continuous_eval_const _).measurable.complex_ofReal.add_const _ - exact @Measurable.comp Driver (ℝ≥0 × (Driver × ℂ)) ℂ - inferInstance inferInstance inferInstance - (fun q => inverseFlow κ q.2.1 q.1 q.2.2) - (fun b => (κ.toNNReal*T,(b,(b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I))) - (measurable_inverseFlow κ) (measurable_const.prodMk (measurable_id.prodMk hw)) - -lemma continuousOn_rawTipAt (κ : ℝ) (hκ : 0 < κ) (T : ℝ≥0) (b : Driver) : - ContinuousOn (fun y => rawTipAt κ T y b) (Ioi 0) := by - apply (continuousOn_inverseFlow κ hκ b (κ.toNNReal*T)).comp - ((continuous_const.add (Complex.continuous_ofReal.mul continuous_const)).continuousOn) - intro y hy - change 0 < ((b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I).im - simpa using hy - -lemma measurableSet_rawTipAt_tendsto (κ : ℝ) (hκ : 0 < κ) (T : ℝ≥0) - (g : Driver → ℂ) (hg : Measurable g) : - MeasurableSet {b : Driver | - Tendsto (fun y => rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (g b))} := - measurableSet_tendsto_right (rawTipAt κ T) g (measurable_rawTipAt κ T) hg - (continuousOn_rawTipAt κ hκ T) - -lemma joint_measurableSet_rawTipAt_tendsto (κ : ℝ) (hκ : 0 < κ) - (g : ℝ≥0 × Driver → ℂ) (hg : Measurable g) : - MeasurableSet {q : ℝ≥0 × Driver | - Tendsto (fun y => rawTipAt κ q.1 y q.2) (𝓝[>] 0) (𝓝 (g q))} := by - have hf : ∀ y : ℝ, Measurable (fun q : ℝ≥0 × Driver => rawTipAt κ q.1 y q.2) := - joint_measurable_rawTipAt κ - have hc : ∀ q : ℝ≥0 × Driver, - ContinuousOn (fun y : ℝ => rawTipAt κ q.1 y q.2) (Ioi 0) := - fun q => continuousOn_rawTipAt κ hκ q.1 q.2 - exact @measurableSet_tendsto_right (ℝ≥0 × Driver) inferInstance - (fun y q => rawTipAt κ q.1 y q.2) g hf hg hc - -lemma rawTipAt_eq_witness - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) - (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) - (hw : IsCapacityTwoWitness U γ G F) - (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) - (T : ℝ≥0) (y : ℝ) (hy : 0 < y) : - rawTipAt κ T y b=F T ((U T:ℂ)+(y:ℂ)*Complex.I) := by - have heTime : (κ*(T:ℝ)).toNNReal=κ.toNNReal*T := by - rw [Real.toNNReal_mul hκ.le,Real.toNNReal_coe] - have hb : b (κ.toNNReal*T)=U T := by - have h := hU T T.coe_nonneg - simpa only [Real.toNNReal_coe,driverValue,heTime] using h.symm - unfold rawTipAt - rw [Real.coe_toNNReal _ hκ.le,hb] - exact inverseFlow_eq_witness κ hκ b U γ G F hw hU T _ (by - change 0 < ((U T:ℂ)+(y:ℂ)*Complex.I).im - simpa using hy) - -lemma rawTipAt_tendsto_witness - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) - (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) - (hw : IsCapacityTwoWitness U γ G F) - (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) - (T : ℝ≥0) : - Tendsto (fun y => rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (γ T)) := by - apply Tendsto.congr' _ (hw.2.2.2.2.2.2.2.2.2.2.2.2 T) - filter_upwards [self_mem_nhdsWithin] with y hy - exact (rawTipAt_eq_witness κ hκ b U γ G F hw hU T y hy).symm - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetInverseContinuity +import OAI.Probability.SLE.Loewner.TargetTraceCandidate +import OAI.Probability.SLE.Geometry.RightLimitMeasurability + +namespace OAI + +/-! The complete vertical boundary limit of the raw inverse flow has a +measurable event, without a trace-generation assumption. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +def rawTipAt (κ : ℝ) (T : ℝ≥0) (y : ℝ) (b : Driver) : ℂ := + inverseFlow κ b (κ.toNNReal*T) + ((b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I) + +lemma joint_measurable_rawTipAt (κ : ℝ) (y : ℝ) : + Measurable (fun q : ℝ≥0 × Driver => rawTipAt κ q.1 y q.2) := by + have ht : Measurable (fun q : ℝ≥0 × Driver => κ.toNNReal*q.1) := + measurable_const.mul measurable_fst + have hw : Measurable (fun q : ℝ≥0 × Driver => + (q.2 (κ.toNNReal*q.1):ℂ)+(y:ℂ)*Complex.I) := + ((continuous_eval.measurable.comp (measurable_snd.prodMk ht)).complex_ofReal).add_const _ + exact @Measurable.comp (ℝ≥0 × Driver) (ℝ≥0 × (Driver × ℂ)) ℂ + inferInstance inferInstance inferInstance + (fun q => inverseFlow κ q.2.1 q.1 q.2.2) + (fun q => (κ.toNNReal*q.1,(q.2,(q.2 (κ.toNNReal*q.1):ℂ)+(y:ℂ)*Complex.I))) + (measurable_inverseFlow κ) (ht.prodMk (measurable_snd.prodMk hw)) + +lemma measurable_rawTipAt (κ : ℝ) (T : ℝ≥0) (y : ℝ) : + Measurable (rawTipAt κ T y) := by + have hw : Measurable (fun b : Driver => + (b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I) := + (continuous_eval_const _).measurable.complex_ofReal.add_const _ + exact @Measurable.comp Driver (ℝ≥0 × (Driver × ℂ)) ℂ + inferInstance inferInstance inferInstance + (fun q => inverseFlow κ q.2.1 q.1 q.2.2) + (fun b => (κ.toNNReal*T,(b,(b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I))) + (measurable_inverseFlow κ) (measurable_const.prodMk (measurable_id.prodMk hw)) + +lemma continuousOn_rawTipAt (κ : ℝ) (hκ : 0 < κ) (T : ℝ≥0) (b : Driver) : + ContinuousOn (fun y => rawTipAt κ T y b) (Ioi 0) := by + apply (continuousOn_inverseFlow κ hκ b (κ.toNNReal*T)).comp + ((continuous_const.add (Complex.continuous_ofReal.mul continuous_const)).continuousOn) + intro y hy + change 0 < ((b (κ.toNNReal*T):ℂ)+(y:ℂ)*Complex.I).im + simpa using hy + +lemma measurableSet_rawTipAt_tendsto (κ : ℝ) (hκ : 0 < κ) (T : ℝ≥0) + (g : Driver → ℂ) (hg : Measurable g) : + MeasurableSet {b : Driver | + Tendsto (fun y => rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (g b))} := + measurableSet_tendsto_right (rawTipAt κ T) g (measurable_rawTipAt κ T) hg + (continuousOn_rawTipAt κ hκ T) + +lemma joint_measurableSet_rawTipAt_tendsto (κ : ℝ) (hκ : 0 < κ) + (g : ℝ≥0 × Driver → ℂ) (hg : Measurable g) : + MeasurableSet {q : ℝ≥0 × Driver | + Tendsto (fun y => rawTipAt κ q.1 y q.2) (𝓝[>] 0) (𝓝 (g q))} := by + have hf : ∀ y : ℝ, Measurable (fun q : ℝ≥0 × Driver => rawTipAt κ q.1 y q.2) := + joint_measurable_rawTipAt κ + have hc : ∀ q : ℝ≥0 × Driver, + ContinuousOn (fun y : ℝ => rawTipAt κ q.1 y q.2) (Ioi 0) := + fun q => continuousOn_rawTipAt κ hκ q.1 q.2 + exact @measurableSet_tendsto_right (ℝ≥0 × Driver) inferInstance + (fun y q => rawTipAt κ q.1 y q.2) g hf hg hc + +lemma rawTipAt_eq_witness + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) + (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) + (hw : IsCapacityTwoWitness U γ G F) + (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) + (T : ℝ≥0) (y : ℝ) (hy : 0 < y) : + rawTipAt κ T y b=F T ((U T:ℂ)+(y:ℂ)*Complex.I) := by + have heTime : (κ*(T:ℝ)).toNNReal=κ.toNNReal*T := by + rw [Real.toNNReal_mul hκ.le,Real.toNNReal_coe] + have hb : b (κ.toNNReal*T)=U T := by + have h := hU T T.coe_nonneg + simpa only [Real.toNNReal_coe,driverValue,heTime] using h.symm + unfold rawTipAt + rw [Real.coe_toNNReal _ hκ.le,hb] + exact inverseFlow_eq_witness κ hκ b U γ G F hw hU T _ (by + change 0 < ((U T:ℂ)+(y:ℂ)*Complex.I).im + simpa using hy) + +lemma rawTipAt_tendsto_witness + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) + (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) + (hw : IsCapacityTwoWitness U γ G F) + (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) + (T : ℝ≥0) : + Tendsto (fun y => rawTipAt κ T y b) (𝓝[>] 0) (𝓝 (γ T)) := by + apply Tendsto.congr' _ (hw.2.2.2.2.2.2.2.2.2.2.2.2 T) + filter_upwards [self_mem_nhdsWithin] with y hy + exact (rawTipAt_eq_witness κ hκ b U γ G F hw hU T y hy).symm + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetContinuationPullback.lean b/lean/OAI/Probability/SLE/Loewner/TargetContinuationPullback.lean index 522102e7a96092633b1857fec7069a563d0e1686..2b41f1f8a641b33203d3b4dededbc02a8fb2d8f6 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetContinuationPullback.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetContinuationPullback.lean @@ -1,99 +1,99 @@ -import OAI.Probability.SLE.Angle.TargetBrownianNormalization -import OAI.Probability.SLE.Angle.TargetPotentialIncrement - -namespace OAI - -/-! The prescribed fresh-driver average pulled back to a regular -Brownian presentation, retaining its full natural past. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -local instance pullbackPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance pullbackPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem continuationMean_comp_condLExp - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - (b : Ω → Driver) (hb : Measurable b) - (hB : IsBrownianReal (fun t ω => b ω t) P) - (W : WienerLaw) (hLaw : HasLaw b W P) (t : ℝ≥0) - (f : Driver → ℝ≥0∞) (hf : Measurable f) : - (fun ω => continuationMean W t f (b ω)) =ᵐ[P] - P⁻[(fun ω => f (b ω)) | - (Filtration.natural (fun t ω => b ω t) - (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t] := by - have h := source_driver_continuation_condLExp hB - (fun t => (continuous_eval_const t).measurable.comp hb) - (fun ω => (b ω).continuous) t (fun p => f (centerDriver p)) (hf.comp measurable_centerDriver) - change (fun ω => driverContinuationMean (P.map (fun ω => (b ω).toContinuousMap)) t - (fun p => f (centerDriver p)) (b ω).toContinuousMap) =ᵐ[P] - P⁻[(fun ω => f (centerDriver (b ω).toContinuousMap)) | - (Filtration.natural (fun t ω => b ω t) - (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t] at h - have he : P.map (fun ω => (b ω).toContinuousMap)=W.map (fun p => p.toContinuousMap) := by - rw [← hLaw.map_eq,Measure.map_map measurable_driverInclusion hb] - rfl - rw [he] at h - have heq : (fun ω => driverContinuationMean (W.map (fun p => p.toContinuousMap)) t - (fun p => f (centerDriver p)) (b ω).toContinuousMap) = - (fun ω => continuationMean W t f (b ω)) := - funext (fun ω => continuationMean_eq_pathMean W t f hf (b ω)) - rw [heq] at h - simpa only [centerDriver_inclusion] using h - -theorem continuationMean_comp_finite_condExp - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - (b : Ω → Driver) (hb : Measurable b) - (hB : IsBrownianReal (fun t ω => b ω t) P) - (W : WienerLaw) (hLaw : HasLaw b W P) (t : ℝ≥0) - (f : Driver → ℝ≥0∞) (hf : Measurable f) - (hfin : (∫⁻ ω, f (b ω) ∂P) ≠ ∞) : - (∀ᵐ ω ∂P, continuationMean W t f (b ω)<∞) ∧ - (fun ω => (continuationMean W t f (b ω)).toReal) =ᵐ[P] - P[(fun ω => (f (b ω)).toReal) | - (Filtration.natural (fun t ω => b ω t) - (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t] := by - have he := continuationMean_comp_condLExp b hb hB W hLaw t f hf - let m := (Filtration.natural (fun t ω => b ω t) - (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t - constructor - · filter_upwards [he,condLExp_lt_top (mΩ := m) hfin] with ω hω hh - exact hω.symm ▸ hh - · exact (he.fun_comp ENNReal.toReal).trans (toReal_condLExp m (hf.comp hb).aemeasurable hfin) - -theorem potentialDifference_comp_eq_current_sub_condExp - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - (b : Ω → Driver) (hb : Measurable b) - (hB : IsBrownianReal (fun t ω => b ω t) P) - (W : WienerLaw) (hLaw : HasLaw b W P) (κ θ : ℝ) (n i : ℕ) - (hm : ∀ j, Measurable (fun b => potential κ θ b (compensatorMeshTime n j))) - (hfin : ∀ j, (∫⁻ ω, potential κ θ (b ω) (compensatorMeshTime n j) ∂P) ≠ ∞) : - (fun ω => potentialDifference κ θ W (b ω) n i) =ᵐ[P] - (fun ω => (potential κ θ (b ω) (compensatorMeshTime n i)).toReal - - condExp ((Filtration.natural (fun t ω => b ω t) - (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) - (compensatorMeshTime n i)) P - (fun ω => (potential κ θ (b ω) (compensatorMeshTime n (i+1))).toReal) ω) := by - have h := continuationMean_comp_finite_condExp b hb hB W hLaw - (compensatorMeshTime n i) - (fun b => potential κ θ b (compensatorMeshTime n (i+1))) (hm (i+1)) (hfin (i+1)) - filter_upwards [ae_lt_top ((hm i).comp hb) (hfin i),h.1,h.2] with ω hv hw he - have hnext : ((i:ℝ)+1)*mesh n=(compensatorMeshTime n (i+1):ℝ) := by - simpa only [Nat.cast_add,Nat.cast_one] using mesh_eq_compensatorMeshTime n (i+1) - simp only [potentialDifference,mesh_eq_compensatorMeshTime,hnext] - change (if potential κ θ (b ω) (compensatorMeshTime n i)<∞ ∧ - continuationMean W (compensatorMeshTime n i) - (fun b => potential κ θ b (compensatorMeshTime n (i+1))) (b ω)<∞ - then _ else _)=_ - rw [ite_eq_left ⟨hv,hw⟩] - dsimp only [continuationMean] at he - rw [he] - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetBrownianNormalization +import OAI.Probability.SLE.Angle.TargetPotentialIncrement + +namespace OAI + +/-! The prescribed fresh-driver average pulled back to a regular +Brownian presentation, retaining its full natural past. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +local instance pullbackPathMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance pullbackPathBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem continuationMean_comp_condLExp + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + (b : Ω → Driver) (hb : Measurable b) + (hB : IsBrownianReal (fun t ω => b ω t) P) + (W : WienerLaw) (hLaw : HasLaw b W P) (t : ℝ≥0) + (f : Driver → ℝ≥0∞) (hf : Measurable f) : + (fun ω => continuationMean W t f (b ω)) =ᵐ[P] + P⁻[(fun ω => f (b ω)) | + (Filtration.natural (fun t ω => b ω t) + (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t] := by + have h := source_driver_continuation_condLExp hB + (fun t => (continuous_eval_const t).measurable.comp hb) + (fun ω => (b ω).continuous) t (fun p => f (centerDriver p)) (hf.comp measurable_centerDriver) + change (fun ω => driverContinuationMean (P.map (fun ω => (b ω).toContinuousMap)) t + (fun p => f (centerDriver p)) (b ω).toContinuousMap) =ᵐ[P] + P⁻[(fun ω => f (centerDriver (b ω).toContinuousMap)) | + (Filtration.natural (fun t ω => b ω t) + (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t] at h + have he : P.map (fun ω => (b ω).toContinuousMap)=W.map (fun p => p.toContinuousMap) := by + rw [← hLaw.map_eq,Measure.map_map measurable_driverInclusion hb] + rfl + rw [he] at h + have heq : (fun ω => driverContinuationMean (W.map (fun p => p.toContinuousMap)) t + (fun p => f (centerDriver p)) (b ω).toContinuousMap) = + (fun ω => continuationMean W t f (b ω)) := + funext (fun ω => continuationMean_eq_pathMean W t f hf (b ω)) + rw [heq] at h + simpa only [centerDriver_inclusion] using h + +theorem continuationMean_comp_finite_condExp + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + (b : Ω → Driver) (hb : Measurable b) + (hB : IsBrownianReal (fun t ω => b ω t) P) + (W : WienerLaw) (hLaw : HasLaw b W P) (t : ℝ≥0) + (f : Driver → ℝ≥0∞) (hf : Measurable f) + (hfin : (∫⁻ ω, f (b ω) ∂P) ≠ ∞) : + (∀ᵐ ω ∂P, continuationMean W t f (b ω)<∞) ∧ + (fun ω => (continuationMean W t f (b ω)).toReal) =ᵐ[P] + P[(fun ω => (f (b ω)).toReal) | + (Filtration.natural (fun t ω => b ω t) + (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t] := by + have he := continuationMean_comp_condLExp b hb hB W hLaw t f hf + let m := (Filtration.natural (fun t ω => b ω t) + (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) t + constructor + · filter_upwards [he,condLExp_lt_top (mΩ := m) hfin] with ω hω hh + exact hω.symm ▸ hh + · exact (he.fun_comp ENNReal.toReal).trans (toReal_condLExp m (hf.comp hb).aemeasurable hfin) + +theorem potentialDifference_comp_eq_current_sub_condExp + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + (b : Ω → Driver) (hb : Measurable b) + (hB : IsBrownianReal (fun t ω => b ω t) P) + (W : WienerLaw) (hLaw : HasLaw b W P) (κ θ : ℝ) (n i : ℕ) + (hm : ∀ j, Measurable (fun b => potential κ θ b (compensatorMeshTime n j))) + (hfin : ∀ j, (∫⁻ ω, potential κ θ (b ω) (compensatorMeshTime n j) ∂P) ≠ ∞) : + (fun ω => potentialDifference κ θ W (b ω) n i) =ᵐ[P] + (fun ω => (potential κ θ (b ω) (compensatorMeshTime n i)).toReal - + condExp ((Filtration.natural (fun t ω => b ω t) + (fun t => ((continuous_eval_const t).measurable.comp hb).stronglyMeasurable)) + (compensatorMeshTime n i)) P + (fun ω => (potential κ θ (b ω) (compensatorMeshTime n (i+1))).toReal) ω) := by + have h := continuationMean_comp_finite_condExp b hb hB W hLaw + (compensatorMeshTime n i) + (fun b => potential κ θ b (compensatorMeshTime n (i+1))) (hm (i+1)) (hfin (i+1)) + filter_upwards [ae_lt_top ((hm i).comp hb) (hfin i),h.1,h.2] with ω hv hw he + have hnext : ((i:ℝ)+1)*mesh n=(compensatorMeshTime n (i+1):ℝ) := by + simpa only [Nat.cast_add,Nat.cast_one] using mesh_eq_compensatorMeshTime n (i+1) + simp only [potentialDifference,mesh_eq_compensatorMeshTime,hnext] + change (if potential κ θ (b ω) (compensatorMeshTime n i)<∞ ∧ + continuationMean W (compensatorMeshTime n i) + (fun b => potential κ θ b (compensatorMeshTime n (i+1))) (b ω)<∞ + then _ else _)=_ + rw [ite_eq_left ⟨hv,hw⟩] + dsimp only [continuationMean] at he + rw [he] + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetCutoffField.lean b/lean/OAI/Probability/SLE/Loewner/TargetCutoffField.lean index 21d85a61481ba06453ec6d38f5c9f66d4417d889..d033f10dd2adbcf407db3b0a3f96a7165d7055c1 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetCutoffField.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetCutoffField.lean @@ -1,125 +1,125 @@ -import OAI.Probability.SLE.MomentGauge -import OAI.Probability.SLE.ClippedCoordinates - -namespace OAI - -/-! Global estimates for the exact reciprocal cutoff used in the -prescribed Euler construction. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Target - -private def reciprocalCutoff (c : ℝ) (z : ℂ) : ℂ := - star z / (max (‖z‖ ^ 2) c : ℝ) - -private lemma cutoff_denom_lower (c : ℝ) (hc : 0 < c) (z : ℂ) : - Real.sqrt c * ‖z‖ ≤ max (‖z‖ ^ 2) c := by - have hs := Real.sq_sqrt hc.le - have hsq := sq_nonneg (‖z‖-Real.sqrt c) - have hz := le_max_left (‖z‖ ^ 2) c - have hc' := le_max_right (‖z‖ ^ 2) c - nlinarith - -private lemma reciprocalCutoff_norm_le (c : ℝ) (hc : 0 < c) (z : ℂ) : - ‖reciprocalCutoff c z‖ ≤ (Real.sqrt c)⁻¹ := by - have hA : 0 < max (‖z‖ ^ 2) c := hc.trans_le (le_max_right _ _) - have hs : 0 < Real.sqrt c := Real.sqrt_pos.2 hc - rw [reciprocalCutoff,norm_div,norm_star,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hA] - apply (div_le_iff₀ hA).2 - simpa only [div_eq_mul_inv,mul_comm] using - (le_div_iff₀ hs).2 (by simpa only [mul_comm] using cutoff_denom_lower c hc z) - -private lemma reciprocalCutoff_dist_le (c : ℝ) (hc : 0 < c) (z w : ℂ) : - dist (reciprocalCutoff c z) (reciprocalCutoff c w) ≤ (3/c)*dist z w := by - let A : ℝ := max (‖z‖^2) c - let B : ℝ := max (‖w‖^2) c - have hA : 0 < A := hc.trans_le (le_max_right _ _) - have hB : 0 < B := hc.trans_le (le_max_right _ _) - have hAB : 0 < A*B := mul_pos hA hB - have hcA : c ≤ A := le_max_right _ _ - have hcB : c ≤ B := le_max_right _ _ - have hwB : ‖w‖^2 ≤ B := le_max_left _ _ - have hzw : c*‖z‖*‖w‖ ≤ A*B := by - have h := mul_le_mul (cutoff_denom_lower c hc z) (cutoff_denom_lower c hc w) - (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _)) hA.le - nlinarith [Real.sq_sqrt hc.le] - have hww : c*‖w‖^2 ≤ A*B := by - exact mul_le_mul hcA hwB (sq_nonneg _) hA.le - have hdiff : |B-A| ≤ (‖z‖+‖w‖)*dist z w := by - have hmax := abs_max_sub_max_le_abs (‖w‖^2) (‖z‖^2) c - have hnorm := abs_norm_sub_norm_le w z - have he : |‖w‖^2-‖z‖^2| = |‖w‖-‖z‖| *(‖z‖+‖w‖) := by - rw [sq_sub_sq,abs_mul,abs_of_nonneg (add_nonneg (norm_nonneg _) (norm_nonneg _))] - ring - refine hmax.trans ?_ - rw [he] - have hnorm' : |‖w‖-‖z‖| ≤ dist z w := by - simpa only [← dist_eq_norm,dist_comm] using hnorm - nlinarith [norm_nonneg z,norm_nonneg w] - have he : reciprocalCutoff c z-reciprocalCutoff c w = - star (z-w)/(A:ℂ)+star w*((B-A:ℝ):ℂ)/((A*B:ℝ):ℂ) := by - simp only [reciprocalCutoff,star_sub,Complex.ofReal_sub,Complex.ofReal_mul] - change star z/(A:ℂ)-star w/(B:ℂ) = _ - field_simp [Complex.ofReal_ne_zero.mpr hA.ne',Complex.ofReal_ne_zero.mpr hB.ne'] - ring - rw [dist_eq_norm,he] - calc - _ ≤ ‖star (z-w)/(A:ℂ)‖+‖star w*((B-A:ℝ):ℂ)/((A*B:ℝ):ℂ)‖ := norm_add_le _ _ - _ = dist z w/A+‖w‖*|B-A|/(A*B) := by - simp only [norm_div,norm_mul,norm_star,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hA, - abs_of_pos hB,dist_eq_norm] - _ ≤ dist z w/A+‖w‖*((‖z‖+‖w‖)*dist z w)/(A*B) := by - gcongr - _ ≤ (3/c)*dist z w := by - have hcoef : (B+‖w‖*(‖z‖+‖w‖))/(A*B) ≤ 3/c := by - apply (div_le_div_iff₀ hAB hc).2 - have hcb := mul_le_mul_of_nonneg_right hcA hB.le - nlinarith - calc - dist z w/A+‖w‖*((‖z‖+‖w‖)*dist z w)/(A*B) = - ((B+‖w‖*(‖z‖+‖w‖))/(A*B))*dist z w := by - field_simp [hA.ne',hB.ne'] - _ ≤ (3/c)*dist z w := mul_le_mul_of_nonneg_right hcoef dist_nonneg - - -lemma cutoffField_bounds (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) : - ∃ L K : ℝ≥0, (∀ b t z, ‖cutoffField κ b j t z‖ ≤ L) ∧ - ∀ b t, LipschitzWith K (cutoffField κ b j t) := by - let c : ℝ := (j:ℝ)^(-(4:ℤ)) - have hc : 0 < c := zpow_pos (by exact_mod_cast (Nat.succ_le_iff.mp hj) : (0:ℝ) cutoffField κ b j p.1 p.2) := by - have hc : 0 < (j:ℝ)^(-(4:ℤ)) := zpow_pos (by exact_mod_cast (Nat.succ_le_iff.mp hj) : (0:ℝ) driverValue b t) := - b.continuous.comp continuous_real_toNNReal - unfold cutoffField - fun_prop (disch := intro x; exact_mod_cast ne_of_gt (hc.trans_le (le_max_right _ _))) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.MomentGauge +import OAI.Probability.SLE.ClippedCoordinates + +namespace OAI + +/-! Global estimates for the exact reciprocal cutoff used in the +prescribed Euler construction. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Target + +private def reciprocalCutoff (c : ℝ) (z : ℂ) : ℂ := + star z / (max (‖z‖ ^ 2) c : ℝ) + +private lemma cutoff_denom_lower (c : ℝ) (hc : 0 < c) (z : ℂ) : + Real.sqrt c * ‖z‖ ≤ max (‖z‖ ^ 2) c := by + have hs := Real.sq_sqrt hc.le + have hsq := sq_nonneg (‖z‖-Real.sqrt c) + have hz := le_max_left (‖z‖ ^ 2) c + have hc' := le_max_right (‖z‖ ^ 2) c + nlinarith + +private lemma reciprocalCutoff_norm_le (c : ℝ) (hc : 0 < c) (z : ℂ) : + ‖reciprocalCutoff c z‖ ≤ (Real.sqrt c)⁻¹ := by + have hA : 0 < max (‖z‖ ^ 2) c := hc.trans_le (le_max_right _ _) + have hs : 0 < Real.sqrt c := Real.sqrt_pos.2 hc + rw [reciprocalCutoff,norm_div,norm_star,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hA] + apply (div_le_iff₀ hA).2 + simpa only [div_eq_mul_inv,mul_comm] using + (le_div_iff₀ hs).2 (by simpa only [mul_comm] using cutoff_denom_lower c hc z) + +private lemma reciprocalCutoff_dist_le (c : ℝ) (hc : 0 < c) (z w : ℂ) : + dist (reciprocalCutoff c z) (reciprocalCutoff c w) ≤ (3/c)*dist z w := by + let A : ℝ := max (‖z‖^2) c + let B : ℝ := max (‖w‖^2) c + have hA : 0 < A := hc.trans_le (le_max_right _ _) + have hB : 0 < B := hc.trans_le (le_max_right _ _) + have hAB : 0 < A*B := mul_pos hA hB + have hcA : c ≤ A := le_max_right _ _ + have hcB : c ≤ B := le_max_right _ _ + have hwB : ‖w‖^2 ≤ B := le_max_left _ _ + have hzw : c*‖z‖*‖w‖ ≤ A*B := by + have h := mul_le_mul (cutoff_denom_lower c hc z) (cutoff_denom_lower c hc w) + (mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _)) hA.le + nlinarith [Real.sq_sqrt hc.le] + have hww : c*‖w‖^2 ≤ A*B := by + exact mul_le_mul hcA hwB (sq_nonneg _) hA.le + have hdiff : |B-A| ≤ (‖z‖+‖w‖)*dist z w := by + have hmax := abs_max_sub_max_le_abs (‖w‖^2) (‖z‖^2) c + have hnorm := abs_norm_sub_norm_le w z + have he : |‖w‖^2-‖z‖^2| = |‖w‖-‖z‖| *(‖z‖+‖w‖) := by + rw [sq_sub_sq,abs_mul,abs_of_nonneg (add_nonneg (norm_nonneg _) (norm_nonneg _))] + ring + refine hmax.trans ?_ + rw [he] + have hnorm' : |‖w‖-‖z‖| ≤ dist z w := by + simpa only [← dist_eq_norm,dist_comm] using hnorm + nlinarith [norm_nonneg z,norm_nonneg w] + have he : reciprocalCutoff c z-reciprocalCutoff c w = + star (z-w)/(A:ℂ)+star w*((B-A:ℝ):ℂ)/((A*B:ℝ):ℂ) := by + simp only [reciprocalCutoff,star_sub,Complex.ofReal_sub,Complex.ofReal_mul] + change star z/(A:ℂ)-star w/(B:ℂ) = _ + field_simp [Complex.ofReal_ne_zero.mpr hA.ne',Complex.ofReal_ne_zero.mpr hB.ne'] + ring + rw [dist_eq_norm,he] + calc + _ ≤ ‖star (z-w)/(A:ℂ)‖+‖star w*((B-A:ℝ):ℂ)/((A*B:ℝ):ℂ)‖ := norm_add_le _ _ + _ = dist z w/A+‖w‖*|B-A|/(A*B) := by + simp only [norm_div,norm_mul,norm_star,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hA, + abs_of_pos hB,dist_eq_norm] + _ ≤ dist z w/A+‖w‖*((‖z‖+‖w‖)*dist z w)/(A*B) := by + gcongr + _ ≤ (3/c)*dist z w := by + have hcoef : (B+‖w‖*(‖z‖+‖w‖))/(A*B) ≤ 3/c := by + apply (div_le_div_iff₀ hAB hc).2 + have hcb := mul_le_mul_of_nonneg_right hcA hB.le + nlinarith + calc + dist z w/A+‖w‖*((‖z‖+‖w‖)*dist z w)/(A*B) = + ((B+‖w‖*(‖z‖+‖w‖))/(A*B))*dist z w := by + field_simp [hA.ne',hB.ne'] + _ ≤ (3/c)*dist z w := mul_le_mul_of_nonneg_right hcoef dist_nonneg + + +lemma cutoffField_bounds (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) : + ∃ L K : ℝ≥0, (∀ b t z, ‖cutoffField κ b j t z‖ ≤ L) ∧ + ∀ b t, LipschitzWith K (cutoffField κ b j t) := by + let c : ℝ := (j:ℝ)^(-(4:ℤ)) + have hc : 0 < c := zpow_pos (by exact_mod_cast (Nat.succ_le_iff.mp hj) : (0:ℝ) cutoffField κ b j p.1 p.2) := by + have hc : 0 < (j:ℝ)^(-(4:ℤ)) := zpow_pos (by exact_mod_cast (Nat.succ_le_iff.mp hj) : (0:ℝ) driverValue b t) := + b.continuous.comp continuous_real_toNNReal + unfold cutoffField + fun_prop (disch := intro x; exact_mod_cast ne_of_gt (hc.trans_le (le_max_right _ _))) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetCutoffMeasurability.lean b/lean/OAI/Probability/SLE/Loewner/TargetCutoffMeasurability.lean index 0488d33125c895531bfb1ada6f5d2950f02e1190..9416cfa5163bfb3cd0eccec0554b6e0389aab954 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetCutoffMeasurability.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetCutoffMeasurability.lean @@ -1,128 +1,128 @@ -import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory -import OAI.Probability.SLE.Loewner.TargetDriverContinuation - -namespace OAI - -/-! Joint measurability and finite-prefix dependence of the prescribed -cutoff flow. The variable initial point is encoded as a displacement in -the already constructed measurable Picard equation. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -lemma measurable_cutoffField (κ : ℝ) (j : ℕ) : - Measurable (fun q : (ℝ × Driver) × ℂ => cutoffField κ q.1.2 j q.1.1 q.2) := by - have hb : Measurable (fun q : (ℝ × Driver) × ℂ => - q.1.2 q.1.1.toNNReal) := - continuous_eval.measurable.comp ((measurable_snd.comp measurable_fst).prodMk - (continuous_real_toNNReal.measurable.comp (measurable_fst.comp measurable_fst))) - have hv : Measurable (fun q : (ℝ × Driver) × ℂ => - q.2 - (q.1.2 q.1.1.toNNReal : ℂ)) := measurable_snd.sub hb.complex_ofReal - exact (measurable_const.mul (Complex.continuous_conj.measurable.comp hv)).div - ((hv.norm.pow_const 2).max measurable_const).complex_ofReal - -lemma measurable_cutoffFlow (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) : - Measurable (fun q : ℝ≥0 × (Driver × ℂ) => cutoffFlow κ q.2.1 j q.2.2 q.1) := by - obtain ⟨L,K,hL,hK⟩ := cutoffField_bounds κ j hj - let V (t : ℝ) (x : Driver × ℂ) (v : ℂ) := cutoffField κ x.1 j t (v+x.2) - have hV : Measurable (fun q : (ℝ × (Driver × ℂ)) × ℂ => V q.1.1 q.1.2 q.2) := by - have hpair : Measurable (fun q : (ℝ × (Driver × ℂ)) × ℂ => - ((q.1.1,q.1.2.1),q.2+q.1.2.2)) := - ((measurable_fst.fst.prodMk measurable_fst.snd.fst).prodMk - (measurable_snd.add measurable_fst.snd.snd)) - exact @Measurable.comp ((ℝ × (Driver × ℂ)) × ℂ) ((ℝ × Driver) × ℂ) ℂ - inferInstance inferInstance inferInstance - (fun q => cutoffField κ q.1.2 j q.1.1 q.2) - (fun q => ((q.1.1,q.1.2.1),q.2+q.1.2.2)) (measurable_cutoffField κ j) hpair - have hc (x : Driver × ℂ) (v : ℂ) : Continuous (fun t => V t x v) := - (continuous_cutoffField κ x.1 j hj).comp (continuous_id.prodMk continuous_const) - have hl (x : Driver × ℂ) (t : ℝ) : LipschitzWith K (V t x) := by - apply LipschitzWith.of_dist_le_mul - intro v w - simpa only [V,dist_add_right] using (hK x.1 t).dist_le_mul (v+x.2) (w+x.2) - obtain ⟨X,hXm,hXl,hX0,hXd,_⟩ := exists_measurable_global_bounded_ode_picard - V 0 L K hV hc hl (fun x t v => hL x.1 t (v+x.2)) - have he (t : ℝ≥0) (x : Driver × ℂ) : - cutoffFlow κ x.1 j x.2 t = X t x+x.2 := by - apply cutoffFlow_eq_solution κ x.1 j hj x.2 L K (hL x.1) (hK x.1) - (fun s => X s x+x.2) ((hXl x).continuous.add continuous_const) - · simp only [hX0,zero_add] - · intro s hs - simpa only [V] using (hXd x s hs).add_const x.2 - · exact t.coe_nonneg - have hm : Measurable (fun q : ℝ≥0 × (Driver × ℂ) => X q.1 q.2+q.2.2) := - (hXm.comp (measurable_fst.coe_nnreal_real.prodMk measurable_snd)).add measurable_snd.snd - convert hm using 1 - exact funext (fun q => he q.1 q.2) - -def stopDriver (b : Driver) (T : ℝ≥0) : Driver where - toFun t := b (min t T) - continuous_toFun := (map_continuous b).comp (continuous_id.min continuous_const) - map_zero' := by simp - -@[simp] lemma stopDriver_apply (b : Driver) (T t : ℝ≥0) : - stopDriver b T t = b (min t T) := rfl - -lemma stopDriver_measurable_past (T : ℝ≥0) : - Measurable[driverFiltration T] (fun b => stopDriver b T) := by - have hm (t : ℝ≥0) : Measurable[driverFiltration T] - (fun b : Driver => b (min t T)) := - ((Filtration.stronglyAdapted_natural - (fun u => (show Measurable (fun b : Driver => b u) from - (continuous_eval_const u).measurable).stronglyMeasurable) (min t T)).measurable).mono - (driverFiltration.mono (min_le_right t T)) le_rfl - apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp - let : MeasurableSpace Driver := driverFiltration T - apply source_path_measurable - intro t - exact hm t - -lemma cutoffFlow_eq_of_driver_prefix (κ : ℝ) (b c : Driver) (j : ℕ) (hj : 1 ≤ j) - (z : ℂ) (T : ℝ) (hT : 0 ≤ T) - (he : ∀ s ∈ Icc 0 T, driverValue b s = driverValue c s) : - ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t = cutoffFlow κ c j z t := by - apply cutoffFlow_eq_of_trajectory κ b j hj z T hT (cutoffFlow κ c j z) - ((cutoffFlow_continuousOn κ c j hj z).mono Icc_subset_Ici_self) - (cutoffFlow_zero κ c j hj z) - intro s hs - have hv : cutoffField κ b j s (cutoffFlow κ c j z s) = - cutoffField κ c j s (cutoffFlow κ c j z s) := by - simp only [cutoffField,he s (Ico_subset_Icc_self hs)] - rw [hv] - exact (cutoffFlow_hasDerivWithinAt κ c j hj z s hs.1).mono - (Ici_subset_Ici.mpr hs.1) - -lemma cutoffFlow_stopDriver (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) - (z : ℂ) (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : - cutoffFlow κ (stopDriver b T) j z t = cutoffFlow κ b j z t := by - apply cutoffFlow_eq_of_driver_prefix κ _ _ j hj z T T.coe_nonneg ?_ t ht - intro s hs - simp only [driverValue,stopDriver_apply, - min_eq_left ((Real.toNNReal_le_iff_le_coe).mpr hs.2)] - -lemma cutoffFlow_measurable_past (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) (T t : ℝ≥0) - (ht : t ≤ T) : - Measurable[(driverFiltration T).prod (borel ℂ)] - (fun q : Driver × ℂ => cutoffFlow κ q.1 j q.2 t) := by - have hpair : Measurable[(driverFiltration T).prod (borel ℂ)] - (fun q : Driver × ℂ => (t,(stopDriver q.1 T,q.2))) := - measurable_const.prodMk (((stopDriver_measurable_past T).comp measurable_fst).prodMk - measurable_snd) - have hm : Measurable[(driverFiltration T).prod (borel ℂ)] - (fun q : Driver × ℂ => cutoffFlow κ (stopDriver q.1 T) j q.2 t) := - @Measurable.comp (Driver × ℂ) (ℝ≥0 × (Driver × ℂ)) ℂ - ((driverFiltration T).prod (borel ℂ)) inferInstance inferInstance - (fun q => cutoffFlow κ q.2.1 j q.2.2 q.1) - (fun q => (t,(stopDriver q.1 T,q.2))) (measurable_cutoffFlow κ j hj) hpair - convert hm using 1 - exact funext (fun q => (cutoffFlow_stopDriver κ q.1 j hj q.2 T t - ⟨t.coe_nonneg,show (t : ℝ) ≤ T from ht⟩).symm) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory +import OAI.Probability.SLE.Loewner.TargetDriverContinuation + +namespace OAI + +/-! Joint measurability and finite-prefix dependence of the prescribed +cutoff flow. The variable initial point is encoded as a displacement in +the already constructed measurable Picard equation. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +lemma measurable_cutoffField (κ : ℝ) (j : ℕ) : + Measurable (fun q : (ℝ × Driver) × ℂ => cutoffField κ q.1.2 j q.1.1 q.2) := by + have hb : Measurable (fun q : (ℝ × Driver) × ℂ => + q.1.2 q.1.1.toNNReal) := + continuous_eval.measurable.comp ((measurable_snd.comp measurable_fst).prodMk + (continuous_real_toNNReal.measurable.comp (measurable_fst.comp measurable_fst))) + have hv : Measurable (fun q : (ℝ × Driver) × ℂ => + q.2 - (q.1.2 q.1.1.toNNReal : ℂ)) := measurable_snd.sub hb.complex_ofReal + exact (measurable_const.mul (Complex.continuous_conj.measurable.comp hv)).div + ((hv.norm.pow_const 2).max measurable_const).complex_ofReal + +lemma measurable_cutoffFlow (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) : + Measurable (fun q : ℝ≥0 × (Driver × ℂ) => cutoffFlow κ q.2.1 j q.2.2 q.1) := by + obtain ⟨L,K,hL,hK⟩ := cutoffField_bounds κ j hj + let V (t : ℝ) (x : Driver × ℂ) (v : ℂ) := cutoffField κ x.1 j t (v+x.2) + have hV : Measurable (fun q : (ℝ × (Driver × ℂ)) × ℂ => V q.1.1 q.1.2 q.2) := by + have hpair : Measurable (fun q : (ℝ × (Driver × ℂ)) × ℂ => + ((q.1.1,q.1.2.1),q.2+q.1.2.2)) := + ((measurable_fst.fst.prodMk measurable_fst.snd.fst).prodMk + (measurable_snd.add measurable_fst.snd.snd)) + exact @Measurable.comp ((ℝ × (Driver × ℂ)) × ℂ) ((ℝ × Driver) × ℂ) ℂ + inferInstance inferInstance inferInstance + (fun q => cutoffField κ q.1.2 j q.1.1 q.2) + (fun q => ((q.1.1,q.1.2.1),q.2+q.1.2.2)) (measurable_cutoffField κ j) hpair + have hc (x : Driver × ℂ) (v : ℂ) : Continuous (fun t => V t x v) := + (continuous_cutoffField κ x.1 j hj).comp (continuous_id.prodMk continuous_const) + have hl (x : Driver × ℂ) (t : ℝ) : LipschitzWith K (V t x) := by + apply LipschitzWith.of_dist_le_mul + intro v w + simpa only [V,dist_add_right] using (hK x.1 t).dist_le_mul (v+x.2) (w+x.2) + obtain ⟨X,hXm,hXl,hX0,hXd,_⟩ := exists_measurable_global_bounded_ode_picard + V 0 L K hV hc hl (fun x t v => hL x.1 t (v+x.2)) + have he (t : ℝ≥0) (x : Driver × ℂ) : + cutoffFlow κ x.1 j x.2 t = X t x+x.2 := by + apply cutoffFlow_eq_solution κ x.1 j hj x.2 L K (hL x.1) (hK x.1) + (fun s => X s x+x.2) ((hXl x).continuous.add continuous_const) + · simp only [hX0,zero_add] + · intro s hs + simpa only [V] using (hXd x s hs).add_const x.2 + · exact t.coe_nonneg + have hm : Measurable (fun q : ℝ≥0 × (Driver × ℂ) => X q.1 q.2+q.2.2) := + (hXm.comp (measurable_fst.coe_nnreal_real.prodMk measurable_snd)).add measurable_snd.snd + convert hm using 1 + exact funext (fun q => he q.1 q.2) + +def stopDriver (b : Driver) (T : ℝ≥0) : Driver where + toFun t := b (min t T) + continuous_toFun := (map_continuous b).comp (continuous_id.min continuous_const) + map_zero' := by simp + +@[simp] lemma stopDriver_apply (b : Driver) (T t : ℝ≥0) : + stopDriver b T t = b (min t T) := rfl + +lemma stopDriver_measurable_past (T : ℝ≥0) : + Measurable[driverFiltration T] (fun b => stopDriver b T) := by + have hm (t : ℝ≥0) : Measurable[driverFiltration T] + (fun b : Driver => b (min t T)) := + ((Filtration.stronglyAdapted_natural + (fun u => (show Measurable (fun b : Driver => b u) from + (continuous_eval_const u).measurable).stronglyMeasurable) (min t T)).measurable).mono + (driverFiltration.mono (min_le_right t T)) le_rfl + apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp + let : MeasurableSpace Driver := driverFiltration T + apply source_path_measurable + intro t + exact hm t + +lemma cutoffFlow_eq_of_driver_prefix (κ : ℝ) (b c : Driver) (j : ℕ) (hj : 1 ≤ j) + (z : ℂ) (T : ℝ) (hT : 0 ≤ T) + (he : ∀ s ∈ Icc 0 T, driverValue b s = driverValue c s) : + ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t = cutoffFlow κ c j z t := by + apply cutoffFlow_eq_of_trajectory κ b j hj z T hT (cutoffFlow κ c j z) + ((cutoffFlow_continuousOn κ c j hj z).mono Icc_subset_Ici_self) + (cutoffFlow_zero κ c j hj z) + intro s hs + have hv : cutoffField κ b j s (cutoffFlow κ c j z s) = + cutoffField κ c j s (cutoffFlow κ c j z s) := by + simp only [cutoffField,he s (Ico_subset_Icc_self hs)] + rw [hv] + exact (cutoffFlow_hasDerivWithinAt κ c j hj z s hs.1).mono + (Ici_subset_Ici.mpr hs.1) + +lemma cutoffFlow_stopDriver (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) + (z : ℂ) (T : ℝ≥0) (t : ℝ) (ht : t ∈ Icc 0 (T : ℝ)) : + cutoffFlow κ (stopDriver b T) j z t = cutoffFlow κ b j z t := by + apply cutoffFlow_eq_of_driver_prefix κ _ _ j hj z T T.coe_nonneg ?_ t ht + intro s hs + simp only [driverValue,stopDriver_apply, + min_eq_left ((Real.toNNReal_le_iff_le_coe).mpr hs.2)] + +lemma cutoffFlow_measurable_past (κ : ℝ) (j : ℕ) (hj : 1 ≤ j) (T t : ℝ≥0) + (ht : t ≤ T) : + Measurable[(driverFiltration T).prod (borel ℂ)] + (fun q : Driver × ℂ => cutoffFlow κ q.1 j q.2 t) := by + have hpair : Measurable[(driverFiltration T).prod (borel ℂ)] + (fun q : Driver × ℂ => (t,(stopDriver q.1 T,q.2))) := + measurable_const.prodMk (((stopDriver_measurable_past T).comp measurable_fst).prodMk + measurable_snd) + have hm : Measurable[(driverFiltration T).prod (borel ℂ)] + (fun q : Driver × ℂ => cutoffFlow κ (stopDriver q.1 T) j q.2 t) := + @Measurable.comp (Driver × ℂ) (ℝ≥0 × (Driver × ℂ)) ℂ + ((driverFiltration T).prod (borel ℂ)) inferInstance inferInstance + (fun q => cutoffFlow κ q.2.1 j q.2.2 q.1) + (fun q => (t,(stopDriver q.1 T,q.2))) (measurable_cutoffFlow κ j hj) hpair + convert hm using 1 + exact funext (fun q => (cutoffFlow_stopDriver κ q.1 j hj q.2 T t + ⟨t.coe_nonneg,show (t : ℝ) ≤ T from ht⟩).symm) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetCutoffTrajectory.lean b/lean/OAI/Probability/SLE/Loewner/TargetCutoffTrajectory.lean index 54d000c589f3a2e0999969d6d5cdf3dd24b90c42..21bfef76f5f44baccc2a1cac9231c59dfc911ffd 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetCutoffTrajectory.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetCutoffTrajectory.lean @@ -1,114 +1,114 @@ -import OAI.Probability.SLE.Loewner.TargetEulerConvergence - -namespace OAI - -/-! The Euler-defined cutoff agrees with every classical trajectory on a -compact time interval. Positive height makes the cutoff inactive. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Target - -lemma cutoffFlow_eq_of_trajectory - (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) - (T : ℝ) (_hT : 0 ≤ T) (Y : ℝ → ℂ) - (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z) - (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y - (cutoffField κ b j t (Y t)) (Ici t) t) : - ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t=Y t := by - obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z - have he := dist_le_of_trajectories_ODE hK hX.continuous.continuousOn - (fun t ht => (hXd t ht.1).mono (Ici_subset_Ici.mpr ht.1)) hY hYd - (show dist (X 0) (Y 0) ≤ 0 by rw [hX0,hY0]; simp) - intro t ht - rw [cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd t ht.1] - exact dist_le_zero.mp (by simpa only [zero_mul] using he t ht) - -lemma cutoffFlow_continuousOn (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) : - ContinuousOn (cutoffFlow κ b j z) (Ici 0) := by - obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z - exact hX.continuous.continuousOn.congr (fun t ht => - cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd t ht) - -lemma cutoffFlow_hasDerivWithinAt (κ : ℝ) (b : Driver) (j : ℕ) - (hj : 1 ≤ j) (z : ℂ) (t : ℝ) (ht : 0 ≤ t) : - HasDerivWithinAt (cutoffFlow κ b j z) - (cutoffField κ b j t (cutoffFlow κ b j z t)) (Ici 0) t := by - obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z - have he (s : ℝ) (hs : 0 ≤ s) := - cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd s hs - rw [he t ht] - exact (hXd t ht).congr_of_eventuallyEq - (by filter_upwards [self_mem_nhdsWithin] with s hs; exact he s hs) (he t ht) - -lemma cutoffFlow_zero (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) : - cutoffFlow κ b j z 0=z := by - obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z - rw [cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd 0 le_rfl,hX0] - -lemma cutoff_inactive_of_height (j : ℕ) (hj : 1 ≤ j) (x : ℝ) (z : ℂ) - (hz : (j:ℝ)⁻¹ ≤ z.im) : - (j:ℝ)^(-(4:ℤ)) ≤ ‖z-(x:ℂ)‖^2 := by - have hj1 : (1:ℝ) ≤ j := by exact_mod_cast hj - have hj0 : (0:ℝ) < j := zero_lt_one.trans_le hj1 - have hi0 : 0 < (j:ℝ)⁻¹ := inv_pos.mpr hj0 - have hi1 : (j:ℝ)⁻¹ ≤ 1 := (inv_le_one₀ hj0).2 hj1 - have hi2 : ((j:ℝ)⁻¹)^2 ≤ 1 := by nlinarith - have hi4 : ((j:ℝ)⁻¹)^4 ≤ ((j:ℝ)⁻¹)^2 := by nlinarith [sq_nonneg (((j:ℝ)⁻¹)^2)] - have hh : ((j:ℝ)⁻¹)^2 ≤ z.im^2 := by nlinarith - have hn := Complex.sq_norm_sub_sq_im (z-(x:ℂ)) - simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] at hn - calc - (j:ℝ)^(-(4:ℤ)) = ((j:ℝ)⁻¹)^4 := by simp only [zpow_neg,zpow_ofNat,inv_pow] - _ ≤ z.im^2 := hi4.trans hh - _ ≤ ‖z-(x:ℂ)‖^2 := by nlinarith [sq_nonneg (z-(x:ℂ)).re] - -lemma cutoffFlow_eq_of_positive_trajectory - (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) - (T : ℝ) (hT : 0 ≤ T) (Y : ℝ → ℂ) - (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z) - (hheight : ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ ≤ (Y t).im) - (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y - ((a κ:ℂ)/(Y t-driverValue b t)) (Ici t) t) : - ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t=Y t := by - apply cutoffFlow_eq_of_trajectory κ b j hj z T hT Y hY hY0 - intro t ht - rw [cutoffField_eq_reciprocal κ b j t (Y t) - (cutoff_inactive_of_height j hj _ _ (hheight t (Ico_subset_Icc_self ht)))] - exact hYd t ht - -lemma surviving_cutoff_height (κ : ℝ) (b : Driver) (z : ℂ) - (T : ℝ) (j : ℕ) (hj : 1 ≤ j) - (hmin : (j:ℝ)⁻¹ < - sInf ((fun t : ℝ => (cutoffFlow κ b j z t).im) '' Icc 0 T)) : - ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im := by - have hc : IsCompact ((fun t : ℝ => (cutoffFlow κ b j z t).im) '' Icc 0 T) := - isCompact_Icc.image_of_continuousOn (Complex.continuous_im.comp_continuousOn - ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self)) - intro t ht - exact hmin.trans_le (csInf_le hc.bddBelow ⟨t,ht,rfl⟩) - -lemma surviving_of_positive_trajectory - (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (hT : 0 ≤ T) - (Y : ℝ → ℂ) (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z) - (m : ℝ) (hm : 0 < m) (hheight : ∀ t ∈ Icc 0 T, m ≤ (Y t).im) - (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y - ((a κ:ℂ)/(Y t-driverValue b t)) (Ici t) t) : Surviving κ b z T := by - obtain ⟨j,hj0,hjm⟩ := Real.exists_nat_pos_inv_lt hm - have hj : 1 ≤ j := hj0 - have he := cutoffFlow_eq_of_positive_trajectory κ b j hj z T hT Y hY hY0 - (fun t ht => hjm.le.trans (hheight t ht)) hYd - refine ⟨j,hj,lt_of_lt_of_le hjm ?_⟩ - apply le_csInf (Set.Nonempty.image _ (nonempty_Icc.mpr hT)) - rintro y ⟨t,ht,rfl⟩ - change m ≤ (cutoffFlow κ b j z t).im - rw [he t ht] - exact hheight t ht - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetEulerConvergence + +namespace OAI + +/-! The Euler-defined cutoff agrees with every classical trajectory on a +compact time interval. Positive height makes the cutoff inactive. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Target + +lemma cutoffFlow_eq_of_trajectory + (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) + (T : ℝ) (_hT : 0 ≤ T) (Y : ℝ → ℂ) + (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z) + (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y + (cutoffField κ b j t (Y t)) (Ici t) t) : + ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t=Y t := by + obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z + have he := dist_le_of_trajectories_ODE hK hX.continuous.continuousOn + (fun t ht => (hXd t ht.1).mono (Ici_subset_Ici.mpr ht.1)) hY hYd + (show dist (X 0) (Y 0) ≤ 0 by rw [hX0,hY0]; simp) + intro t ht + rw [cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd t ht.1] + exact dist_le_zero.mp (by simpa only [zero_mul] using he t ht) + +lemma cutoffFlow_continuousOn (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) : + ContinuousOn (cutoffFlow κ b j z) (Ici 0) := by + obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z + exact hX.continuous.continuousOn.congr (fun t ht => + cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd t ht) + +lemma cutoffFlow_hasDerivWithinAt (κ : ℝ) (b : Driver) (j : ℕ) + (hj : 1 ≤ j) (z : ℂ) (t : ℝ) (ht : 0 ≤ t) : + HasDerivWithinAt (cutoffFlow κ b j z) + (cutoffField κ b j t (cutoffFlow κ b j z t)) (Ici 0) t := by + obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z + have he (s : ℝ) (hs : 0 ≤ s) := + cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd s hs + rw [he t ht] + exact (hXd t ht).congr_of_eventuallyEq + (by filter_upwards [self_mem_nhdsWithin] with s hs; exact he s hs) (he t ht) + +lemma cutoffFlow_zero (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) : + cutoffFlow κ b j z 0=z := by + obtain ⟨L,K,X,hL,hK,hX,hX0,hXd⟩ := exists_cutoff_solution κ b j hj z + rw [cutoffFlow_eq_solution κ b j hj z L K hL hK X hX.continuous hX0 hXd 0 le_rfl,hX0] + +lemma cutoff_inactive_of_height (j : ℕ) (hj : 1 ≤ j) (x : ℝ) (z : ℂ) + (hz : (j:ℝ)⁻¹ ≤ z.im) : + (j:ℝ)^(-(4:ℤ)) ≤ ‖z-(x:ℂ)‖^2 := by + have hj1 : (1:ℝ) ≤ j := by exact_mod_cast hj + have hj0 : (0:ℝ) < j := zero_lt_one.trans_le hj1 + have hi0 : 0 < (j:ℝ)⁻¹ := inv_pos.mpr hj0 + have hi1 : (j:ℝ)⁻¹ ≤ 1 := (inv_le_one₀ hj0).2 hj1 + have hi2 : ((j:ℝ)⁻¹)^2 ≤ 1 := by nlinarith + have hi4 : ((j:ℝ)⁻¹)^4 ≤ ((j:ℝ)⁻¹)^2 := by nlinarith [sq_nonneg (((j:ℝ)⁻¹)^2)] + have hh : ((j:ℝ)⁻¹)^2 ≤ z.im^2 := by nlinarith + have hn := Complex.sq_norm_sub_sq_im (z-(x:ℂ)) + simp only [Complex.sub_im,Complex.ofReal_im,sub_zero] at hn + calc + (j:ℝ)^(-(4:ℤ)) = ((j:ℝ)⁻¹)^4 := by simp only [zpow_neg,zpow_ofNat,inv_pow] + _ ≤ z.im^2 := hi4.trans hh + _ ≤ ‖z-(x:ℂ)‖^2 := by nlinarith [sq_nonneg (z-(x:ℂ)).re] + +lemma cutoffFlow_eq_of_positive_trajectory + (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) + (T : ℝ) (hT : 0 ≤ T) (Y : ℝ → ℂ) + (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z) + (hheight : ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ ≤ (Y t).im) + (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y + ((a κ:ℂ)/(Y t-driverValue b t)) (Ici t) t) : + ∀ t ∈ Icc 0 T, cutoffFlow κ b j z t=Y t := by + apply cutoffFlow_eq_of_trajectory κ b j hj z T hT Y hY hY0 + intro t ht + rw [cutoffField_eq_reciprocal κ b j t (Y t) + (cutoff_inactive_of_height j hj _ _ (hheight t (Ico_subset_Icc_self ht)))] + exact hYd t ht + +lemma surviving_cutoff_height (κ : ℝ) (b : Driver) (z : ℂ) + (T : ℝ) (j : ℕ) (hj : 1 ≤ j) + (hmin : (j:ℝ)⁻¹ < + sInf ((fun t : ℝ => (cutoffFlow κ b j z t).im) '' Icc 0 T)) : + ∀ t ∈ Icc 0 T, (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im := by + have hc : IsCompact ((fun t : ℝ => (cutoffFlow κ b j z t).im) '' Icc 0 T) := + isCompact_Icc.image_of_continuousOn (Complex.continuous_im.comp_continuousOn + ((cutoffFlow_continuousOn κ b j hj z).mono Icc_subset_Ici_self)) + intro t ht + exact hmin.trans_le (csInf_le hc.bddBelow ⟨t,ht,rfl⟩) + +lemma surviving_of_positive_trajectory + (κ : ℝ) (b : Driver) (z : ℂ) (T : ℝ) (hT : 0 ≤ T) + (Y : ℝ → ℂ) (hY : ContinuousOn Y (Icc 0 T)) (hY0 : Y 0=z) + (m : ℝ) (hm : 0 < m) (hheight : ∀ t ∈ Icc 0 T, m ≤ (Y t).im) + (hYd : ∀ t ∈ Ico 0 T, HasDerivWithinAt Y + ((a κ:ℂ)/(Y t-driverValue b t)) (Ici t) t) : Surviving κ b z T := by + obtain ⟨j,hj0,hjm⟩ := Real.exists_nat_pos_inv_lt hm + have hj : 1 ≤ j := hj0 + have he := cutoffFlow_eq_of_positive_trajectory κ b j hj z T hT Y hY hY0 + (fun t ht => hjm.le.trans (hheight t ht)) hYd + refine ⟨j,hj,lt_of_lt_of_le hjm ?_⟩ + apply le_csInf (Set.Nonempty.image _ (nonempty_Icc.mpr hT)) + rintro y ⟨t,ht,rfl⟩ + change m ≤ (cutoffFlow κ b j z t).im + rw [he t ht] + exact hheight t ht + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetDensityBound.lean b/lean/OAI/Probability/SLE/Loewner/TargetDensityBound.lean index 4af2b68e7fbccd5fe8310f9f8b0a64bdef276989..37f7d578ee0f526316ffa5bee70309e1181f2a7f 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetDensityBound.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetDensityBound.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.Mass.DiskMassMomentBound -import OAI.Probability.SLE.Stochastic.WienerCoefficientLimit - -namespace OAI - -/-! The exact change-of-law density is supported on the physical center -hit and bounded by its compensating exponential factor. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal - -namespace SLEExactGauge.Target - -lemma levelDensity_le_centerHit - (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) - (b : Driver) (U : ℝ≥0 → ℝ) (hc : Continuous U) - (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) - (hw : IsCapacityTwoWitness U γ G F) - (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) : - levelDensity κ θ b ℓ ≤ - (Multipoint.diskCenterHit (fun _ : Unit => γ) (fun _ : Unit => G) θ - (Real.exp (-(4*ℓ/κ)))).indicator - (fun _ => Real.exp ((4*ℓ/κ)*codimension κ)/(Real.sin θ)^greenExponent κ) () := by - let z := diskMapInv θ 0 - let u := 4*ℓ/κ - have hz : z∈upperHalfPlane := diskMapInv_maps hθ (by simp) - have hR : 0<2*z.im := mul_pos (by norm_num) hz - have hs : 0 γ) (fun _ : Unit => G) θ (Real.exp (-u)) - · rw [indicator_of_mem hh] - have hb := Multipoint.stoppedGreen_level_bound hc hw hκ hκ8 z hz - ((2*z.im)*Real.exp (-u)) (mul_pos hR (Real.exp_pos _)) - rw [Multipoint.common_level_log_target z hz u] at hb - apply (div_le_div_of_nonneg_right hb (initialGreen_pos κ z hz).le).trans_eq - have hi : initialGreen κ z=(2*z.im)^(-codimension κ)*(Real.sin θ)^greenExponent κ := by - rw [initialGreen] - exact congrArg (fun s : ℝ => (2*z.im)^(-codimension κ)*s^greenExponent κ) - (disk_center_sin_arg θ) - rw [hi,Real.mul_rpow hR.le (Real.exp_pos _).le,← Real.exp_mul,neg_mul_neg] - field_simp [(Real.rpow_pos_of_pos hR (-codimension κ)).ne', - (Real.rpow_pos_of_pos hs (greenExponent κ)).ne'] - congr 1 - dsimp only [u] - ring - · rw [indicator_of_notMem hh] - have ht : ¬ radiusLevelTime γ G z u<∞ := by - intro ht - apply hh - apply (physical_radius_hit_iff_finite_level hc hw (diskMap θ) z 1 (Real.exp (-u)) - zero_lt_one (Real.exp_pos _) he1 (initialRadius_zero hθ).symm).mpr - rwa [hlog] - simp only [stoppedGreen,ite_eq_right ht,zero_div,le_refl] - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Mass.DiskMassMomentBound +import OAI.Probability.SLE.Stochastic.WienerCoefficientLimit + +namespace OAI + +/-! The exact change-of-law density is supported on the physical center +hit and bounded by its compensating exponential factor. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal + +namespace SLEExactGauge.Target + +lemma levelDensity_le_centerHit + (κ : ℝ) (hκ : 0<κ) (hκ8 : κ<8) + (b : Driver) (U : ℝ≥0 → ℝ) (hc : Continuous U) + (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) + (hw : IsCapacityTwoWitness U γ G F) + (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) (ℓ : ℝ) (hℓ : 0≤ℓ) : + levelDensity κ θ b ℓ ≤ + (Multipoint.diskCenterHit (fun _ : Unit => γ) (fun _ : Unit => G) θ + (Real.exp (-(4*ℓ/κ)))).indicator + (fun _ => Real.exp ((4*ℓ/κ)*codimension κ)/(Real.sin θ)^greenExponent κ) () := by + let z := diskMapInv θ 0 + let u := 4*ℓ/κ + have hz : z∈upperHalfPlane := diskMapInv_maps hθ (by simp) + have hR : 0<2*z.im := mul_pos (by norm_num) hz + have hs : 0 γ) (fun _ : Unit => G) θ (Real.exp (-u)) + · rw [indicator_of_mem hh] + have hb := Multipoint.stoppedGreen_level_bound hc hw hκ hκ8 z hz + ((2*z.im)*Real.exp (-u)) (mul_pos hR (Real.exp_pos _)) + rw [Multipoint.common_level_log_target z hz u] at hb + apply (div_le_div_of_nonneg_right hb (initialGreen_pos κ z hz).le).trans_eq + have hi : initialGreen κ z=(2*z.im)^(-codimension κ)*(Real.sin θ)^greenExponent κ := by + rw [initialGreen] + exact congrArg (fun s : ℝ => (2*z.im)^(-codimension κ)*s^greenExponent κ) + (disk_center_sin_arg θ) + rw [hi,Real.mul_rpow hR.le (Real.exp_pos _).le,← Real.exp_mul,neg_mul_neg] + field_simp [(Real.rpow_pos_of_pos hR (-codimension κ)).ne', + (Real.rpow_pos_of_pos hs (greenExponent κ)).ne'] + congr 1 + dsimp only [u] + ring + · rw [indicator_of_notMem hh] + have ht : ¬ radiusLevelTime γ G z u<∞ := by + intro ht + apply hh + apply (physical_radius_hit_iff_finite_level hc hw (diskMap θ) z 1 (Real.exp (-u)) + zero_lt_one (Real.exp_pos _) he1 (initialRadius_zero hθ).symm).mpr + rwa [hlog] + simp only [stoppedGreen,ite_eq_right ht,zero_div,le_refl] + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetDriverContinuation.lean b/lean/OAI/Probability/SLE/Loewner/TargetDriverContinuation.lean index 2bbd660500adfbffb7b72ea46f07233c50b77e6c..7c5a141008a9370a7426a64e20377bd2ce962b87 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetDriverContinuation.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetDriverContinuation.lean @@ -1,154 +1,154 @@ -import OAI.Probability.SLE.MomentGauge -import OAI.Probability.SLE.Geometry.RestartContinuationKernel - -namespace OAI - -/-! The prescribed based-driver concatenation is the ordinary Brownian -continuation kernel, including its finite real-valued conditional mean. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -local instance targetContinuationPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance targetContinuationPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ - -def centerDriver (p : C(ℝ≥0, ℝ)) : Driver where - toFun t := p t - p 0 - continuous_toFun := p.continuous.sub continuous_const - map_zero' := sub_self _ - -lemma measurable_centerDriver : Measurable centerDriver := by - apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp - apply source_path_measurable - intro t - exact (continuous_eval_const t).measurable.sub (continuous_eval_const 0).measurable - -lemma measurable_driverInclusion : - Measurable (fun b : Driver => b.toContinuousMap) := - ContinuousMapZero.isClosedEmbedding_toContinuousMap.continuous.measurable - -@[simp] lemma centerDriver_inclusion (b : Driver) : centerDriver b.toContinuousMap = b := by - ext t - change b t - b 0 = b t - simp - -lemma measurable_concat : - Measurable (fun x : ℝ × (Driver × Driver) => concat x.2.1 x.2.2 x.1) := by - apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp - apply source_path_measurable - intro u - have ht : Measurable (fun x : ℝ × (Driver × Driver) => x.1.toNNReal) := - continuous_real_toNNReal.measurable.comp measurable_fst - have hb : Measurable (fun x : ℝ × (Driver × Driver) => x.2.1 u) := - (continuous_eval_const u).measurable.comp (measurable_fst.comp measurable_snd) - have hbt : Measurable (fun x : ℝ × (Driver × Driver) => x.2.1 x.1.toNNReal) := - continuous_eval.measurable.comp ((measurable_fst.comp measurable_snd).prodMk ht) - have hc : Measurable (fun x : ℝ × (Driver × Driver) => x.2.2 (u-x.1.toNNReal)) := - continuous_eval.measurable.comp - ((measurable_snd.comp measurable_snd).prodMk (measurable_const.sub ht)) - exact hb.ite (measurableSet_le measurable_const ht) (hbt.add hc) - -lemma centerDriver_concatenate (b c : Driver) (t : ℝ≥0) : - centerDriver (concatenateDriverPath t (driverPathStopped t b.toContinuousMap) - c.toContinuousMap) = concat b c t := by - ext u - change (if u ≤ t then b (min u t) else b (min t t) + c (u-t) - c 0) - - (if 0 ≤ t then b (min 0 t) else b (min t t) + c (0-t) - c 0) = - (if u ≤ (t : ℝ).toNNReal then b u else b (t : ℝ).toNNReal + c (u-(t : ℝ).toNNReal)) - by_cases hu : u ≤ t - · simp [hu] - · simp [hu] - -def driverFiltration : Filtration ℝ≥0 (borel Driver) := - Filtration.natural (fun t (b : Driver) => b t) - (fun t => (continuous_eval_const t).measurable.stronglyMeasurable) - -def continuationMean (W : WienerLaw) (t : ℝ) (f : Driver → ℝ≥0∞) - (b : Driver) : ℝ≥0∞ := ∫⁻ c, f (concat b c t) ∂W - -lemma measurable_continuationMean (W : WienerLaw) [SFinite W] - (t : ℝ) (f : Driver → ℝ≥0∞) (hf : Measurable f) : - Measurable (continuationMean W t f) := by - exact (hf.comp (measurable_concat.comp - (measurable_const.prodMk (measurable_fst.prodMk measurable_snd)))).lintegral_prod_right' - -lemma continuationMean_eq_pathMean (W : WienerLaw) (t : ℝ≥0) - (f : Driver → ℝ≥0∞) (hf : Measurable f) (b : Driver) : - driverContinuationMean (W.map (fun c : Driver => c.toContinuousMap)) t - (fun p => f (centerDriver p)) b.toContinuousMap = continuationMean W t f b := by - have hm : Measurable (fun q : C(ℝ≥0, ℝ) => - f (centerDriver (concatenateDriverPath t (driverPathStopped t b.toContinuousMap) q))) := - (hf.comp measurable_centerDriver).comp (measurable_concatenateDriverPath.comp - (measurable_const.prodMk (measurable_const.prodMk measurable_id))) - unfold driverContinuationMean - rw [lintegral_map hm measurable_driverInclusion] - simp only [centerDriver_concatenate, continuationMean] - -lemma continuationMean_measurable_past (W : WienerLaw) [IsProbabilityMeasure W] - (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) : - Measurable[driverFiltration t] (continuationMean W t f) := by - let ν := W.map (fun c : Driver => c.toContinuousMap) - let : IsProbabilityMeasure ν := - (Measure.isProbabilityMeasure_map_iff measurable_driverInclusion.aemeasurable).mpr inferInstance - have hpath : Measurable[driverFiltration t, driverPathFiltration t] - (fun b : Driver => b.toContinuousMap) := - driverPath_measurable_past_of_adapted driverFiltration _ - (fun s => (Filtration.stronglyAdapted_natural - (fun u => (show Measurable (fun b : Driver => b u) from - (continuous_eval_const u).measurable).stronglyMeasurable) s).measurable) t - have hh := (driverContinuationMean_measurable ν t (fun p => f (centerDriver p)) - (hf.comp measurable_centerDriver)).comp hpath - have he : (fun b : Driver => driverContinuationMean ν t - (fun p => f (centerDriver p)) b.toContinuousMap) = continuationMean W t f := - funext (fun b => continuationMean_eq_pathMean W t f hf b) - change Measurable[driverFiltration t] (fun b : Driver => driverContinuationMean ν t - (fun p => f (centerDriver p)) b.toContinuousMap) at hh - rwa [he] at hh - -theorem continuationMean_condLExp (W : WienerLaw) (hW : IsWiener W) - (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) : - continuationMean W t f =ᵐ[W] W⁻[f | driverFiltration t] := by - let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure - have h := source_driver_continuation_condLExp hW - (fun s => (continuous_eval_const s).measurable) - (fun b => map_continuous b) t (fun p => f (centerDriver p)) - (hf.comp measurable_centerDriver) - change (fun b : Driver => driverContinuationMean - (W.map (fun c : Driver => c.toContinuousMap)) t - (fun p => f (centerDriver p)) b.toContinuousMap) =ᵐ[W] - W⁻[(fun b : Driver => f (centerDriver b.toContinuousMap)) | driverFiltration t] at h - have he : (fun b : Driver => driverContinuationMean - (W.map (fun c : Driver => c.toContinuousMap)) t - (fun p => f (centerDriver p)) b.toContinuousMap) = continuationMean W t f := - funext (fun b => continuationMean_eq_pathMean W t f hf b) - rw [he] at h - simpa only [centerDriver_inclusion] using h - -theorem continuationMean_setLIntegral (W : WienerLaw) (hW : IsWiener W) - (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) - (A : Set Driver) (hA : MeasurableSet[driverFiltration t] A) : - (∫⁻ b in A, continuationMean W t f b ∂W) = ∫⁻ b in A, f b ∂W := by - let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure - rw [lintegral_congr_ae (ae_restrict_of_ae (continuationMean_condLExp W hW t f hf))] - exact setLIntegral_condLExp (driverFiltration.le t) W f hA - -theorem continuationMean_finite_condExp (W : WienerLaw) (hW : IsWiener W) - (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) - (hfin : (∫⁻ b, f b ∂W) ≠ ∞) : - (∀ᵐ b ∂W, continuationMean W t f b < ∞) ∧ - (fun b => (continuationMean W t f b).toReal) =ᵐ[W] - W[fun b => (f b).toReal | driverFiltration t] := by - have he := continuationMean_condLExp W hW t f hf - constructor - · filter_upwards [he, condLExp_lt_top (mΩ := driverFiltration t) hfin] with b hb hbfin - exact hb.symm ▸ hbfin - · exact (he.fun_comp ENNReal.toReal).trans (toReal_condLExp _ hf.aemeasurable hfin) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.MomentGauge +import OAI.Probability.SLE.Geometry.RestartContinuationKernel + +namespace OAI + +/-! The prescribed based-driver concatenation is the ordinary Brownian +continuation kernel, including its finite real-valued conditional mean. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +local instance targetContinuationPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance targetContinuationPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ + +def centerDriver (p : C(ℝ≥0, ℝ)) : Driver where + toFun t := p t - p 0 + continuous_toFun := p.continuous.sub continuous_const + map_zero' := sub_self _ + +lemma measurable_centerDriver : Measurable centerDriver := by + apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp + apply source_path_measurable + intro t + exact (continuous_eval_const t).measurable.sub (continuous_eval_const 0).measurable + +lemma measurable_driverInclusion : + Measurable (fun b : Driver => b.toContinuousMap) := + ContinuousMapZero.isClosedEmbedding_toContinuousMap.continuous.measurable + +@[simp] lemma centerDriver_inclusion (b : Driver) : centerDriver b.toContinuousMap = b := by + ext t + change b t - b 0 = b t + simp + +lemma measurable_concat : + Measurable (fun x : ℝ × (Driver × Driver) => concat x.2.1 x.2.2 x.1) := by + apply ContinuousMapZero.isClosedEmbedding_toContinuousMap.measurableEmbedding.measurable_comp_iff.mp + apply source_path_measurable + intro u + have ht : Measurable (fun x : ℝ × (Driver × Driver) => x.1.toNNReal) := + continuous_real_toNNReal.measurable.comp measurable_fst + have hb : Measurable (fun x : ℝ × (Driver × Driver) => x.2.1 u) := + (continuous_eval_const u).measurable.comp (measurable_fst.comp measurable_snd) + have hbt : Measurable (fun x : ℝ × (Driver × Driver) => x.2.1 x.1.toNNReal) := + continuous_eval.measurable.comp ((measurable_fst.comp measurable_snd).prodMk ht) + have hc : Measurable (fun x : ℝ × (Driver × Driver) => x.2.2 (u-x.1.toNNReal)) := + continuous_eval.measurable.comp + ((measurable_snd.comp measurable_snd).prodMk (measurable_const.sub ht)) + exact hb.ite (measurableSet_le measurable_const ht) (hbt.add hc) + +lemma centerDriver_concatenate (b c : Driver) (t : ℝ≥0) : + centerDriver (concatenateDriverPath t (driverPathStopped t b.toContinuousMap) + c.toContinuousMap) = concat b c t := by + ext u + change (if u ≤ t then b (min u t) else b (min t t) + c (u-t) - c 0) - + (if 0 ≤ t then b (min 0 t) else b (min t t) + c (0-t) - c 0) = + (if u ≤ (t : ℝ).toNNReal then b u else b (t : ℝ).toNNReal + c (u-(t : ℝ).toNNReal)) + by_cases hu : u ≤ t + · simp [hu] + · simp [hu] + +def driverFiltration : Filtration ℝ≥0 (borel Driver) := + Filtration.natural (fun t (b : Driver) => b t) + (fun t => (continuous_eval_const t).measurable.stronglyMeasurable) + +def continuationMean (W : WienerLaw) (t : ℝ) (f : Driver → ℝ≥0∞) + (b : Driver) : ℝ≥0∞ := ∫⁻ c, f (concat b c t) ∂W + +lemma measurable_continuationMean (W : WienerLaw) [SFinite W] + (t : ℝ) (f : Driver → ℝ≥0∞) (hf : Measurable f) : + Measurable (continuationMean W t f) := by + exact (hf.comp (measurable_concat.comp + (measurable_const.prodMk (measurable_fst.prodMk measurable_snd)))).lintegral_prod_right' + +lemma continuationMean_eq_pathMean (W : WienerLaw) (t : ℝ≥0) + (f : Driver → ℝ≥0∞) (hf : Measurable f) (b : Driver) : + driverContinuationMean (W.map (fun c : Driver => c.toContinuousMap)) t + (fun p => f (centerDriver p)) b.toContinuousMap = continuationMean W t f b := by + have hm : Measurable (fun q : C(ℝ≥0, ℝ) => + f (centerDriver (concatenateDriverPath t (driverPathStopped t b.toContinuousMap) q))) := + (hf.comp measurable_centerDriver).comp (measurable_concatenateDriverPath.comp + (measurable_const.prodMk (measurable_const.prodMk measurable_id))) + unfold driverContinuationMean + rw [lintegral_map hm measurable_driverInclusion] + simp only [centerDriver_concatenate, continuationMean] + +lemma continuationMean_measurable_past (W : WienerLaw) [IsProbabilityMeasure W] + (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) : + Measurable[driverFiltration t] (continuationMean W t f) := by + let ν := W.map (fun c : Driver => c.toContinuousMap) + let : IsProbabilityMeasure ν := + (Measure.isProbabilityMeasure_map_iff measurable_driverInclusion.aemeasurable).mpr inferInstance + have hpath : Measurable[driverFiltration t, driverPathFiltration t] + (fun b : Driver => b.toContinuousMap) := + driverPath_measurable_past_of_adapted driverFiltration _ + (fun s => (Filtration.stronglyAdapted_natural + (fun u => (show Measurable (fun b : Driver => b u) from + (continuous_eval_const u).measurable).stronglyMeasurable) s).measurable) t + have hh := (driverContinuationMean_measurable ν t (fun p => f (centerDriver p)) + (hf.comp measurable_centerDriver)).comp hpath + have he : (fun b : Driver => driverContinuationMean ν t + (fun p => f (centerDriver p)) b.toContinuousMap) = continuationMean W t f := + funext (fun b => continuationMean_eq_pathMean W t f hf b) + change Measurable[driverFiltration t] (fun b : Driver => driverContinuationMean ν t + (fun p => f (centerDriver p)) b.toContinuousMap) at hh + rwa [he] at hh + +theorem continuationMean_condLExp (W : WienerLaw) (hW : IsWiener W) + (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) : + continuationMean W t f =ᵐ[W] W⁻[f | driverFiltration t] := by + let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure + have h := source_driver_continuation_condLExp hW + (fun s => (continuous_eval_const s).measurable) + (fun b => map_continuous b) t (fun p => f (centerDriver p)) + (hf.comp measurable_centerDriver) + change (fun b : Driver => driverContinuationMean + (W.map (fun c : Driver => c.toContinuousMap)) t + (fun p => f (centerDriver p)) b.toContinuousMap) =ᵐ[W] + W⁻[(fun b : Driver => f (centerDriver b.toContinuousMap)) | driverFiltration t] at h + have he : (fun b : Driver => driverContinuationMean + (W.map (fun c : Driver => c.toContinuousMap)) t + (fun p => f (centerDriver p)) b.toContinuousMap) = continuationMean W t f := + funext (fun b => continuationMean_eq_pathMean W t f hf b) + rw [he] at h + simpa only [centerDriver_inclusion] using h + +theorem continuationMean_setLIntegral (W : WienerLaw) (hW : IsWiener W) + (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) + (A : Set Driver) (hA : MeasurableSet[driverFiltration t] A) : + (∫⁻ b in A, continuationMean W t f b ∂W) = ∫⁻ b in A, f b ∂W := by + let : IsProbabilityMeasure W := hW.isGaussianProcess.isProbabilityMeasure + rw [lintegral_congr_ae (ae_restrict_of_ae (continuationMean_condLExp W hW t f hf))] + exact setLIntegral_condLExp (driverFiltration.le t) W f hA + +theorem continuationMean_finite_condExp (W : WienerLaw) (hW : IsWiener W) + (t : ℝ≥0) (f : Driver → ℝ≥0∞) (hf : Measurable f) + (hfin : (∫⁻ b, f b ∂W) ≠ ∞) : + (∀ᵐ b ∂W, continuationMean W t f b < ∞) ∧ + (fun b => (continuationMean W t f b).toReal) =ᵐ[W] + W[fun b => (f b).toReal | driverFiltration t] := by + have he := continuationMean_condLExp W hW t f hf + constructor + · filter_upwards [he, condLExp_lt_top (mΩ := driverFiltration t) hfin] with b hb hbfin + exact hb.symm ▸ hbfin + · exact (he.fun_comp ENNReal.toReal).trans (toReal_condLExp _ hf.aemeasurable hfin) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetEulerBridge.lean b/lean/OAI/Probability/SLE/Loewner/TargetEulerBridge.lean index ee6561bda84c9a20117f0030ab8d6a0428d3cfb3..dd30d1708a6f740a18a0aaaff8b6d16c0db19070 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetEulerBridge.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetEulerBridge.lean @@ -1,50 +1,50 @@ -import OAI.Probability.SLE.Loewner.TargetCutoffField -import Mathlib.Analysis.ODE.DiscreteGronwall - -namespace OAI - -/-! The bounded equation underlying the explicitly prescribed Euler flow. -The cutoff is inactive on any path whose height stays above its threshold. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Target - -lemma cutoffField_eq_reciprocal - (κ : ℝ) (b : Driver) (j : ℕ) (t : ℝ) (z : ℂ) - (hcut : (j:ℝ)^(-(4:ℤ)) ≤ ‖z-driverValue b t‖^2) : - cutoffField κ b j t z = (a κ:ℂ)/(z-driverValue b t) := by - rw [cutoffField,max_eq_left hcut,mul_div_assoc] - congr 1 - rw [Complex.inv_def,Complex.normSq_eq_norm_sq] - simp only [div_eq_mul_inv,starRingEnd_apply,Complex.ofReal_inv] - -lemma exists_cutoff_solution - (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) : - ∃ (L K : ℝ≥0) (X : ℝ → ℂ), - (∀ t w, ‖cutoffField κ b j t w‖ ≤ L) ∧ - (∀ t, LipschitzWith K (cutoffField κ b j t)) ∧ - LipschitzWith L X ∧ X 0=z ∧ - (∀ t, 0 ≤ t → HasDerivWithinAt X (cutoffField κ b j t (X t)) (Ici 0) t) := by - obtain ⟨L,K,hL,hK⟩ := cutoffField_bounds κ j hj - let V (t : ℝ) (_ : Unit) (w : ℂ) := cutoffField κ b j t w - have hF : Measurable (fun p : ℝ × ℂ => cutoffField κ b j p.1 p.2) := - (continuous_cutoffField κ b j hj).measurable - have hpair : Measurable (fun q : (ℝ × Unit) × ℂ => (q.1.1,q.2)) := - measurable_fst.fst.prodMk measurable_snd - have hm : Measurable (fun q : (ℝ × Unit) × ℂ => V q.1.1 q.1.2 q.2) := - @Measurable.comp ((ℝ × Unit) × ℂ) (ℝ × ℂ) ℂ inferInstance inferInstance inferInstance - (fun p => cutoffField κ b j p.1 p.2) (fun q => (q.1.1,q.2)) hF hpair - have hc (_ : Unit) (w : ℂ) : Continuous (fun t => V t () w) := - (continuous_cutoffField κ b j hj).comp (continuous_id.prodMk continuous_const) - obtain ⟨X,hXm,hXl,hX0,hXd,hXconv⟩ := exists_measurable_global_bounded_ode_picard - V z L K hm hc (fun _ t => hK b t) (fun _ t w => hL b t w) - exact ⟨L,K,(fun t => X t ()),hL b,hK b,hXl (),hX0 (),fun t ht => hXd () t ht⟩ - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetCutoffField +import Mathlib.Analysis.ODE.DiscreteGronwall + +namespace OAI + +/-! The bounded equation underlying the explicitly prescribed Euler flow. +The cutoff is inactive on any path whose height stays above its threshold. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Target + +lemma cutoffField_eq_reciprocal + (κ : ℝ) (b : Driver) (j : ℕ) (t : ℝ) (z : ℂ) + (hcut : (j:ℝ)^(-(4:ℤ)) ≤ ‖z-driverValue b t‖^2) : + cutoffField κ b j t z = (a κ:ℂ)/(z-driverValue b t) := by + rw [cutoffField,max_eq_left hcut,mul_div_assoc] + congr 1 + rw [Complex.inv_def,Complex.normSq_eq_norm_sq] + simp only [div_eq_mul_inv,starRingEnd_apply,Complex.ofReal_inv] + +lemma exists_cutoff_solution + (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) : + ∃ (L K : ℝ≥0) (X : ℝ → ℂ), + (∀ t w, ‖cutoffField κ b j t w‖ ≤ L) ∧ + (∀ t, LipschitzWith K (cutoffField κ b j t)) ∧ + LipschitzWith L X ∧ X 0=z ∧ + (∀ t, 0 ≤ t → HasDerivWithinAt X (cutoffField κ b j t (X t)) (Ici 0) t) := by + obtain ⟨L,K,hL,hK⟩ := cutoffField_bounds κ j hj + let V (t : ℝ) (_ : Unit) (w : ℂ) := cutoffField κ b j t w + have hF : Measurable (fun p : ℝ × ℂ => cutoffField κ b j p.1 p.2) := + (continuous_cutoffField κ b j hj).measurable + have hpair : Measurable (fun q : (ℝ × Unit) × ℂ => (q.1.1,q.2)) := + measurable_fst.fst.prodMk measurable_snd + have hm : Measurable (fun q : (ℝ × Unit) × ℂ => V q.1.1 q.1.2 q.2) := + @Measurable.comp ((ℝ × Unit) × ℂ) (ℝ × ℂ) ℂ inferInstance inferInstance inferInstance + (fun p => cutoffField κ b j p.1 p.2) (fun q => (q.1.1,q.2)) hF hpair + have hc (_ : Unit) (w : ℂ) : Continuous (fun t => V t () w) := + (continuous_cutoffField κ b j hj).comp (continuous_id.prodMk continuous_const) + obtain ⟨X,hXm,hXl,hX0,hXd,hXconv⟩ := exists_measurable_global_bounded_ode_picard + V z L K hm hc (fun _ t => hK b t) (fun _ t w => hL b t w) + exact ⟨L,K,(fun t => X t ()),hL b,hK b,hXl (),hX0 (),fun t ht => hXd () t ht⟩ + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetEulerConvergence.lean b/lean/OAI/Probability/SLE/Loewner/TargetEulerConvergence.lean index 071b71bae0b9db4fe0a5e30ac0a0b4ee52991d90..580bdc26a9ed9ff7071598eca63e6c600e9404fc 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetEulerConvergence.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetEulerConvergence.lean @@ -1,127 +1,127 @@ -import OAI.Probability.SLE.Loewner.TargetEulerInterpolation - -namespace OAI - -/-! The prescribed dyadic Euler definition converges to the bounded cutoff ODE. -Only continuity in time and a global Lipschitz bound in space are used. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Target - -private def eulerSlope (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (t : ℝ) : ℂ := - cutoffField κ b j (((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N) - (eulerSteps κ b j N z ⌊t/mesh N⌋₊) - -private lemma euler_pairs_mem (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) - (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) - (T t : ℝ) (ht : t ∈ Icc 0 T) : - (((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N, - eulerSteps κ b j N z ⌊t/mesh N⌋₊) ∈ Icc (0:ℝ) T ×ˢ closedBall z (L*T) ∧ - (t,eulerFlow κ b j N z t) ∈ Icc (0:ℝ) T ×ˢ closedBall z (L*T) := by - let k := ⌊t/mesh N⌋₊ - have hs0 : 0 ≤ (k:ℝ)*mesh N := mul_nonneg (Nat.cast_nonneg _) (mesh_pos N).le - have hst : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ (mesh_pos N)).1 - (Nat.floor_le (div_nonneg ht.1 (mesh_pos N).le)) - refine ⟨⟨⟨hs0,hst.trans ht.2⟩,?_⟩,⟨ht,?_⟩⟩ - · apply (eulerSteps_dist_le κ b j N z L hL k).trans - calc - (k:ℝ)*mesh N*L = L*((k:ℝ)*mesh N) := by ring - _ ≤ L*T := mul_le_mul_of_nonneg_left (hst.trans ht.2) L.coe_nonneg - · exact (eulerFlow_dist_initial_le κ b j N z L hL t ht.1).trans - (mul_le_mul_of_nonneg_left ht.2 L.coe_nonneg) - -private lemma euler_pairs_dist_lt (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) - (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) - (t δ : ℝ) (ht : 0 ≤ t) (hmesh : mesh N*(L+1)<δ) : - dist ((((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N), - eulerSteps κ b j N z ⌊t/mesh N⌋₊) (t,eulerFlow κ b j N z t) < δ := by - let k := ⌊t/mesh N⌋₊ - have hst : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ (mesh_pos N)).1 - (Nat.floor_le (div_nonneg ht (mesh_pos N).le)) - have hts : t-(k:ℝ)*mesh N < mesh N := by - have hh := (div_lt_iff₀ (mesh_pos N)).1 (Nat.lt_floor_add_one (t/mesh N)) - change t < ((k:ℝ)+1)*mesh N at hh - linarith - rw [Prod.dist_eq] - apply max_lt - · rw [Real.dist_eq,abs_sub_comm,abs_of_nonneg (sub_nonneg.mpr hst)] - have hh : mesh N ≤ mesh N*(L+1) := by nlinarith [mesh_pos N,L.coe_nonneg] - exact hts.trans_le hh |>.trans hmesh - · rw [dist_comm] - have hh : L*mesh N < δ := by nlinarith [mesh_pos N] - exact (eulerFlow_dist_grid_le κ b j N z L hL t ht).trans_lt hh - -lemma eulerSlope_error_tendsto - (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) - (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) - (T : ℝ) (_hT : 0 ≤ T) (ε : ℝ) (hε : 0 < ε) : - ∀ᶠ N : ℕ in atTop, ∀ t ∈ Icc 0 T, - dist (eulerSlope κ b j N z t) (cutoffField κ b j t (eulerFlow κ b j N z t)) < ε := by - let C := Icc (0:ℝ) T ×ˢ closedBall z (L*T) - have hC : IsCompact C := isCompact_Icc.prod (isCompact_closedBall z (L*T)) - obtain ⟨δ,hδ,huniform⟩ := Metric.uniformContinuousOn_iff.mp - (hC.uniformContinuousOn_of_continuous (continuous_cutoffField κ b j hj).continuousOn) ε hε - have hd : 0 < δ/(L+1) := div_pos hδ (by positivity) - have hmesh : ∀ᶠ N : ℕ in atTop, mesh N < δ/(L+1) := - (tendsto_order.1 mesh_tendsto_zero).2 _ hd - filter_upwards [hmesh] with N hN - intro t ht - obtain ⟨hgrid,hflow⟩ := euler_pairs_mem κ b j N z L hL T t ht - have hhδ : mesh N*(L+1) < δ := (lt_div_iff₀ (by positivity : (0:ℝ) eulerFlow κ b j N z T) atTop (𝓝 (X T)) := by - apply Metric.tendsto_nhds.mpr - intro ε hε - have hcont : ContinuousAt (fun ε => gronwallBound 0 K ε T) 0 := - (gronwallBound_continuous_ε 0 K T).continuousAt - obtain ⟨δ,hδ,hsmall⟩ := Metric.continuousAt_iff.mp hcont ε hε - let η := δ/2 - have hη : 0 < η := half_pos hδ - have hbound : gronwallBound 0 K η T < ε := by - have h := hsmall (show dist η 0 < δ by - rw [Real.dist_eq,sub_zero,abs_of_pos hη] - exact half_lt_self hδ) - rw [gronwallBound_ε0_δ0,Real.dist_eq,sub_zero] at h - exact (le_abs_self _).trans_lt h - filter_upwards [eulerSlope_error_tendsto κ b j hj z L hL T hT η hη] with N hN - have hestimate := dist_le_of_approx_trajectories_ODE (v := cutoffField κ b j) - (f := eulerFlow κ b j N z) (g := X) - (f' := eulerSlope κ b j N z) (g' := fun t => cutoffField κ b j t (X t)) - (a := 0) (b := T) (δ := 0) (εf := η) (εg := 0) hK - (eulerFlow_continuousOn κ b j N z T) - (fun t ht => eulerFlow_right_deriv κ b j N z t ht.1) - (fun t ht => (hN t (Ico_subset_Icc_self ht)).le) - hX.continuousOn - (fun t ht => (hXd t ht.1).mono (Ici_subset_Ici.mpr ht.1)) - (fun _ _ => by simp) - (by simp only [eulerFlow_zero,hX0,dist_self]; exact le_rfl) T ⟨hT,le_rfl⟩ - rw [add_zero,sub_zero] at hestimate - exact hestimate.trans_lt hbound - -lemma cutoffFlow_eq_solution - (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) - (L K : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) - (hK : ∀ t, LipschitzWith K (cutoffField κ b j t)) - (X : ℝ → ℂ) (hX : Continuous X) (hX0 : X 0=z) - (hXd : ∀ t, 0 ≤ t → HasDerivWithinAt X (cutoffField κ b j t (X t)) (Ici 0) t) - (T : ℝ) (hT : 0 ≤ T) : cutoffFlow κ b j z T = X T := by - exact (eulerFlow_tendsto_solution κ b j hj z L K hL hK X hX hX0 hXd T hT).limUnder_eq - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetEulerInterpolation + +namespace OAI + +/-! The prescribed dyadic Euler definition converges to the bounded cutoff ODE. +Only continuity in time and a global Lipschitz bound in space are used. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Target + +private def eulerSlope (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (t : ℝ) : ℂ := + cutoffField κ b j (((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N) + (eulerSteps κ b j N z ⌊t/mesh N⌋₊) + +private lemma euler_pairs_mem (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) + (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) + (T t : ℝ) (ht : t ∈ Icc 0 T) : + (((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N, + eulerSteps κ b j N z ⌊t/mesh N⌋₊) ∈ Icc (0:ℝ) T ×ˢ closedBall z (L*T) ∧ + (t,eulerFlow κ b j N z t) ∈ Icc (0:ℝ) T ×ˢ closedBall z (L*T) := by + let k := ⌊t/mesh N⌋₊ + have hs0 : 0 ≤ (k:ℝ)*mesh N := mul_nonneg (Nat.cast_nonneg _) (mesh_pos N).le + have hst : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ (mesh_pos N)).1 + (Nat.floor_le (div_nonneg ht.1 (mesh_pos N).le)) + refine ⟨⟨⟨hs0,hst.trans ht.2⟩,?_⟩,⟨ht,?_⟩⟩ + · apply (eulerSteps_dist_le κ b j N z L hL k).trans + calc + (k:ℝ)*mesh N*L = L*((k:ℝ)*mesh N) := by ring + _ ≤ L*T := mul_le_mul_of_nonneg_left (hst.trans ht.2) L.coe_nonneg + · exact (eulerFlow_dist_initial_le κ b j N z L hL t ht.1).trans + (mul_le_mul_of_nonneg_left ht.2 L.coe_nonneg) + +private lemma euler_pairs_dist_lt (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) + (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) + (t δ : ℝ) (ht : 0 ≤ t) (hmesh : mesh N*(L+1)<δ) : + dist ((((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N), + eulerSteps κ b j N z ⌊t/mesh N⌋₊) (t,eulerFlow κ b j N z t) < δ := by + let k := ⌊t/mesh N⌋₊ + have hst : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ (mesh_pos N)).1 + (Nat.floor_le (div_nonneg ht (mesh_pos N).le)) + have hts : t-(k:ℝ)*mesh N < mesh N := by + have hh := (div_lt_iff₀ (mesh_pos N)).1 (Nat.lt_floor_add_one (t/mesh N)) + change t < ((k:ℝ)+1)*mesh N at hh + linarith + rw [Prod.dist_eq] + apply max_lt + · rw [Real.dist_eq,abs_sub_comm,abs_of_nonneg (sub_nonneg.mpr hst)] + have hh : mesh N ≤ mesh N*(L+1) := by nlinarith [mesh_pos N,L.coe_nonneg] + exact hts.trans_le hh |>.trans hmesh + · rw [dist_comm] + have hh : L*mesh N < δ := by nlinarith [mesh_pos N] + exact (eulerFlow_dist_grid_le κ b j N z L hL t ht).trans_lt hh + +lemma eulerSlope_error_tendsto + (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) + (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) + (T : ℝ) (_hT : 0 ≤ T) (ε : ℝ) (hε : 0 < ε) : + ∀ᶠ N : ℕ in atTop, ∀ t ∈ Icc 0 T, + dist (eulerSlope κ b j N z t) (cutoffField κ b j t (eulerFlow κ b j N z t)) < ε := by + let C := Icc (0:ℝ) T ×ˢ closedBall z (L*T) + have hC : IsCompact C := isCompact_Icc.prod (isCompact_closedBall z (L*T)) + obtain ⟨δ,hδ,huniform⟩ := Metric.uniformContinuousOn_iff.mp + (hC.uniformContinuousOn_of_continuous (continuous_cutoffField κ b j hj).continuousOn) ε hε + have hd : 0 < δ/(L+1) := div_pos hδ (by positivity) + have hmesh : ∀ᶠ N : ℕ in atTop, mesh N < δ/(L+1) := + (tendsto_order.1 mesh_tendsto_zero).2 _ hd + filter_upwards [hmesh] with N hN + intro t ht + obtain ⟨hgrid,hflow⟩ := euler_pairs_mem κ b j N z L hL T t ht + have hhδ : mesh N*(L+1) < δ := (lt_div_iff₀ (by positivity : (0:ℝ) eulerFlow κ b j N z T) atTop (𝓝 (X T)) := by + apply Metric.tendsto_nhds.mpr + intro ε hε + have hcont : ContinuousAt (fun ε => gronwallBound 0 K ε T) 0 := + (gronwallBound_continuous_ε 0 K T).continuousAt + obtain ⟨δ,hδ,hsmall⟩ := Metric.continuousAt_iff.mp hcont ε hε + let η := δ/2 + have hη : 0 < η := half_pos hδ + have hbound : gronwallBound 0 K η T < ε := by + have h := hsmall (show dist η 0 < δ by + rw [Real.dist_eq,sub_zero,abs_of_pos hη] + exact half_lt_self hδ) + rw [gronwallBound_ε0_δ0,Real.dist_eq,sub_zero] at h + exact (le_abs_self _).trans_lt h + filter_upwards [eulerSlope_error_tendsto κ b j hj z L hL T hT η hη] with N hN + have hestimate := dist_le_of_approx_trajectories_ODE (v := cutoffField κ b j) + (f := eulerFlow κ b j N z) (g := X) + (f' := eulerSlope κ b j N z) (g' := fun t => cutoffField κ b j t (X t)) + (a := 0) (b := T) (δ := 0) (εf := η) (εg := 0) hK + (eulerFlow_continuousOn κ b j N z T) + (fun t ht => eulerFlow_right_deriv κ b j N z t ht.1) + (fun t ht => (hN t (Ico_subset_Icc_self ht)).le) + hX.continuousOn + (fun t ht => (hXd t ht.1).mono (Ici_subset_Ici.mpr ht.1)) + (fun _ _ => by simp) + (by simp only [eulerFlow_zero,hX0,dist_self]; exact le_rfl) T ⟨hT,le_rfl⟩ + rw [add_zero,sub_zero] at hestimate + exact hestimate.trans_lt hbound + +lemma cutoffFlow_eq_solution + (κ : ℝ) (b : Driver) (j : ℕ) (hj : 1 ≤ j) (z : ℂ) + (L K : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) + (hK : ∀ t, LipschitzWith K (cutoffField κ b j t)) + (X : ℝ → ℂ) (hX : Continuous X) (hX0 : X 0=z) + (hXd : ∀ t, 0 ≤ t → HasDerivWithinAt X (cutoffField κ b j t (X t)) (Ici 0) t) + (T : ℝ) (hT : 0 ≤ T) : cutoffFlow κ b j z T = X T := by + exact (eulerFlow_tendsto_solution κ b j hj z L K hL hK X hX hX0 hXd T hT).limUnder_eq + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetEulerInterpolation.lean b/lean/OAI/Probability/SLE/Loewner/TargetEulerInterpolation.lean index d437fa56f18dbe2a14d0a6cc36a62a50bb5e86ce..d45f8b9d4dfc7484f02522e5937a75c4f480fcf7 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetEulerInterpolation.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetEulerInterpolation.lean @@ -1,165 +1,165 @@ -import OAI.Probability.SLE.Loewner.TargetEulerBridge - -namespace OAI - -/-! The prescribed Euler interpolants are continuous piecewise-affine -paths with the exact right derivative at every nonnegative time. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Target - -lemma mesh_pos (N : ℕ) : 0 < mesh N := zpow_pos (by norm_num) _ - -lemma mesh_tendsto_zero : Tendsto mesh atTop (𝓝 0) := by - have he (N : ℕ) : mesh N = ((2:ℝ)⁻¹)^N := by - simp only [mesh,zpow_neg,zpow_natCast,inv_pow] - have hm : mesh = fun N : ℕ => ((2:ℝ)⁻¹)^N := funext he - rw [hm] - exact tendsto_pow_atTop_nhds_zero_of_lt_one - (by norm_num : (0:ℝ) ≤ (2:ℝ)⁻¹) (by norm_num : (2:ℝ)⁻¹ < 1) - -lemma eulerFlow_on_cell (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) - (k : ℕ) (t : ℝ) - (ht : t ∈ Icc ((k:ℝ)*mesh N) (((k:ℝ)+1)*mesh N)) : - eulerFlow κ b j N z t = eulerSteps κ b j N z k + - ((t-(k:ℝ)*mesh N : ℝ):ℂ) * cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k) := by - have hh := mesh_pos N - have ht0 : 0 ≤ t := (mul_nonneg (Nat.cast_nonneg k) hh.le).trans ht.1 - by_cases hu : t=((k:ℝ)+1)*mesh N - · have hd : t/mesh N=((k+1:ℕ):ℝ) := by rw [hu,mul_div_cancel_right₀ _ hh.ne']; norm_num - simp only [eulerFlow,hd,Nat.floor_natCast,sub_self,Complex.ofReal_zero,zero_mul, - sub_zero,add_zero] - rw [eulerSteps] - have he : t-(k:ℝ)*mesh N=mesh N := by rw [hu]; ring - rw [he] - simp only [Complex.ofReal_one,one_mul] - · have hlt : t<((k:ℝ)+1)*mesh N := lt_of_le_of_ne ht.2 hu - have hk : ⌊t/mesh N⌋₊=k := (Nat.floor_eq_iff (div_nonneg ht0 hh.le)).2 - ⟨(le_div_iff₀ hh).2 ht.1,(div_lt_iff₀ hh).2 hlt⟩ - simp only [eulerFlow,hk,eulerSteps] - push_cast - field_simp [Complex.ofReal_ne_zero.mpr hh.ne'] - ring - -lemma eulerFlow_continuousOn_cell (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (k : ℕ) : - ContinuousOn (eulerFlow κ b j N z) (Icc ((k:ℝ)*mesh N) (((k:ℝ)+1)*mesh N)) := by - have hc : Continuous (fun t : ℝ => eulerSteps κ b j N z k + - ((t-(k:ℝ)*mesh N : ℝ):ℂ) * cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) := by - fun_prop - exact hc.continuousOn.congr (fun t ht => eulerFlow_on_cell κ b j N z k t ht) - -lemma eulerFlow_continuousOn_grid (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (k : ℕ) : - ContinuousOn (eulerFlow κ b j N z) (Icc 0 ((k:ℝ)*mesh N)) := by - induction k with - | zero => simpa only [Nat.cast_zero,zero_mul,Icc_self] using - (continuousOn_singleton (eulerFlow κ b j N z) 0) - | succ k ih => - have hnonneg : 0 ≤ (k:ℝ)*mesh N := mul_nonneg (Nat.cast_nonneg _) (mesh_pos N).le - have hle : (k:ℝ)*mesh N ≤ ((k:ℝ)+1)*mesh N := by nlinarith [mesh_pos N] - have hu : Icc (0:ℝ) (((k:ℝ)+1)*mesh N) = - Icc 0 ((k:ℝ)*mesh N) ∪ Icc ((k:ℝ)*mesh N) (((k:ℝ)+1)*mesh N) := by - ext t - simp only [mem_Icc,mem_union] - constructor - · rintro ⟨ht0,ht1⟩ - rcases le_total t ((k:ℝ)*mesh N) with h|h - · exact Or.inl ⟨ht0,h⟩ - · exact Or.inr ⟨h,ht1⟩ - · rintro (⟨ht0,ht1⟩|⟨ht0,ht1⟩) - · exact ⟨ht0,ht1.trans hle⟩ - · exact ⟨hnonneg.trans ht0,ht1⟩ - simp only [Nat.cast_add,Nat.cast_one] - rw [hu] - exact ih.union_of_isClosed (eulerFlow_continuousOn_cell κ b j N z k) isClosed_Icc isClosed_Icc - -lemma eulerFlow_continuousOn (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (T : ℝ) : - ContinuousOn (eulerFlow κ b j N z) (Icc 0 T) := by - obtain ⟨k,hk⟩ := exists_nat_ge (T/mesh N) - have hT : T ≤ (k:ℝ)*mesh N := (div_le_iff₀ (mesh_pos N)).1 hk - exact (eulerFlow_continuousOn_grid κ b j N z k).mono (Icc_subset_Icc le_rfl hT) - -lemma eulerFlow_right_deriv (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) - (t : ℝ) (ht : 0 ≤ t) : - HasDerivWithinAt (eulerFlow κ b j N z) - (cutoffField κ b j (((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N) - (eulerSteps κ b j N z ⌊t/mesh N⌋₊)) (Ici t) t := by - let k := ⌊t/mesh N⌋₊ - let v := cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k) - have hh := mesh_pos N - have hk : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ hh).1 (Nat.floor_le (div_nonneg ht hh.le)) - have hnext : t<((k:ℝ)+1)*mesh N := (div_lt_iff₀ hh).1 (Nat.lt_floor_add_one (t/mesh N)) - have he : eulerFlow κ b j N z =ᶠ[𝓝[Ici t] t] - (fun s => eulerSteps κ b j N z k + ((s-(k:ℝ)*mesh N : ℝ):ℂ)*v) := by - filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds hnext)] with s hs hu - exact eulerFlow_on_cell κ b j N z k s ⟨hk.trans hs,hu.le⟩ - have hd : HasDerivAt (fun s : ℝ => eulerSteps κ b j N z k + ((s-(k:ℝ)*mesh N : ℝ):ℂ)*v) v t := by - convert ((((hasDerivAt_id t).sub_const ((k:ℝ)*mesh N)).ofReal_comp).mul_const v).const_add - (eulerSteps κ b j N z k) using 1 <;> simp - exact hd.hasDerivWithinAt.congr_of_eventuallyEq he - (eulerFlow_on_cell κ b j N z k t ⟨hk,hnext.le⟩) - -lemma eulerFlow_zero (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) : - eulerFlow κ b j N z 0=z := by - simp [eulerFlow,eulerSteps] - -lemma eulerSteps_dist_le (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) - (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) (k : ℕ) : - dist (eulerSteps κ b j N z k) z ≤ (k:ℝ)*mesh N*L := by - induction k with - | zero => simp [eulerSteps] - | succ k ih => - rw [eulerSteps] - have hs : dist (eulerSteps κ b j N z k + mesh N * - cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) - (eulerSteps κ b j N z k) ≤ mesh N*L := by - rw [dist_eq_norm,add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (mesh_pos N)] - exact mul_le_mul_of_nonneg_left (hL _ _) (mesh_pos N).le - have ht := (dist_triangle (eulerSteps κ b j N z k + mesh N * - cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) - (eulerSteps κ b j N z k) z).trans (add_le_add hs ih) - exact ht.trans_eq (by push_cast; ring) - -lemma eulerFlow_dist_grid_le (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) - (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) - (t : ℝ) (ht : 0 ≤ t) : - dist (eulerFlow κ b j N z t) (eulerSteps κ b j N z ⌊t/mesh N⌋₊) ≤ L*mesh N := by - let k := ⌊t/mesh N⌋₊ - have hh := mesh_pos N - have hk : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ hh).1 (Nat.floor_le (div_nonneg ht hh.le)) - have hnext : t<((k:ℝ)+1)*mesh N := (div_lt_iff₀ hh).1 (Nat.lt_floor_add_one (t/mesh N)) - change dist (eulerFlow κ b j N z t) (eulerSteps κ b j N z k) ≤ L*mesh N - rw [eulerFlow_on_cell κ b j N z k t ⟨hk,hnext.le⟩,dist_eq_norm,add_sub_cancel_left,norm_mul, - Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hk)] - have hlag : t-(k:ℝ)*mesh N ≤ mesh N := by nlinarith - exact (mul_le_mul_of_nonneg_left (hL _ _) (sub_nonneg.mpr hk)).trans - (by nlinarith [L.coe_nonneg]) - -lemma eulerFlow_dist_initial_le (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) - (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) - (t : ℝ) (ht : 0 ≤ t) : - dist (eulerFlow κ b j N z t) z ≤ L*t := by - let k := ⌊t/mesh N⌋₊ - have hh := mesh_pos N - have hk : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ hh).1 (Nat.floor_le (div_nonneg ht hh.le)) - have hnext : t<((k:ℝ)+1)*mesh N := (div_lt_iff₀ hh).1 (Nat.lt_floor_add_one (t/mesh N)) - have he := eulerFlow_on_cell κ b j N z k t ⟨hk,hnext.le⟩ - rw [he] - have hb : dist (eulerSteps κ b j N z k+((t-(k:ℝ)*mesh N : ℝ):ℂ)* - cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) - (eulerSteps κ b j N z k) ≤ (t-(k:ℝ)*mesh N)*L := by - rw [dist_eq_norm,add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hk)] - exact mul_le_mul_of_nonneg_left (hL _ _) (sub_nonneg.mpr hk) - have htriangle := (dist_triangle (eulerSteps κ b j N z k+((t-(k:ℝ)*mesh N : ℝ):ℂ)* - cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) - (eulerSteps κ b j N z k) z).trans - (add_le_add hb (eulerSteps_dist_le κ b j N z L hL k)) - exact htriangle.trans_eq (by ring) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetEulerBridge + +namespace OAI + +/-! The prescribed Euler interpolants are continuous piecewise-affine +paths with the exact right derivative at every nonnegative time. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Target + +lemma mesh_pos (N : ℕ) : 0 < mesh N := zpow_pos (by norm_num) _ + +lemma mesh_tendsto_zero : Tendsto mesh atTop (𝓝 0) := by + have he (N : ℕ) : mesh N = ((2:ℝ)⁻¹)^N := by + simp only [mesh,zpow_neg,zpow_natCast,inv_pow] + have hm : mesh = fun N : ℕ => ((2:ℝ)⁻¹)^N := funext he + rw [hm] + exact tendsto_pow_atTop_nhds_zero_of_lt_one + (by norm_num : (0:ℝ) ≤ (2:ℝ)⁻¹) (by norm_num : (2:ℝ)⁻¹ < 1) + +lemma eulerFlow_on_cell (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) + (k : ℕ) (t : ℝ) + (ht : t ∈ Icc ((k:ℝ)*mesh N) (((k:ℝ)+1)*mesh N)) : + eulerFlow κ b j N z t = eulerSteps κ b j N z k + + ((t-(k:ℝ)*mesh N : ℝ):ℂ) * cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k) := by + have hh := mesh_pos N + have ht0 : 0 ≤ t := (mul_nonneg (Nat.cast_nonneg k) hh.le).trans ht.1 + by_cases hu : t=((k:ℝ)+1)*mesh N + · have hd : t/mesh N=((k+1:ℕ):ℝ) := by rw [hu,mul_div_cancel_right₀ _ hh.ne']; norm_num + simp only [eulerFlow,hd,Nat.floor_natCast,sub_self,Complex.ofReal_zero,zero_mul, + sub_zero,add_zero] + rw [eulerSteps] + have he : t-(k:ℝ)*mesh N=mesh N := by rw [hu]; ring + rw [he] + simp only [Complex.ofReal_one,one_mul] + · have hlt : t<((k:ℝ)+1)*mesh N := lt_of_le_of_ne ht.2 hu + have hk : ⌊t/mesh N⌋₊=k := (Nat.floor_eq_iff (div_nonneg ht0 hh.le)).2 + ⟨(le_div_iff₀ hh).2 ht.1,(div_lt_iff₀ hh).2 hlt⟩ + simp only [eulerFlow,hk,eulerSteps] + push_cast + field_simp [Complex.ofReal_ne_zero.mpr hh.ne'] + ring + +lemma eulerFlow_continuousOn_cell (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (k : ℕ) : + ContinuousOn (eulerFlow κ b j N z) (Icc ((k:ℝ)*mesh N) (((k:ℝ)+1)*mesh N)) := by + have hc : Continuous (fun t : ℝ => eulerSteps κ b j N z k + + ((t-(k:ℝ)*mesh N : ℝ):ℂ) * cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) := by + fun_prop + exact hc.continuousOn.congr (fun t ht => eulerFlow_on_cell κ b j N z k t ht) + +lemma eulerFlow_continuousOn_grid (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (k : ℕ) : + ContinuousOn (eulerFlow κ b j N z) (Icc 0 ((k:ℝ)*mesh N)) := by + induction k with + | zero => simpa only [Nat.cast_zero,zero_mul,Icc_self] using + (continuousOn_singleton (eulerFlow κ b j N z) 0) + | succ k ih => + have hnonneg : 0 ≤ (k:ℝ)*mesh N := mul_nonneg (Nat.cast_nonneg _) (mesh_pos N).le + have hle : (k:ℝ)*mesh N ≤ ((k:ℝ)+1)*mesh N := by nlinarith [mesh_pos N] + have hu : Icc (0:ℝ) (((k:ℝ)+1)*mesh N) = + Icc 0 ((k:ℝ)*mesh N) ∪ Icc ((k:ℝ)*mesh N) (((k:ℝ)+1)*mesh N) := by + ext t + simp only [mem_Icc,mem_union] + constructor + · rintro ⟨ht0,ht1⟩ + rcases le_total t ((k:ℝ)*mesh N) with h|h + · exact Or.inl ⟨ht0,h⟩ + · exact Or.inr ⟨h,ht1⟩ + · rintro (⟨ht0,ht1⟩|⟨ht0,ht1⟩) + · exact ⟨ht0,ht1.trans hle⟩ + · exact ⟨hnonneg.trans ht0,ht1⟩ + simp only [Nat.cast_add,Nat.cast_one] + rw [hu] + exact ih.union_of_isClosed (eulerFlow_continuousOn_cell κ b j N z k) isClosed_Icc isClosed_Icc + +lemma eulerFlow_continuousOn (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) (T : ℝ) : + ContinuousOn (eulerFlow κ b j N z) (Icc 0 T) := by + obtain ⟨k,hk⟩ := exists_nat_ge (T/mesh N) + have hT : T ≤ (k:ℝ)*mesh N := (div_le_iff₀ (mesh_pos N)).1 hk + exact (eulerFlow_continuousOn_grid κ b j N z k).mono (Icc_subset_Icc le_rfl hT) + +lemma eulerFlow_right_deriv (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) + (t : ℝ) (ht : 0 ≤ t) : + HasDerivWithinAt (eulerFlow κ b j N z) + (cutoffField κ b j (((⌊t/mesh N⌋₊:ℕ):ℝ)*mesh N) + (eulerSteps κ b j N z ⌊t/mesh N⌋₊)) (Ici t) t := by + let k := ⌊t/mesh N⌋₊ + let v := cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k) + have hh := mesh_pos N + have hk : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ hh).1 (Nat.floor_le (div_nonneg ht hh.le)) + have hnext : t<((k:ℝ)+1)*mesh N := (div_lt_iff₀ hh).1 (Nat.lt_floor_add_one (t/mesh N)) + have he : eulerFlow κ b j N z =ᶠ[𝓝[Ici t] t] + (fun s => eulerSteps κ b j N z k + ((s-(k:ℝ)*mesh N : ℝ):ℂ)*v) := by + filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Iio_mem_nhds hnext)] with s hs hu + exact eulerFlow_on_cell κ b j N z k s ⟨hk.trans hs,hu.le⟩ + have hd : HasDerivAt (fun s : ℝ => eulerSteps κ b j N z k + ((s-(k:ℝ)*mesh N : ℝ):ℂ)*v) v t := by + convert ((((hasDerivAt_id t).sub_const ((k:ℝ)*mesh N)).ofReal_comp).mul_const v).const_add + (eulerSteps κ b j N z k) using 1 <;> simp + exact hd.hasDerivWithinAt.congr_of_eventuallyEq he + (eulerFlow_on_cell κ b j N z k t ⟨hk,hnext.le⟩) + +lemma eulerFlow_zero (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) : + eulerFlow κ b j N z 0=z := by + simp [eulerFlow,eulerSteps] + +lemma eulerSteps_dist_le (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) + (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) (k : ℕ) : + dist (eulerSteps κ b j N z k) z ≤ (k:ℝ)*mesh N*L := by + induction k with + | zero => simp [eulerSteps] + | succ k ih => + rw [eulerSteps] + have hs : dist (eulerSteps κ b j N z k + mesh N * + cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) + (eulerSteps κ b j N z k) ≤ mesh N*L := by + rw [dist_eq_norm,add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (mesh_pos N)] + exact mul_le_mul_of_nonneg_left (hL _ _) (mesh_pos N).le + have ht := (dist_triangle (eulerSteps κ b j N z k + mesh N * + cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) + (eulerSteps κ b j N z k) z).trans (add_le_add hs ih) + exact ht.trans_eq (by push_cast; ring) + +lemma eulerFlow_dist_grid_le (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) + (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) + (t : ℝ) (ht : 0 ≤ t) : + dist (eulerFlow κ b j N z t) (eulerSteps κ b j N z ⌊t/mesh N⌋₊) ≤ L*mesh N := by + let k := ⌊t/mesh N⌋₊ + have hh := mesh_pos N + have hk : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ hh).1 (Nat.floor_le (div_nonneg ht hh.le)) + have hnext : t<((k:ℝ)+1)*mesh N := (div_lt_iff₀ hh).1 (Nat.lt_floor_add_one (t/mesh N)) + change dist (eulerFlow κ b j N z t) (eulerSteps κ b j N z k) ≤ L*mesh N + rw [eulerFlow_on_cell κ b j N z k t ⟨hk,hnext.le⟩,dist_eq_norm,add_sub_cancel_left,norm_mul, + Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hk)] + have hlag : t-(k:ℝ)*mesh N ≤ mesh N := by nlinarith + exact (mul_le_mul_of_nonneg_left (hL _ _) (sub_nonneg.mpr hk)).trans + (by nlinarith [L.coe_nonneg]) + +lemma eulerFlow_dist_initial_le (κ : ℝ) (b : Driver) (j N : ℕ) (z : ℂ) + (L : ℝ≥0) (hL : ∀ t w, ‖cutoffField κ b j t w‖ ≤ L) + (t : ℝ) (ht : 0 ≤ t) : + dist (eulerFlow κ b j N z t) z ≤ L*t := by + let k := ⌊t/mesh N⌋₊ + have hh := mesh_pos N + have hk : (k:ℝ)*mesh N ≤ t := (le_div_iff₀ hh).1 (Nat.floor_le (div_nonneg ht hh.le)) + have hnext : t<((k:ℝ)+1)*mesh N := (div_lt_iff₀ hh).1 (Nat.lt_floor_add_one (t/mesh N)) + have he := eulerFlow_on_cell κ b j N z k t ⟨hk,hnext.le⟩ + rw [he] + have hb : dist (eulerSteps κ b j N z k+((t-(k:ℝ)*mesh N : ℝ):ℂ)* + cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) + (eulerSteps κ b j N z k) ≤ (t-(k:ℝ)*mesh N)*L := by + rw [dist_eq_norm,add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (sub_nonneg.mpr hk)] + exact mul_le_mul_of_nonneg_left (hL _ _) (sub_nonneg.mpr hk) + have htriangle := (dist_triangle (eulerSteps κ b j N z k+((t-(k:ℝ)*mesh N : ℝ):ℂ)* + cutoffField κ b j ((k:ℝ)*mesh N) (eulerSteps κ b j N z k)) + (eulerSteps κ b j N z k) z).trans + (add_le_add hb (eulerSteps_dist_le κ b j N z L hL k)) + exact htriangle.trans_eq (by ring) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetForwardAnalytic.lean b/lean/OAI/Probability/SLE/Loewner/TargetForwardAnalytic.lean index 29a300ed91343ef8470e4daf9cf11d2136e8a4d7..60196d57918eeb4a7dc53c10612e316babd102df 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetForwardAnalytic.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetForwardAnalytic.lean @@ -1,37 +1,37 @@ -import OAI.Probability.SLE.Loewner.TargetInverseProperties -import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv -import Mathlib.Analysis.Calculus.FDeriv.Analytic - -namespace OAI - -/-! Spatial analyticity of the raw forward map follows from the analytic -inverse and its nonzero derivative. -/ -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge.Target - -lemma differentiableOn_flow_of_inverse - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) - (hF : DifferentiableOn ℂ (inverseFlow κ b T) upperHalfPlane) - (hF' : ∀ w ∈ upperHalfPlane, deriv (inverseFlow κ b T) w ≠ 0) : - DifferentiableOn ℂ (fun z => flow κ b z T) {z | Surviving κ b z T} := by - intro z hz - let w := flow κ b z T - have hw : w ∈ upperHalfPlane := surviving_flow_mem_upperHalfPlane κ b z T T.coe_nonneg hz - have han := hF.analyticAt (isOpen_upperHalfPlane.mem_nhds hw) - have hinv : inverseFlow κ b T w=z := inverseFlow_eq_of_surviving κ b T T.coe_nonneg z w hz rfl - have hlocal : ∀ᶠ v in 𝓝 w, flow κ b (inverseFlow κ b T v) T=v := by - filter_upwards [isOpen_upperHalfPlane.mem_nhds hw] with v hv - exact flow_inverseFlow κ hκ b T v hv - have hg := han.hasStrictDerivAt.to_local_left_inverse - (g := fun z => flow κ b z T) (hF' w hw) hlocal - rw [hinv] at hg - exact hg.hasDerivAt.differentiableAt.differentiableWithinAt - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetInverseProperties +import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv +import Mathlib.Analysis.Calculus.FDeriv.Analytic + +namespace OAI + +/-! Spatial analyticity of the raw forward map follows from the analytic +inverse and its nonzero derivative. -/ +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge.Target + +lemma differentiableOn_flow_of_inverse + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) + (hF : DifferentiableOn ℂ (inverseFlow κ b T) upperHalfPlane) + (hF' : ∀ w ∈ upperHalfPlane, deriv (inverseFlow κ b T) w ≠ 0) : + DifferentiableOn ℂ (fun z => flow κ b z T) {z | Surviving κ b z T} := by + intro z hz + let w := flow κ b z T + have hw : w ∈ upperHalfPlane := surviving_flow_mem_upperHalfPlane κ b z T T.coe_nonneg hz + have han := hF.analyticAt (isOpen_upperHalfPlane.mem_nhds hw) + have hinv : inverseFlow κ b T w=z := inverseFlow_eq_of_surviving κ b T T.coe_nonneg z w hz rfl + have hlocal : ∀ᶠ v in 𝓝 w, flow κ b (inverseFlow κ b T v) T=v := by + filter_upwards [isOpen_upperHalfPlane.mem_nhds hw] with v hv + exact flow_inverseFlow κ hκ b T v hv + have hg := han.hasStrictDerivAt.to_local_left_inverse + (g := fun z => flow κ b z T) (hF' w hw) hlocal + rw [hinv] at hg + exact hg.hasDerivAt.differentiableAt.differentiableWithinAt + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetInverseContinuity.lean b/lean/OAI/Probability/SLE/Loewner/TargetInverseContinuity.lean index 77148a741bf8efaf02d6b4cd330d5a29fe1f2405..9268281de85b21d5be51c4aae29ce874f7a6b0eb 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetInverseContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetInverseContinuity.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Loewner.TargetInverseProperties - -namespace OAI - -/-! Local stability and continuity of the raw inverse flow, independently of -the existence of a geometric trace. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -lemma inverseFlow_eq_reverse_trajectory - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) (w : ℂ) - (η : ℝ) (hη : 0 < η) (hηw : η ≤ w.im) (Q : ℝ → ℂ) - (hQ0 : Q 0=w) - (hQd : ∀ s, 0 ≤ s → HasDerivWithinAt Q - (reverseField η (driverValue b ((T:ℝ)-κ*s)) (Q s)) (Ici 0) s) : - inverseFlow κ b T w=Q ((T:ℝ)/κ) := by - have h := reverse_trajectory_preimage κ hκ b T w η hη hηw Q hQ0 hQd - exact inverseFlow_eq_of_surviving κ b T T.coe_nonneg _ w h.1 h.2 - -lemma inverseFlow_lipschitzOn_height - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) - (η : ℝ) (hη : 0 < η) : - LipschitzOnWith ⟨Real.exp ((4*η⁻¹^2)*((T:ℝ)/κ)), (Real.exp_pos _).le⟩ - (inverseFlow κ b T) {w : ℂ | η ≤ w.im} := by - let U (s : ℝ) (_ : ℂ) := driverValue b ((T:ℝ)-κ*s) - have hUc : Continuous (fun s : ℝ => driverValue b ((T:ℝ)-κ*s)) := by - exact b.continuous.comp (continuous_real_toNNReal.comp - (continuous_const.sub (continuous_const.mul continuous_id))) - have hUm : Measurable (fun q : ℝ × ℂ => U q.1 q.2) := - hUc.measurable.comp measurable_fst - obtain ⟨Q,_,hQ0,hQd⟩ := exists_measurable_reverse_flow U hUm - (fun _ => hUc) id measurable_id hη - have hQc (w : ℂ) : ContinuousOn (fun s => Q s w) (Icc 0 ((T:ℝ)/κ)) := - fun s hs => ((hQd w s hs.1).continuousWithinAt).mono Icc_subset_Ici_self - have hQd' (w : ℂ) (s : ℝ) (hs : s ∈ Ico 0 ((T:ℝ)/κ)) : - HasDerivWithinAt (fun r => Q r w) - (reverseField η (driverValue b ((T:ℝ)-κ*s)) (Q s w)) (Ici s) s := - (hQd w s hs.1).mono (Ici_subset_Ici.mpr hs.1) - have he (w : ℂ) (hw : η ≤ w.im) : inverseFlow κ b T w=Q ((T:ℝ)/κ) w := - inverseFlow_eq_reverse_trajectory κ hκ b T w η hη hw - (fun s => Q s w) (hQ0 w) (hQd w) - apply LipschitzOnWith.of_dist_le_mul - intro z hz w hw - rw [he z hz,he w hw] - have h := dist_le_of_trajectories_ODE - (fun s => reverseField_lipschitz hη (driverValue b ((T:ℝ)-κ*s))) - (hQc z) (hQd' z) (hQc w) (hQd' w) - (show dist (Q 0 z) (Q 0 w) ≤ dist z w by simp only [hQ0,id_eq]; exact le_rfl) - ((T:ℝ)/κ) ⟨div_nonneg T.coe_nonneg hκ.le,le_rfl⟩ - change dist (Q ((T:ℝ)/κ) z) (Q ((T:ℝ)/κ) w) ≤ - dist z w * Real.exp ((4*η⁻¹^2)*((T:ℝ)/κ-0)) at h - change dist (Q ((T:ℝ)/κ) z) (Q ((T:ℝ)/κ) w) ≤ - Real.exp ((4*η⁻¹^2)*((T:ℝ)/κ)) * dist z w - simpa only [sub_zero,mul_comm] using h - -lemma continuousOn_inverseFlow - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) : - ContinuousOn (inverseFlow κ b T) upperHalfPlane := by - intro w hw - have hη : 0 < w.im/2 := half_pos hw - have hLip := inverseFlow_lipschitzOn_height κ hκ b T (w.im/2) hη - have hn : {z : ℂ | w.im/2 ≤ z.im} ∈ 𝓝 w := by - have hn0 : {z : ℂ | w.im/2 < z.im} ∈ 𝓝 w := - (isOpen_lt continuous_const Complex.continuous_im).mem_nhds - (by change w.im/2 < w.im; exact half_lt_self hw) - exact mem_of_superset hn0 (fun z hz => by - change w.im/2 ≤ z.im - exact (show w.im/2 < z.im from hz).le) - exact (hLip.continuousOn.continuousAt hn).continuousWithinAt - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetInverseProperties + +namespace OAI + +/-! Local stability and continuity of the raw inverse flow, independently of +the existence of a geometric trace. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +lemma inverseFlow_eq_reverse_trajectory + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) (w : ℂ) + (η : ℝ) (hη : 0 < η) (hηw : η ≤ w.im) (Q : ℝ → ℂ) + (hQ0 : Q 0=w) + (hQd : ∀ s, 0 ≤ s → HasDerivWithinAt Q + (reverseField η (driverValue b ((T:ℝ)-κ*s)) (Q s)) (Ici 0) s) : + inverseFlow κ b T w=Q ((T:ℝ)/κ) := by + have h := reverse_trajectory_preimage κ hκ b T w η hη hηw Q hQ0 hQd + exact inverseFlow_eq_of_surviving κ b T T.coe_nonneg _ w h.1 h.2 + +lemma inverseFlow_lipschitzOn_height + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) + (η : ℝ) (hη : 0 < η) : + LipschitzOnWith ⟨Real.exp ((4*η⁻¹^2)*((T:ℝ)/κ)), (Real.exp_pos _).le⟩ + (inverseFlow κ b T) {w : ℂ | η ≤ w.im} := by + let U (s : ℝ) (_ : ℂ) := driverValue b ((T:ℝ)-κ*s) + have hUc : Continuous (fun s : ℝ => driverValue b ((T:ℝ)-κ*s)) := by + exact b.continuous.comp (continuous_real_toNNReal.comp + (continuous_const.sub (continuous_const.mul continuous_id))) + have hUm : Measurable (fun q : ℝ × ℂ => U q.1 q.2) := + hUc.measurable.comp measurable_fst + obtain ⟨Q,_,hQ0,hQd⟩ := exists_measurable_reverse_flow U hUm + (fun _ => hUc) id measurable_id hη + have hQc (w : ℂ) : ContinuousOn (fun s => Q s w) (Icc 0 ((T:ℝ)/κ)) := + fun s hs => ((hQd w s hs.1).continuousWithinAt).mono Icc_subset_Ici_self + have hQd' (w : ℂ) (s : ℝ) (hs : s ∈ Ico 0 ((T:ℝ)/κ)) : + HasDerivWithinAt (fun r => Q r w) + (reverseField η (driverValue b ((T:ℝ)-κ*s)) (Q s w)) (Ici s) s := + (hQd w s hs.1).mono (Ici_subset_Ici.mpr hs.1) + have he (w : ℂ) (hw : η ≤ w.im) : inverseFlow κ b T w=Q ((T:ℝ)/κ) w := + inverseFlow_eq_reverse_trajectory κ hκ b T w η hη hw + (fun s => Q s w) (hQ0 w) (hQd w) + apply LipschitzOnWith.of_dist_le_mul + intro z hz w hw + rw [he z hz,he w hw] + have h := dist_le_of_trajectories_ODE + (fun s => reverseField_lipschitz hη (driverValue b ((T:ℝ)-κ*s))) + (hQc z) (hQd' z) (hQc w) (hQd' w) + (show dist (Q 0 z) (Q 0 w) ≤ dist z w by simp only [hQ0,id_eq]; exact le_rfl) + ((T:ℝ)/κ) ⟨div_nonneg T.coe_nonneg hκ.le,le_rfl⟩ + change dist (Q ((T:ℝ)/κ) z) (Q ((T:ℝ)/κ) w) ≤ + dist z w * Real.exp ((4*η⁻¹^2)*((T:ℝ)/κ-0)) at h + change dist (Q ((T:ℝ)/κ) z) (Q ((T:ℝ)/κ) w) ≤ + Real.exp ((4*η⁻¹^2)*((T:ℝ)/κ)) * dist z w + simpa only [sub_zero,mul_comm] using h + +lemma continuousOn_inverseFlow + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) : + ContinuousOn (inverseFlow κ b T) upperHalfPlane := by + intro w hw + have hη : 0 < w.im/2 := half_pos hw + have hLip := inverseFlow_lipschitzOn_height κ hκ b T (w.im/2) hη + have hn : {z : ℂ | w.im/2 ≤ z.im} ∈ 𝓝 w := by + have hn0 : {z : ℂ | w.im/2 < z.im} ∈ 𝓝 w := + (isOpen_lt continuous_const Complex.continuous_im).mem_nhds + (by change w.im/2 < w.im; exact half_lt_self hw) + exact mem_of_superset hn0 (fun z hz => by + change w.im/2 ≤ z.im + exact (show w.im/2 < z.im from hz).le) + exact (hLip.continuousOn.continuousAt hn).continuousWithinAt + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetInverseExistence.lean b/lean/OAI/Probability/SLE/Loewner/TargetInverseExistence.lean index 15072f368a7dee2034c36d49dbdff9c21db77406..631a57b8e03d5e8801e08f4636dab235b8d9243e 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetInverseExistence.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetInverseExistence.lean @@ -1,96 +1,96 @@ -import OAI.Probability.SLE.Angle.TargetPositiveFlow -import OAI.Probability.SLE.Loewner.TargetInverseMeasurability - -namespace OAI - -/-! Reverse flow supplies every point of the upper half-plane as an -image of the raw surviving flow, for every continuous based driver. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric Complex -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma reverseField_im_nonneg {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : - 0 ≤ (reverseField a u z).im := by - have hm : 0 ≤ max a z.im := ha.le.trans (le_max_left _ _) - have hd : 0 ≤ Complex.normSq (reverseCenter a u z) := Complex.normSq_nonneg _ - have he : (reverseField a u z).im=2*(max a z.im / Complex.normSq (reverseCenter a u z)) := by - simp [reverseField,Complex.mul_im,Complex.inv_im,reverseCenter_im] - ring - rw [he] - exact mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (div_nonneg hm hd) - -namespace Target - -theorem reverse_trajectory_preimage - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) (w : ℂ) - (η : ℝ) (hη : 0 < η) (hηw : η ≤ w.im) (Q : ℝ → ℂ) - (hQ0 : Q 0=w) - (hQd : ∀ s, 0 ≤ s → HasDerivWithinAt Q - (reverseField η (driverValue b ((T:ℝ)-κ*s)) (Q s)) (Ici 0) s) : - Surviving κ b (Q ((T:ℝ)/κ)) T ∧ flow κ b (Q ((T:ℝ)/κ)) T=w := by - have hQc : ContinuousOn Q (Ici 0) := - fun s hs => (hQd s hs).continuousWithinAt - have hQi : MonotoneOn (fun s => (Q s).im) (Ici 0) := by - apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Ici 0) - (Complex.continuous_im.comp_continuousOn hQc) - (fun s hs => (Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt s (hQd s (interior_subset hs))).mono interior_subset) - intro s hs - exact reverseField_im_nonneg hη _ _ - have hheightQ (s : ℝ) (hs : 0 ≤ s) : w.im ≤ (Q s).im := by - simpa only [hQ0] using hQi (show (0:ℝ) ∈ Ici (0:ℝ) from le_refl (0:ℝ)) hs hs - let Y (s : ℝ) := Q (((T:ℝ)-s)/κ) - let z := Y 0 - have hmaps : MapsTo (fun s : ℝ => ((T:ℝ)-s)/κ) (Icc 0 (T:ℝ)) (Ici 0) := - fun s hs => div_nonneg (sub_nonneg.mpr hs.2) hκ.le - have hY : ContinuousOn Y (Icc 0 (T:ℝ)) := - hQc.comp ((continuous_const.sub continuous_id).div_const κ).continuousOn hmaps - have hYheight : ∀ s ∈ Icc 0 (T:ℝ), w.im ≤ (Y s).im := - fun s hs => hheightQ _ (hmaps hs) - have hYd (s : ℝ) (hs : s ∈ Ico 0 (T:ℝ)) : - HasDerivWithinAt Y ((a κ:ℂ)/(Y s-driverValue b s)) (Ici s) s := by - have ht : 0 ≤ ((T:ℝ)-s)/κ := hmaps (Ico_subset_Icc_self hs) - have hi : HasDerivWithinAt (fun r : ℝ => ((T:ℝ)-r)/κ) (-1/κ) (Icc 0 (T:ℝ)) s := by - convert (((hasDerivAt_id s).const_sub (T:ℝ)).div_const κ).hasDerivWithinAt using 1 - simp - have hd := (hQd _ ht).scomp s hi hmaps - have he : (T:ℝ)-κ*(((T:ℝ)-s)/κ)=s := by field_simp [hκ.ne']; ring - have hh : η ≤ (Y s).im := hηw.trans (hYheight s (Ico_subset_Icc_self hs)) - have hc : (-1/κ) • reverseField η (driverValue b ((T:ℝ)-κ*(((T:ℝ)-s)/κ))) (Y s) = - (a κ:ℂ)/(Y s-driverValue b s) := by - have hUt : driverValue b ((T:ℝ)-κ*(((T:ℝ)-s)/κ))=driverValue b s := by rw [he] - rw [hUt,reverseField,reverseCenter_eq hh] - simp only [Complex.real_smul,a,div_eq_mul_inv] - push_cast - ring - exact hasDerivWithinAt_Ici_of_Icc hs (hd.congr_deriv hc) - obtain ⟨hz,hflow⟩ := flow_eq_of_positive_trajectory κ b z T T.coe_nonneg Y hY rfl - w.im (hη.trans_le hηw) hYheight hYd - constructor - · simpa only [z,Y,sub_zero] using hz - · simpa only [z,Y,sub_zero,sub_self,zero_div,hQ0] using hflow - -theorem exists_surviving_preimage - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) (w : ℂ) - (hw : w ∈ upperHalfPlane) : - ∃ z, Surviving κ b z T ∧ flow κ b z T=w := by - let η := w.im/2 - have hη : 0 < η := half_pos hw - let U (s : ℝ) (_ : Unit) := driverValue b ((T:ℝ)-κ*s) - have hUc : Continuous (fun s : ℝ => U s ()) := by - exact b.continuous.comp (continuous_real_toNNReal.comp - (continuous_const.sub (continuous_const.mul continuous_id))) - have hUm : Measurable (fun q : ℝ × Unit => U q.1 q.2) := - hUc.measurable.comp measurable_fst - obtain ⟨Q,hQm,hQ0,hQd⟩ := exists_measurable_reverse_flow U hUm - (fun _ => hUc) (fun _ => w) measurable_const hη - exact ⟨Q ((T:ℝ)/κ) (), reverse_trajectory_preimage κ hκ b T w η hη - (half_le_self (show 0 ≤ w.im from hw.le)) (fun s => Q s ()) (hQ0 ()) (hQd ())⟩ - -end Target -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetPositiveFlow +import OAI.Probability.SLE.Loewner.TargetInverseMeasurability + +namespace OAI + +/-! Reverse flow supplies every point of the upper half-plane as an +image of the raw surviving flow, for every continuous based driver. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric Complex +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma reverseField_im_nonneg {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : + 0 ≤ (reverseField a u z).im := by + have hm : 0 ≤ max a z.im := ha.le.trans (le_max_left _ _) + have hd : 0 ≤ Complex.normSq (reverseCenter a u z) := Complex.normSq_nonneg _ + have he : (reverseField a u z).im=2*(max a z.im / Complex.normSq (reverseCenter a u z)) := by + simp [reverseField,Complex.mul_im,Complex.inv_im,reverseCenter_im] + ring + rw [he] + exact mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (div_nonneg hm hd) + +namespace Target + +theorem reverse_trajectory_preimage + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) (w : ℂ) + (η : ℝ) (hη : 0 < η) (hηw : η ≤ w.im) (Q : ℝ → ℂ) + (hQ0 : Q 0=w) + (hQd : ∀ s, 0 ≤ s → HasDerivWithinAt Q + (reverseField η (driverValue b ((T:ℝ)-κ*s)) (Q s)) (Ici 0) s) : + Surviving κ b (Q ((T:ℝ)/κ)) T ∧ flow κ b (Q ((T:ℝ)/κ)) T=w := by + have hQc : ContinuousOn Q (Ici 0) := + fun s hs => (hQd s hs).continuousWithinAt + have hQi : MonotoneOn (fun s => (Q s).im) (Ici 0) := by + apply monotoneOn_of_hasDerivWithinAt_nonneg (convex_Ici 0) + (Complex.continuous_im.comp_continuousOn hQc) + (fun s hs => (Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt s (hQd s (interior_subset hs))).mono interior_subset) + intro s hs + exact reverseField_im_nonneg hη _ _ + have hheightQ (s : ℝ) (hs : 0 ≤ s) : w.im ≤ (Q s).im := by + simpa only [hQ0] using hQi (show (0:ℝ) ∈ Ici (0:ℝ) from le_refl (0:ℝ)) hs hs + let Y (s : ℝ) := Q (((T:ℝ)-s)/κ) + let z := Y 0 + have hmaps : MapsTo (fun s : ℝ => ((T:ℝ)-s)/κ) (Icc 0 (T:ℝ)) (Ici 0) := + fun s hs => div_nonneg (sub_nonneg.mpr hs.2) hκ.le + have hY : ContinuousOn Y (Icc 0 (T:ℝ)) := + hQc.comp ((continuous_const.sub continuous_id).div_const κ).continuousOn hmaps + have hYheight : ∀ s ∈ Icc 0 (T:ℝ), w.im ≤ (Y s).im := + fun s hs => hheightQ _ (hmaps hs) + have hYd (s : ℝ) (hs : s ∈ Ico 0 (T:ℝ)) : + HasDerivWithinAt Y ((a κ:ℂ)/(Y s-driverValue b s)) (Ici s) s := by + have ht : 0 ≤ ((T:ℝ)-s)/κ := hmaps (Ico_subset_Icc_self hs) + have hi : HasDerivWithinAt (fun r : ℝ => ((T:ℝ)-r)/κ) (-1/κ) (Icc 0 (T:ℝ)) s := by + convert (((hasDerivAt_id s).const_sub (T:ℝ)).div_const κ).hasDerivWithinAt using 1 + simp + have hd := (hQd _ ht).scomp s hi hmaps + have he : (T:ℝ)-κ*(((T:ℝ)-s)/κ)=s := by field_simp [hκ.ne']; ring + have hh : η ≤ (Y s).im := hηw.trans (hYheight s (Ico_subset_Icc_self hs)) + have hc : (-1/κ) • reverseField η (driverValue b ((T:ℝ)-κ*(((T:ℝ)-s)/κ))) (Y s) = + (a κ:ℂ)/(Y s-driverValue b s) := by + have hUt : driverValue b ((T:ℝ)-κ*(((T:ℝ)-s)/κ))=driverValue b s := by rw [he] + rw [hUt,reverseField,reverseCenter_eq hh] + simp only [Complex.real_smul,a,div_eq_mul_inv] + push_cast + ring + exact hasDerivWithinAt_Ici_of_Icc hs (hd.congr_deriv hc) + obtain ⟨hz,hflow⟩ := flow_eq_of_positive_trajectory κ b z T T.coe_nonneg Y hY rfl + w.im (hη.trans_le hηw) hYheight hYd + constructor + · simpa only [z,Y,sub_zero] using hz + · simpa only [z,Y,sub_zero,sub_self,zero_div,hQ0] using hflow + +theorem exists_surviving_preimage + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (T : ℝ≥0) (w : ℂ) + (hw : w ∈ upperHalfPlane) : + ∃ z, Surviving κ b z T ∧ flow κ b z T=w := by + let η := w.im/2 + have hη : 0 < η := half_pos hw + let U (s : ℝ) (_ : Unit) := driverValue b ((T:ℝ)-κ*s) + have hUc : Continuous (fun s : ℝ => U s ()) := by + exact b.continuous.comp (continuous_real_toNNReal.comp + (continuous_const.sub (continuous_const.mul continuous_id))) + have hUm : Measurable (fun q : ℝ × Unit => U q.1 q.2) := + hUc.measurable.comp measurable_fst + obtain ⟨Q,hQm,hQ0,hQd⟩ := exists_measurable_reverse_flow U hUm + (fun _ => hUc) (fun _ => w) measurable_const hη + exact ⟨Q ((T:ℝ)/κ) (), reverse_trajectory_preimage κ hκ b T w η hη + (half_le_self (show 0 ≤ w.im from hw.le)) (fun s => Q s ()) (hQ0 ()) (hQd ())⟩ + +end Target +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetInverseMeasurability.lean b/lean/OAI/Probability/SLE/Loewner/TargetInverseMeasurability.lean index c83be449c24ae7065dc93293206874a0c7213ae2..38f94045c0a0f6fc7df16ee335345f364eda9c33 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetInverseMeasurability.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetInverseMeasurability.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Angle.TargetFlowInjectivity - -namespace OAI - -/-! A measurable inverse of the raw surviving flow. The inverse is the -prescribed `inverseFlow`, including its value zero outside its range. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -local instance inverseDriverPolish : PolishSpace Driver := - ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace - -def survivingState (κ : ℝ) : Set (ℝ≥0 × (Driver × ℂ)) := - {q | Surviving κ q.2.1 q.2.2 q.1} - -def survivingStateMap (κ : ℝ) (q : survivingState κ) : ℝ≥0 × (Driver × ℂ) := - (q.1.1,(q.1.2.1,flow κ q.1.2.1 q.1.2.2 q.1.1)) - -lemma measurable_survivingStateMap (κ : ℝ) : Measurable (survivingStateMap κ) := - measurable_subtype_coe.fst.prodMk (measurable_subtype_coe.snd.fst.prodMk - ((measurable_flow κ).comp measurable_subtype_coe)) - -lemma survivingStateMap_injective (κ : ℝ) : Function.Injective (survivingStateMap κ) := by - rintro ⟨⟨t,b,z⟩,hz⟩ ⟨⟨s,c,w⟩,hw⟩ he - change (t,(b,flow κ b z t))=(s,(c,flow κ c w s)) at he - rcases Prod.mk.inj he with ⟨rfl,he⟩ - rcases Prod.mk.inj he with ⟨rfl,he⟩ - have hzw := flow_injOn_surviving κ b t t.coe_nonneg hz hw he - change z=w at hzw - subst w - rfl - -lemma measurableEmbedding_survivingStateMap (κ : ℝ) : - MeasurableEmbedding (survivingStateMap κ) := by - let : StandardBorelSpace (survivingState κ) := (measurableSet_surviving κ).standardBorel - exact (measurable_survivingStateMap κ).measurableEmbedding (survivingStateMap_injective κ) - -lemma inverseFlow_eq_extend (κ : ℝ) (q : ℝ≥0 × (Driver × ℂ)) : - inverseFlow κ q.2.1 q.1 q.2.2 = - Function.extend (survivingStateMap κ) (fun p => p.1.2.2) (fun _ => (0:ℂ)) q := by - classical - by_cases h : ∃ z, Surviving κ q.2.1 z q.1 ∧ flow κ q.2.1 z q.1=q.2.2 - · let z := Classical.epsilon (fun z => Surviving κ q.2.1 z q.1 ∧ flow κ q.2.1 z q.1=q.2.2) - have hz : Surviving κ q.2.1 z q.1 ∧ flow κ q.2.1 z q.1=q.2.2 := Classical.epsilon_spec h - let p : survivingState κ := ⟨(q.1,(q.2.1,z)),hz.1⟩ - have hp : survivingStateMap κ p=q := by - rcases q with ⟨t,b,w⟩ - change (t,(b,flow κ b z t))=(t,(b,w)) - rw [hz.2] - rw [inverseFlow,ite_eq_left h] - change z = _ - calc - z = p.1.2.2 := rfl - _ = Function.extend (survivingStateMap κ) (fun p => p.1.2.2) (fun _ => (0:ℂ)) - (survivingStateMap κ p) := ((survivingStateMap_injective κ).extend_apply - (fun p => p.1.2.2) (fun _ => (0:ℂ)) p).symm - _ = _ := by rw [hp] - · have hn : ¬∃ p, survivingStateMap κ p=q := by - rintro ⟨p,hp⟩ - apply h - have ht := congrArg Prod.fst hp - have hb := congrArg (fun x : ℝ≥0 × (Driver × ℂ) => x.2.1) hp - have hf := congrArg (fun x : ℝ≥0 × (Driver × ℂ) => x.2.2) hp - change p.1.1=q.1 at ht - change p.1.2.1=q.2.1 at hb - change flow κ p.1.2.1 p.1.2.2 p.1.1=q.2.2 at hf - refine ⟨p.1.2.2,?_,?_⟩ - · simpa only [ht,hb] using (show Surviving κ p.1.2.1 p.1.2.2 p.1.1 from p.2) - · simpa only [ht,hb] using hf - rw [inverseFlow,ite_eq_right h,Function.extend_apply' _ _ _ hn] - -lemma measurable_inverseFlow (κ : ℝ) : - Measurable (fun q : ℝ≥0 × (Driver × ℂ) => inverseFlow κ q.2.1 q.1 q.2.2) := by - have hm := (measurableEmbedding_survivingStateMap κ).measurable_extend - measurable_subtype_coe.snd.snd (measurable_const (a := (0:ℂ))) - convert hm using 1 - funext q - exact inverseFlow_eq_extend κ q - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetFlowInjectivity + +namespace OAI + +/-! A measurable inverse of the raw surviving flow. The inverse is the +prescribed `inverseFlow`, including its value zero outside its range. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +local instance inverseDriverPolish : PolishSpace Driver := + ContinuousMapZero.isClosedEmbedding_toContinuousMap.polishSpace + +def survivingState (κ : ℝ) : Set (ℝ≥0 × (Driver × ℂ)) := + {q | Surviving κ q.2.1 q.2.2 q.1} + +def survivingStateMap (κ : ℝ) (q : survivingState κ) : ℝ≥0 × (Driver × ℂ) := + (q.1.1,(q.1.2.1,flow κ q.1.2.1 q.1.2.2 q.1.1)) + +lemma measurable_survivingStateMap (κ : ℝ) : Measurable (survivingStateMap κ) := + measurable_subtype_coe.fst.prodMk (measurable_subtype_coe.snd.fst.prodMk + ((measurable_flow κ).comp measurable_subtype_coe)) + +lemma survivingStateMap_injective (κ : ℝ) : Function.Injective (survivingStateMap κ) := by + rintro ⟨⟨t,b,z⟩,hz⟩ ⟨⟨s,c,w⟩,hw⟩ he + change (t,(b,flow κ b z t))=(s,(c,flow κ c w s)) at he + rcases Prod.mk.inj he with ⟨rfl,he⟩ + rcases Prod.mk.inj he with ⟨rfl,he⟩ + have hzw := flow_injOn_surviving κ b t t.coe_nonneg hz hw he + change z=w at hzw + subst w + rfl + +lemma measurableEmbedding_survivingStateMap (κ : ℝ) : + MeasurableEmbedding (survivingStateMap κ) := by + let : StandardBorelSpace (survivingState κ) := (measurableSet_surviving κ).standardBorel + exact (measurable_survivingStateMap κ).measurableEmbedding (survivingStateMap_injective κ) + +lemma inverseFlow_eq_extend (κ : ℝ) (q : ℝ≥0 × (Driver × ℂ)) : + inverseFlow κ q.2.1 q.1 q.2.2 = + Function.extend (survivingStateMap κ) (fun p => p.1.2.2) (fun _ => (0:ℂ)) q := by + classical + by_cases h : ∃ z, Surviving κ q.2.1 z q.1 ∧ flow κ q.2.1 z q.1=q.2.2 + · let z := Classical.epsilon (fun z => Surviving κ q.2.1 z q.1 ∧ flow κ q.2.1 z q.1=q.2.2) + have hz : Surviving κ q.2.1 z q.1 ∧ flow κ q.2.1 z q.1=q.2.2 := Classical.epsilon_spec h + let p : survivingState κ := ⟨(q.1,(q.2.1,z)),hz.1⟩ + have hp : survivingStateMap κ p=q := by + rcases q with ⟨t,b,w⟩ + change (t,(b,flow κ b z t))=(t,(b,w)) + rw [hz.2] + rw [inverseFlow,ite_eq_left h] + change z = _ + calc + z = p.1.2.2 := rfl + _ = Function.extend (survivingStateMap κ) (fun p => p.1.2.2) (fun _ => (0:ℂ)) + (survivingStateMap κ p) := ((survivingStateMap_injective κ).extend_apply + (fun p => p.1.2.2) (fun _ => (0:ℂ)) p).symm + _ = _ := by rw [hp] + · have hn : ¬∃ p, survivingStateMap κ p=q := by + rintro ⟨p,hp⟩ + apply h + have ht := congrArg Prod.fst hp + have hb := congrArg (fun x : ℝ≥0 × (Driver × ℂ) => x.2.1) hp + have hf := congrArg (fun x : ℝ≥0 × (Driver × ℂ) => x.2.2) hp + change p.1.1=q.1 at ht + change p.1.2.1=q.2.1 at hb + change flow κ p.1.2.1 p.1.2.2 p.1.1=q.2.2 at hf + refine ⟨p.1.2.2,?_,?_⟩ + · simpa only [ht,hb] using (show Surviving κ p.1.2.1 p.1.2.2 p.1.1 from p.2) + · simpa only [ht,hb] using hf + rw [inverseFlow,ite_eq_right h,Function.extend_apply' _ _ _ hn] + +lemma measurable_inverseFlow (κ : ℝ) : + Measurable (fun q : ℝ≥0 × (Driver × ℂ) => inverseFlow κ q.2.1 q.1 q.2.2) := by + have hm := (measurableEmbedding_survivingStateMap κ).measurable_extend + measurable_subtype_coe.snd.snd (measurable_const (a := (0:ℂ))) + convert hm using 1 + funext q + exact inverseFlow_eq_extend κ q + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetInverseProperties.lean b/lean/OAI/Probability/SLE/Loewner/TargetInverseProperties.lean index daa3455503059da7a90897c5afb1c82287947c94..b277f089ec1dd8f2703b9111cd9c7481589281a8 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetInverseProperties.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetInverseProperties.lean @@ -1,61 +1,61 @@ -import OAI.Probability.SLE.Loewner.TargetInverseExistence - -namespace OAI - -/-! Exact inverse identities for the raw flow on its surviving domain. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -lemma surviving_mem_upperHalfPlane (κ : ℝ) (b : Driver) (z : ℂ) - (T : ℝ) (hT : 0 ≤ T) (hs : Surviving κ b z T) : z ∈ upperHalfPlane := by - obtain ⟨j,hj,hmin⟩ := hs - have hh := surviving_cutoff_height κ b z T j hj hmin 0 ⟨le_rfl,hT⟩ - rw [cutoffFlow_zero κ b j hj] at hh - have hj0 : (0:ℝ) diskMapInv q.1 q.2) := by - unfold diskMapInv - fun_prop - -private lemma measurable_joint_clockIntegrand (κ : ℝ) : - Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => - (flow κ q.1.2.2.1 (diskMapInv q.1.2.1 q.1.2.2.2) q.2.toNNReal).im ^ 2 / - ‖flow κ q.1.2.2.1 (diskMapInv q.1.2.1 q.1.2.2.2) q.2.toNNReal - - (driverValue q.1.2.2.1 q.2 : ℂ)‖ ^ 4) := by - have hz : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => - diskMapInv q.1.2.1 q.1.2.2.2) := - @Measurable.comp ((ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ) (ℝ × ℂ) ℂ - inferInstance inferInstance inferInstance (fun q => diskMapInv q.1 q.2) - (fun q => (q.1.2.1,q.1.2.2.2)) measurable_diskMapInv_joint - (measurable_fst.snd.fst.prodMk measurable_fst.snd.snd.snd) - have hp : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => - (q.2.toNNReal,(q.1.2.2.1,diskMapInv q.1.2.1 q.1.2.2.2))) := - (continuous_real_toNNReal.measurable.comp measurable_snd).prodMk - (measurable_fst.snd.snd.fst.prodMk hz) - have hF : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => - flow κ q.1.2.2.1 (diskMapInv q.1.2.1 q.1.2.2.2) q.2.toNNReal) := - @Measurable.comp ((ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ) (ℝ≥0 × (Driver × ℂ)) ℂ - inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1) - (fun q => (q.2.toNNReal,(q.1.2.2.1,diskMapInv q.1.2.1 q.1.2.2.2))) - (measurable_flow κ) hp - have hB : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => - driverValue q.1.2.2.1 q.2) := - continuous_eval.measurable.comp (measurable_fst.snd.snd.fst.prodMk - (continuous_real_toNNReal.measurable.comp measurable_snd)) - exact ((Complex.measurable_im.comp hF).pow_const 2).div - ((hF.sub hB.complex_ofReal).norm.pow_const 4) - -lemma measurable_clock_joint (κ : ℝ) : - Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - clock κ q.2.1 q.2.2.1 q.2.2.2 q.1) := by - let A : Set ((ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ) := - {q | 0 ≤ q.2 ∧ q.2 ≤ (q.1.1 : ℝ)} - have hA : MeasurableSet A := - (measurableSet_le measurable_const measurable_snd).inter - (measurableSet_le measurable_snd measurable_fst.fst.coe_nnreal_real) - have hm : StronglyMeasurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => ∫ u : ℝ, - A.indicator (fun p : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => - (flow κ p.1.2.2.1 (diskMapInv p.1.2.1 p.1.2.2.2) p.2.toNNReal).im ^ 2 / - ‖flow κ p.1.2.2.1 (diskMapInv p.1.2.1 p.1.2.2.2) p.2.toNNReal - - (driverValue p.1.2.2.1 p.2 : ℂ)‖ ^ 4) (q,u)) := - ((measurable_joint_clockIntegrand κ).indicator hA).stronglyMeasurable.integral_prod_right' - convert hm.measurable using 1 - funext q - rw [clock,← integral_indicator measurableSet_Icc] - apply integral_congr_ae - filter_upwards [] with u - by_cases hu : u ∈ Icc 0 (q.1 : ℝ) - · have hu' : (q,u) ∈ A := hu - simp only [indicator_of_mem hu,indicator_of_mem hu'] - simp only [Real.toNNReal_of_nonneg hu.1,NNReal.coe_mk,centeredFlow] - · have hu' : (q,u) ∉ A := hu - simp only [indicator_of_notMem hu,indicator_of_notMem hu'] - -lemma measurable_angle_joint (κ : ℝ) : - Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - angle κ q.2.1 q.2.2.1 q.2.2.2 q.1) := by - have hz : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - diskMapInv q.2.1 q.2.2.2) := - @Measurable.comp (ℝ≥0 × (ℝ × (Driver × ℂ))) (ℝ × ℂ) ℂ - inferInstance inferInstance inferInstance (fun q => diskMapInv q.1 q.2) - (fun q => (q.2.1,q.2.2.2)) measurable_diskMapInv_joint - (measurable_snd.fst.prodMk measurable_snd.snd.snd) - have hp : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - (q.1,(q.2.2.1,diskMapInv q.2.1 q.2.2.2))) := - measurable_fst.prodMk (measurable_snd.snd.fst.prodMk hz) - have hF : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - flow κ q.2.2.1 (diskMapInv q.2.1 q.2.2.2) q.1) := - @Measurable.comp (ℝ≥0 × (ℝ × (Driver × ℂ))) (ℝ≥0 × (Driver × ℂ)) ℂ - inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1) - (fun q => (q.1,(q.2.2.1,diskMapInv q.2.1 q.2.2.2))) (measurable_flow κ) hp - have hB : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - driverValue q.2.2.1 q.1) := by - simpa only [driverValue,Real.toNNReal_coe,Function.comp_def] using - (continuous_eval.measurable.comp (measurable_snd.snd.fst.prodMk measurable_fst) : - Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => q.2.2.1 q.1)) - exact (hF.sub hB.complex_ofReal).carg - -lemma measurableSet_disk_surviving_joint (κ : ℝ) : - MeasurableSet {q : ℝ≥0 × (ℝ × (Driver × ℂ)) | - Surviving κ q.2.2.1 (diskMapInv q.2.1 q.2.2.2) q.1} := by - have hz : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - diskMapInv q.2.1 q.2.2.2) := - @Measurable.comp (ℝ≥0 × (ℝ × (Driver × ℂ))) (ℝ × ℂ) ℂ - inferInstance inferInstance inferInstance (fun q => diskMapInv q.1 q.2) - (fun q => (q.2.1,q.2.2.2)) measurable_diskMapInv_joint - (measurable_snd.fst.prodMk measurable_snd.snd.snd) - have hp : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - (q.1,(q.2.2.1,diskMapInv q.2.1 q.2.2.2))) := - measurable_fst.prodMk (measurable_snd.snd.fst.prodMk hz) - exact hp (measurableSet_surviving κ) - -private def diskGreenFormula (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ≥0) : ℝ := - if Surviving κ b (diskMapInv θ x) t then - ((1-‖x‖^2)*Real.exp (-2*a κ*clock κ θ b x t))^(-α κ) * - Real.sin (angle κ θ b x t)^p κ - else 0 - -private lemma measurable_diskGreenFormula (κ : ℝ) : - Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - diskGreenFormula κ q.2.1 q.2.2.1 q.2.2.2 q.1) := by - have hR : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => - (1-‖q.2.2.2‖^2)*Real.exp (-2*a κ*clock κ q.2.1 q.2.2.1 q.2.2.2 q.1)) := - (measurable_const.sub (measurable_snd.snd.snd.norm.pow_const 2)).mul - ((measurable_const.mul (measurable_clock_joint κ)).exp) - exact ((hR.pow_const (-α κ)).mul - ((measurable_angle_joint κ).sin.pow_const (p κ))).ite - (measurableSet_disk_surviving_joint κ) measurable_const - -private lemma diskGreenFormula_eq {κ θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (b : Driver) {x : ℂ} (hx : x ∈ diskK) (t : ℝ≥0) : - diskGreenFormula κ θ b x t=green κ θ b x t := by - have hx1 : x ∈ ball (0:ℂ) 1 := - (closedBall_subset_ball (by norm_num : (1:ℝ)/4<1)) hx - simp only [diskGreenFormula,green,radius,initialRadius_disk hθ hx1] - -private lemma measurable_diskPotentialFormula (κ : ℝ) : - Measurable (fun q : ℝ≥0 × (ℝ × Driver) => - ∫⁻ x in diskK, ENNReal.ofReal (diskGreenFormula κ q.2.1 q.2.2 x q.1)) := by - have hp : Measurable (fun q : (ℝ≥0 × (ℝ × Driver)) × ℂ => - (q.1.1,(q.1.2.1,(q.1.2.2,q.2)))) := - measurable_fst.fst.prodMk (measurable_fst.snd.fst.prodMk - (measurable_fst.snd.snd.prodMk measurable_snd)) - have hm : Measurable (fun q : (ℝ≥0 × (ℝ × Driver)) × ℂ => - ENNReal.ofReal (diskGreenFormula κ q.1.2.1 q.1.2.2 q.2 q.1.1)) := - ((measurable_diskGreenFormula κ).comp hp).ennreal_ofReal - exact @Measurable.lintegral_prod_right' _ _ _ _ (volume.restrict diskK) _ _ hm - -lemma measurable_potential_angle (κ : ℝ) : - Measurable (fun q : ℝ≥0 × ({θ : ℝ // θ ∈ Ioo (0:ℝ) Real.pi} × Driver) => - potential κ q.2.1 q.2.2 q.1) := by - have hp : Measurable (fun q : ℝ≥0 × ({θ : ℝ // θ ∈ Ioo (0:ℝ) Real.pi} × Driver) => - (q.1,((q.2.1:ℝ),q.2.2))) := - measurable_fst.prodMk ((measurable_subtype_coe.comp measurable_snd.fst).prodMk - measurable_snd.snd) - convert (measurable_diskPotentialFormula κ).comp hp using 1 - funext q - apply setLIntegral_congr_fun (isClosed_closedBall.measurableSet) - intro x hx - exact congrArg ENNReal.ofReal (diskGreenFormula_eq q.2.1.property q.2.2 hx q.1).symm - -lemma measurable_potential_restricted_angle (κ : ℝ) : - Measurable (fun q : ℝ≥0 × (ℝ × Driver) => - if q.2.1 ∈ Ioo (0:ℝ) Real.pi then potential κ q.2.1 q.2.2 q.1 else 0) := by - classical - have hA : MeasurableSet {q : ℝ≥0 × (ℝ × Driver) | q.2.1 ∈ Ioo (0:ℝ) Real.pi} := - measurable_snd.fst measurableSet_Ioo - have hm : Measurable (fun q : ℝ≥0 × (ℝ × Driver) => - if q.2.1 ∈ Ioo (0:ℝ) Real.pi then - ∫⁻ x in diskK, ENNReal.ofReal (diskGreenFormula κ q.2.1 q.2.2 x q.1) - else (0:ℝ≥0∞)) := - (measurable_diskPotentialFormula κ).ite hA measurable_const - convert hm using 1 - funext q - by_cases hq : q.2.1 ∈ Ioo (0:ℝ) Real.pi - · simp only [ite_eq_left hq] - apply setLIntegral_congr_fun isClosed_closedBall.measurableSet - intro x hx - exact congrArg ENNReal.ofReal (diskGreenFormula_eq hq q.2.2 hx q.1).symm - · simp only [ite_eq_right hq] - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetPotentialMeasurability +import OAI.Probability.SLE.Angle.TargetDiskRadius + +namespace OAI + +/-! Joint measurability in the initial disk angle. The potential is needed +only on the open angle interval, where the exact initial disk radius is +`1 - ‖x‖²`; this avoids imposing any regularity at degenerate disk charts. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +lemma measurable_diskMapInv_joint : + Measurable (fun q : ℝ × ℂ => diskMapInv q.1 q.2) := by + unfold diskMapInv + fun_prop + +private lemma measurable_joint_clockIntegrand (κ : ℝ) : + Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => + (flow κ q.1.2.2.1 (diskMapInv q.1.2.1 q.1.2.2.2) q.2.toNNReal).im ^ 2 / + ‖flow κ q.1.2.2.1 (diskMapInv q.1.2.1 q.1.2.2.2) q.2.toNNReal - + (driverValue q.1.2.2.1 q.2 : ℂ)‖ ^ 4) := by + have hz : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => + diskMapInv q.1.2.1 q.1.2.2.2) := + @Measurable.comp ((ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ) (ℝ × ℂ) ℂ + inferInstance inferInstance inferInstance (fun q => diskMapInv q.1 q.2) + (fun q => (q.1.2.1,q.1.2.2.2)) measurable_diskMapInv_joint + (measurable_fst.snd.fst.prodMk measurable_fst.snd.snd.snd) + have hp : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => + (q.2.toNNReal,(q.1.2.2.1,diskMapInv q.1.2.1 q.1.2.2.2))) := + (continuous_real_toNNReal.measurable.comp measurable_snd).prodMk + (measurable_fst.snd.snd.fst.prodMk hz) + have hF : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => + flow κ q.1.2.2.1 (diskMapInv q.1.2.1 q.1.2.2.2) q.2.toNNReal) := + @Measurable.comp ((ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ) (ℝ≥0 × (Driver × ℂ)) ℂ + inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1) + (fun q => (q.2.toNNReal,(q.1.2.2.1,diskMapInv q.1.2.1 q.1.2.2.2))) + (measurable_flow κ) hp + have hB : Measurable (fun q : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => + driverValue q.1.2.2.1 q.2) := + continuous_eval.measurable.comp (measurable_fst.snd.snd.fst.prodMk + (continuous_real_toNNReal.measurable.comp measurable_snd)) + exact ((Complex.measurable_im.comp hF).pow_const 2).div + ((hF.sub hB.complex_ofReal).norm.pow_const 4) + +lemma measurable_clock_joint (κ : ℝ) : + Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + clock κ q.2.1 q.2.2.1 q.2.2.2 q.1) := by + let A : Set ((ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ) := + {q | 0 ≤ q.2 ∧ q.2 ≤ (q.1.1 : ℝ)} + have hA : MeasurableSet A := + (measurableSet_le measurable_const measurable_snd).inter + (measurableSet_le measurable_snd measurable_fst.fst.coe_nnreal_real) + have hm : StronglyMeasurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => ∫ u : ℝ, + A.indicator (fun p : (ℝ≥0 × (ℝ × (Driver × ℂ))) × ℝ => + (flow κ p.1.2.2.1 (diskMapInv p.1.2.1 p.1.2.2.2) p.2.toNNReal).im ^ 2 / + ‖flow κ p.1.2.2.1 (diskMapInv p.1.2.1 p.1.2.2.2) p.2.toNNReal - + (driverValue p.1.2.2.1 p.2 : ℂ)‖ ^ 4) (q,u)) := + ((measurable_joint_clockIntegrand κ).indicator hA).stronglyMeasurable.integral_prod_right' + convert hm.measurable using 1 + funext q + rw [clock,← integral_indicator measurableSet_Icc] + apply integral_congr_ae + filter_upwards [] with u + by_cases hu : u ∈ Icc 0 (q.1 : ℝ) + · have hu' : (q,u) ∈ A := hu + simp only [indicator_of_mem hu,indicator_of_mem hu'] + simp only [Real.toNNReal_of_nonneg hu.1,NNReal.coe_mk,centeredFlow] + · have hu' : (q,u) ∉ A := hu + simp only [indicator_of_notMem hu,indicator_of_notMem hu'] + +lemma measurable_angle_joint (κ : ℝ) : + Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + angle κ q.2.1 q.2.2.1 q.2.2.2 q.1) := by + have hz : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + diskMapInv q.2.1 q.2.2.2) := + @Measurable.comp (ℝ≥0 × (ℝ × (Driver × ℂ))) (ℝ × ℂ) ℂ + inferInstance inferInstance inferInstance (fun q => diskMapInv q.1 q.2) + (fun q => (q.2.1,q.2.2.2)) measurable_diskMapInv_joint + (measurable_snd.fst.prodMk measurable_snd.snd.snd) + have hp : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + (q.1,(q.2.2.1,diskMapInv q.2.1 q.2.2.2))) := + measurable_fst.prodMk (measurable_snd.snd.fst.prodMk hz) + have hF : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + flow κ q.2.2.1 (diskMapInv q.2.1 q.2.2.2) q.1) := + @Measurable.comp (ℝ≥0 × (ℝ × (Driver × ℂ))) (ℝ≥0 × (Driver × ℂ)) ℂ + inferInstance inferInstance inferInstance (fun q => flow κ q.2.1 q.2.2 q.1) + (fun q => (q.1,(q.2.2.1,diskMapInv q.2.1 q.2.2.2))) (measurable_flow κ) hp + have hB : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + driverValue q.2.2.1 q.1) := by + simpa only [driverValue,Real.toNNReal_coe,Function.comp_def] using + (continuous_eval.measurable.comp (measurable_snd.snd.fst.prodMk measurable_fst) : + Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => q.2.2.1 q.1)) + exact (hF.sub hB.complex_ofReal).carg + +lemma measurableSet_disk_surviving_joint (κ : ℝ) : + MeasurableSet {q : ℝ≥0 × (ℝ × (Driver × ℂ)) | + Surviving κ q.2.2.1 (diskMapInv q.2.1 q.2.2.2) q.1} := by + have hz : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + diskMapInv q.2.1 q.2.2.2) := + @Measurable.comp (ℝ≥0 × (ℝ × (Driver × ℂ))) (ℝ × ℂ) ℂ + inferInstance inferInstance inferInstance (fun q => diskMapInv q.1 q.2) + (fun q => (q.2.1,q.2.2.2)) measurable_diskMapInv_joint + (measurable_snd.fst.prodMk measurable_snd.snd.snd) + have hp : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + (q.1,(q.2.2.1,diskMapInv q.2.1 q.2.2.2))) := + measurable_fst.prodMk (measurable_snd.snd.fst.prodMk hz) + exact hp (measurableSet_surviving κ) + +private def diskGreenFormula (κ θ : ℝ) (b : Driver) (x : ℂ) (t : ℝ≥0) : ℝ := + if Surviving κ b (diskMapInv θ x) t then + ((1-‖x‖^2)*Real.exp (-2*a κ*clock κ θ b x t))^(-α κ) * + Real.sin (angle κ θ b x t)^p κ + else 0 + +private lemma measurable_diskGreenFormula (κ : ℝ) : + Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + diskGreenFormula κ q.2.1 q.2.2.1 q.2.2.2 q.1) := by + have hR : Measurable (fun q : ℝ≥0 × (ℝ × (Driver × ℂ)) => + (1-‖q.2.2.2‖^2)*Real.exp (-2*a κ*clock κ q.2.1 q.2.2.1 q.2.2.2 q.1)) := + (measurable_const.sub (measurable_snd.snd.snd.norm.pow_const 2)).mul + ((measurable_const.mul (measurable_clock_joint κ)).exp) + exact ((hR.pow_const (-α κ)).mul + ((measurable_angle_joint κ).sin.pow_const (p κ))).ite + (measurableSet_disk_surviving_joint κ) measurable_const + +private lemma diskGreenFormula_eq {κ θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (b : Driver) {x : ℂ} (hx : x ∈ diskK) (t : ℝ≥0) : + diskGreenFormula κ θ b x t=green κ θ b x t := by + have hx1 : x ∈ ball (0:ℂ) 1 := + (closedBall_subset_ball (by norm_num : (1:ℝ)/4<1)) hx + simp only [diskGreenFormula,green,radius,initialRadius_disk hθ hx1] + +private lemma measurable_diskPotentialFormula (κ : ℝ) : + Measurable (fun q : ℝ≥0 × (ℝ × Driver) => + ∫⁻ x in diskK, ENNReal.ofReal (diskGreenFormula κ q.2.1 q.2.2 x q.1)) := by + have hp : Measurable (fun q : (ℝ≥0 × (ℝ × Driver)) × ℂ => + (q.1.1,(q.1.2.1,(q.1.2.2,q.2)))) := + measurable_fst.fst.prodMk (measurable_fst.snd.fst.prodMk + (measurable_fst.snd.snd.prodMk measurable_snd)) + have hm : Measurable (fun q : (ℝ≥0 × (ℝ × Driver)) × ℂ => + ENNReal.ofReal (diskGreenFormula κ q.1.2.1 q.1.2.2 q.2 q.1.1)) := + ((measurable_diskGreenFormula κ).comp hp).ennreal_ofReal + exact @Measurable.lintegral_prod_right' _ _ _ _ (volume.restrict diskK) _ _ hm + +lemma measurable_potential_angle (κ : ℝ) : + Measurable (fun q : ℝ≥0 × ({θ : ℝ // θ ∈ Ioo (0:ℝ) Real.pi} × Driver) => + potential κ q.2.1 q.2.2 q.1) := by + have hp : Measurable (fun q : ℝ≥0 × ({θ : ℝ // θ ∈ Ioo (0:ℝ) Real.pi} × Driver) => + (q.1,((q.2.1:ℝ),q.2.2))) := + measurable_fst.prodMk ((measurable_subtype_coe.comp measurable_snd.fst).prodMk + measurable_snd.snd) + convert (measurable_diskPotentialFormula κ).comp hp using 1 + funext q + apply setLIntegral_congr_fun (isClosed_closedBall.measurableSet) + intro x hx + exact congrArg ENNReal.ofReal (diskGreenFormula_eq q.2.1.property q.2.2 hx q.1).symm + +lemma measurable_potential_restricted_angle (κ : ℝ) : + Measurable (fun q : ℝ≥0 × (ℝ × Driver) => + if q.2.1 ∈ Ioo (0:ℝ) Real.pi then potential κ q.2.1 q.2.2 q.1 else 0) := by + classical + have hA : MeasurableSet {q : ℝ≥0 × (ℝ × Driver) | q.2.1 ∈ Ioo (0:ℝ) Real.pi} := + measurable_snd.fst measurableSet_Ioo + have hm : Measurable (fun q : ℝ≥0 × (ℝ × Driver) => + if q.2.1 ∈ Ioo (0:ℝ) Real.pi then + ∫⁻ x in diskK, ENNReal.ofReal (diskGreenFormula κ q.2.1 q.2.2 x q.1) + else (0:ℝ≥0∞)) := + (measurable_diskPotentialFormula κ).ite hA measurable_const + convert hm using 1 + funext q + by_cases hq : q.2.1 ∈ Ioo (0:ℝ) Real.pi + · simp only [ite_eq_left hq] + apply setLIntegral_congr_fun isClosed_closedBall.measurableSet + intro x hx + exact congrArg ENNReal.ofReal (diskGreenFormula_eq hq q.2.2 hx q.1).symm + · simp only [ite_eq_right hq] + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetOuterMeasurability.lean b/lean/OAI/Probability/SLE/Loewner/TargetOuterMeasurability.lean index cf65a00a9d94be8f0fc45c18e2d25985f53adb3e..9b43f692c4c34e4b754e77a65b13639fe42a524e 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetOuterMeasurability.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetOuterMeasurability.lean @@ -1,109 +1,109 @@ -import OAI.Probability.SLE.Angle.TargetLevelMeasurability -import OAI.Probability.SLE.Angle.TargetAngleLaw - -namespace OAI - -/-! Measurability for the outer angle integral in the prescribed moments. -The exact potential is extended by zero only outside the support of the -angle law; the original coefficient functions agree almost everywhere. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped NNReal ENNReal - -namespace SLEExactGauge.Target - -lemma measurable_potentialDifference_restricted_angle - (κ : ℝ) (W : WienerLaw) [SFinite W] (h i : ℕ) : - Measurable (fun p : ℝ × Driver => if p.1 ∈ Ioo (0:ℝ) Real.pi then - potentialDifference κ p.1 W p.2 h i else 0) := by - classical - let t : ℝ≥0 := NNReal.mk ((i:ℝ)*mesh h) - (mul_nonneg (Nat.cast_nonneg i) (mesh_pos h).le) - let T : ℝ≥0 := NNReal.mk (((i:ℝ)+1)*mesh h) - (mul_nonneg (by positivity) (mesh_pos h).le) - let v (p : ℝ × Driver) : ℝ≥0∞ := - if p.1 ∈ Ioo (0:ℝ) Real.pi then potential κ p.1 p.2 t else 0 - let w (p : ℝ × Driver) : ℝ≥0∞ := ∫⁻ c, - if p.1 ∈ Ioo (0:ℝ) Real.pi then potential κ p.1 (concat p.2 c t) T else 0 ∂W - have hv : Measurable v := - (measurable_potential_restricted_angle κ).comp (measurable_const.prodMk measurable_id) - have hc : Measurable (fun p : (ℝ × Driver) × Driver => concat p.1.2 p.2 t) := - measurable_concat.comp - (measurable_const.prodMk (measurable_fst.snd.prodMk measurable_snd)) - have hp : Measurable (fun p : (ℝ × Driver) × Driver => - (T,(p.1.1,concat p.1.2 p.2 t))) := - measurable_const.prodMk (measurable_fst.fst.prodMk hc) - have hw : Measurable w := - ((measurable_potential_restricted_angle κ).comp hp).lintegral_prod_right' - have hm : Measurable (fun p : ℝ × Driver => - if v p < ∞ ∧ w p < ∞ then (v p).toReal-(w p).toReal else (0:ℝ)) := - (hv.ennreal_toReal.sub hw.ennreal_toReal).ite - ((measurableSet_lt hv measurable_const).inter (measurableSet_lt hw measurable_const)) - measurable_const - convert hm using 1 - funext p - by_cases hθ : p.1 ∈ Ioo (0:ℝ) Real.pi - · have hθ' : 0 < p.1 ∧ p.1 < Real.pi := hθ - simp [v,w,potentialDifference,hθ,hθ',t,T] - · have hθ' : ¬ (0 < p.1 ∧ p.1 < Real.pi) := hθ - simp [v,w,hθ,hθ',t,T] - -lemma measurable_discreteMass_restricted_angle - (κ ℓ : ℝ) (W : WienerLaw) [SFinite W] (h : ℕ) : - Measurable (fun p : ℝ × Driver => if p.1 ∈ Ioo (0:ℝ) Real.pi then - discreteMass κ p.1 W p.2 ℓ h else 0) := by - have hm : Measurable (fun p : ℝ × Driver => ∑ i ∈ Finset.range (h*2^h), - if ENNReal.ofReal (((i:ℝ)+1)*mesh h) ≤ levelTime κ p.1 p.2 ℓ - then (if p.1 ∈ Ioo (0:ℝ) Real.pi then potentialDifference κ p.1 W p.2 h i else 0) - else 0) := by - apply Finset.measurable_sum - intro i _ - exact (measurable_potentialDifference_restricted_angle κ W h i).ite - (measurableSet_le measurable_const (measurable_levelTime_angle κ ℓ)) measurable_const - convert hm using 1 - funext p - by_cases hθ : p.1 ∈ Ioo (0:ℝ) Real.pi - · simp only [ite_eq_left hθ,discreteMass] - · simp only [ite_eq_right hθ,ite_self,Finset.sum_const_zero] - -lemma angleLaw_prod_ae_mem (κ : ℝ) (W : WienerLaw) [SFinite W] : - ∀ᵐ p ∂(angleLaw κ).prod W, p.1 ∈ Ioo (0:ℝ) Real.pi := by - apply (Measure.ae_prod_iff_ae_ae (measurable_fst measurableSet_Ioo)).2 - filter_upwards [angleLaw_ae_mem κ] with θ hθ - exact ae_of_all _ (fun _ => hθ) - -lemma aemeasurable_potential_angle (κ : ℝ) (W : WienerLaw) [SFinite W] (t : ℝ≥0) : - AEMeasurable (fun p : ℝ × Driver => potential κ p.1 p.2 t) ((angleLaw κ).prod W) := by - have hm : AEMeasurable (fun p : ℝ × Driver => - if p.1 ∈ Ioo (0:ℝ) Real.pi then potential κ p.1 p.2 t else 0) - ((angleLaw κ).prod W) := - ((measurable_potential_restricted_angle κ).comp - ((measurable_const : Measurable (fun _ : ℝ × Driver => t)).prodMk measurable_id)).aemeasurable - apply hm.congr - filter_upwards [angleLaw_prod_ae_mem κ W] with p hp - exact ite_eq_left hp - -lemma aemeasurable_discreteMass_angle - (κ ℓ : ℝ) (W : WienerLaw) [SFinite W] (h : ℕ) : - AEMeasurable (fun p : ℝ × Driver => discreteMass κ p.1 W p.2 ℓ h) - ((angleLaw κ).prod W) := by - apply (measurable_discreteMass_restricted_angle κ ℓ W h).aemeasurable.congr - filter_upwards [angleLaw_prod_ae_mem κ W] with p hp - exact ite_eq_left hp - -lemma aemeasurable_coefficient_integrand - (κ ℓ u : ℝ) (W : WienerLaw) [SFinite W] (h k : ℕ) : - AEMeasurable (fun p : ℝ × Driver => levelDensity κ p.1 p.2 ℓ * - (clip u (discreteMass κ p.1 W p.2 ℓ h))^k) ((angleLaw κ).prod W) := by - have hc : Continuous (fun x : ℝ => (clip u x)^k) := by - unfold clip - fun_prop - exact (measurable_levelDensity_angle κ ℓ).aemeasurable.mul - (hc.measurable.comp_aemeasurable (aemeasurable_discreteMass_angle κ ℓ W h)) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetLevelMeasurability +import OAI.Probability.SLE.Angle.TargetAngleLaw + +namespace OAI + +/-! Measurability for the outer angle integral in the prescribed moments. +The exact potential is extended by zero only outside the support of the +angle law; the original coefficient functions agree almost everywhere. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped NNReal ENNReal + +namespace SLEExactGauge.Target + +lemma measurable_potentialDifference_restricted_angle + (κ : ℝ) (W : WienerLaw) [SFinite W] (h i : ℕ) : + Measurable (fun p : ℝ × Driver => if p.1 ∈ Ioo (0:ℝ) Real.pi then + potentialDifference κ p.1 W p.2 h i else 0) := by + classical + let t : ℝ≥0 := NNReal.mk ((i:ℝ)*mesh h) + (mul_nonneg (Nat.cast_nonneg i) (mesh_pos h).le) + let T : ℝ≥0 := NNReal.mk (((i:ℝ)+1)*mesh h) + (mul_nonneg (by positivity) (mesh_pos h).le) + let v (p : ℝ × Driver) : ℝ≥0∞ := + if p.1 ∈ Ioo (0:ℝ) Real.pi then potential κ p.1 p.2 t else 0 + let w (p : ℝ × Driver) : ℝ≥0∞ := ∫⁻ c, + if p.1 ∈ Ioo (0:ℝ) Real.pi then potential κ p.1 (concat p.2 c t) T else 0 ∂W + have hv : Measurable v := + (measurable_potential_restricted_angle κ).comp (measurable_const.prodMk measurable_id) + have hc : Measurable (fun p : (ℝ × Driver) × Driver => concat p.1.2 p.2 t) := + measurable_concat.comp + (measurable_const.prodMk (measurable_fst.snd.prodMk measurable_snd)) + have hp : Measurable (fun p : (ℝ × Driver) × Driver => + (T,(p.1.1,concat p.1.2 p.2 t))) := + measurable_const.prodMk (measurable_fst.fst.prodMk hc) + have hw : Measurable w := + ((measurable_potential_restricted_angle κ).comp hp).lintegral_prod_right' + have hm : Measurable (fun p : ℝ × Driver => + if v p < ∞ ∧ w p < ∞ then (v p).toReal-(w p).toReal else (0:ℝ)) := + (hv.ennreal_toReal.sub hw.ennreal_toReal).ite + ((measurableSet_lt hv measurable_const).inter (measurableSet_lt hw measurable_const)) + measurable_const + convert hm using 1 + funext p + by_cases hθ : p.1 ∈ Ioo (0:ℝ) Real.pi + · have hθ' : 0 < p.1 ∧ p.1 < Real.pi := hθ + simp [v,w,potentialDifference,hθ,hθ',t,T] + · have hθ' : ¬ (0 < p.1 ∧ p.1 < Real.pi) := hθ + simp [v,w,hθ,hθ',t,T] + +lemma measurable_discreteMass_restricted_angle + (κ ℓ : ℝ) (W : WienerLaw) [SFinite W] (h : ℕ) : + Measurable (fun p : ℝ × Driver => if p.1 ∈ Ioo (0:ℝ) Real.pi then + discreteMass κ p.1 W p.2 ℓ h else 0) := by + have hm : Measurable (fun p : ℝ × Driver => ∑ i ∈ Finset.range (h*2^h), + if ENNReal.ofReal (((i:ℝ)+1)*mesh h) ≤ levelTime κ p.1 p.2 ℓ + then (if p.1 ∈ Ioo (0:ℝ) Real.pi then potentialDifference κ p.1 W p.2 h i else 0) + else 0) := by + apply Finset.measurable_sum + intro i _ + exact (measurable_potentialDifference_restricted_angle κ W h i).ite + (measurableSet_le measurable_const (measurable_levelTime_angle κ ℓ)) measurable_const + convert hm using 1 + funext p + by_cases hθ : p.1 ∈ Ioo (0:ℝ) Real.pi + · simp only [ite_eq_left hθ,discreteMass] + · simp only [ite_eq_right hθ,ite_self,Finset.sum_const_zero] + +lemma angleLaw_prod_ae_mem (κ : ℝ) (W : WienerLaw) [SFinite W] : + ∀ᵐ p ∂(angleLaw κ).prod W, p.1 ∈ Ioo (0:ℝ) Real.pi := by + apply (Measure.ae_prod_iff_ae_ae (measurable_fst measurableSet_Ioo)).2 + filter_upwards [angleLaw_ae_mem κ] with θ hθ + exact ae_of_all _ (fun _ => hθ) + +lemma aemeasurable_potential_angle (κ : ℝ) (W : WienerLaw) [SFinite W] (t : ℝ≥0) : + AEMeasurable (fun p : ℝ × Driver => potential κ p.1 p.2 t) ((angleLaw κ).prod W) := by + have hm : AEMeasurable (fun p : ℝ × Driver => + if p.1 ∈ Ioo (0:ℝ) Real.pi then potential κ p.1 p.2 t else 0) + ((angleLaw κ).prod W) := + ((measurable_potential_restricted_angle κ).comp + ((measurable_const : Measurable (fun _ : ℝ × Driver => t)).prodMk measurable_id)).aemeasurable + apply hm.congr + filter_upwards [angleLaw_prod_ae_mem κ W] with p hp + exact ite_eq_left hp + +lemma aemeasurable_discreteMass_angle + (κ ℓ : ℝ) (W : WienerLaw) [SFinite W] (h : ℕ) : + AEMeasurable (fun p : ℝ × Driver => discreteMass κ p.1 W p.2 ℓ h) + ((angleLaw κ).prod W) := by + apply (measurable_discreteMass_restricted_angle κ ℓ W h).aemeasurable.congr + filter_upwards [angleLaw_prod_ae_mem κ W] with p hp + exact ite_eq_left hp + +lemma aemeasurable_coefficient_integrand + (κ ℓ u : ℝ) (W : WienerLaw) [SFinite W] (h k : ℕ) : + AEMeasurable (fun p : ℝ × Driver => levelDensity κ p.1 p.2 ℓ * + (clip u (discreteMass κ p.1 W p.2 ℓ h))^k) ((angleLaw κ).prod W) := by + have hc : Continuous (fun x : ℝ => (clip u x)^k) := by + unfold clip + fun_prop + exact (measurable_levelDensity_angle κ ℓ).aemeasurable.mul + (hc.measurable.comp_aemeasurable (aemeasurable_discreteMass_angle κ ℓ W h)) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetSurvivalExtension.lean b/lean/OAI/Probability/SLE/Loewner/TargetSurvivalExtension.lean index 087b358761296b9b44fe4a6f313cca15a495a4ea..ab482502a01e3f5c63c204e470de9e043a3c0606 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetSurvivalExtension.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetSurvivalExtension.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory - -namespace OAI - -/-! A surviving raw trajectory continues for a strictly longer time. -This follows directly from the strict cutoff-height certificate. -/ -noncomputable section -open Set Filter Metric Complex -open scoped Topology - -namespace SLEExactGauge.Target - -theorem exists_later_survival (κ : ℝ) (b : Driver) (z : ℂ) - {T : ℝ} (hT : 0 ≤ T) (hs : Surviving κ b z T) : - ∃ S : ℝ, T < S ∧ Surviving κ b z S := by - obtain ⟨j,hj,hmin⟩ := hs - have hc := cutoffFlow_continuousOn κ b j hj z - have hh := surviving_cutoff_height κ b z T j hj hmin - have hTj := hh T ⟨hT,le_rfl⟩ - have he : ∀ᶠ t in 𝓝[Ici 0] T, (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im := - (Complex.continuous_im.comp_continuousOn hc T hT).eventually (Ioi_mem_nhds hTj) - obtain ⟨δ,hδ,hδb⟩ := Metric.mem_nhdsWithin_iff.mp he - let S : ℝ := T+δ/2 - have hTS : T < S := by dsimp [S]; linarith - have hS : 0 ≤ S := hT.trans hTS.le - have hpoint (t : ℝ) (ht : t ∈ Icc 0 S) : - (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im := by - by_cases htT : t ≤ T - · exact hh t ⟨ht.1,htT⟩ - · apply hδb - refine ⟨?_,ht.1⟩ - rw [mem_ball,Real.dist_eq,abs_of_pos (sub_pos.mpr (lt_of_not_ge htT))] - have hts := ht.2 - dsimp only [S] at hts - linarith - obtain ⟨t,ht,htmin⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr hS) - (Complex.continuous_im.comp_continuousOn (hc.mono Icc_subset_Ici_self)) - refine ⟨S,hTS,j,hj,(hpoint t ht).trans_le ?_⟩ - apply le_csInf (Set.Nonempty.image _ (nonempty_Icc.mpr hS)) - rintro y ⟨s,hs,rfl⟩ - exact htmin hs - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetCutoffTrajectory + +namespace OAI + +/-! A surviving raw trajectory continues for a strictly longer time. +This follows directly from the strict cutoff-height certificate. -/ +noncomputable section +open Set Filter Metric Complex +open scoped Topology + +namespace SLEExactGauge.Target + +theorem exists_later_survival (κ : ℝ) (b : Driver) (z : ℂ) + {T : ℝ} (hT : 0 ≤ T) (hs : Surviving κ b z T) : + ∃ S : ℝ, T < S ∧ Surviving κ b z S := by + obtain ⟨j,hj,hmin⟩ := hs + have hc := cutoffFlow_continuousOn κ b j hj z + have hh := surviving_cutoff_height κ b z T j hj hmin + have hTj := hh T ⟨hT,le_rfl⟩ + have he : ∀ᶠ t in 𝓝[Ici 0] T, (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im := + (Complex.continuous_im.comp_continuousOn hc T hT).eventually (Ioi_mem_nhds hTj) + obtain ⟨δ,hδ,hδb⟩ := Metric.mem_nhdsWithin_iff.mp he + let S : ℝ := T+δ/2 + have hTS : T < S := by dsimp [S]; linarith + have hS : 0 ≤ S := hT.trans hTS.le + have hpoint (t : ℝ) (ht : t ∈ Icc 0 S) : + (j:ℝ)⁻¹ < (cutoffFlow κ b j z t).im := by + by_cases htT : t ≤ T + · exact hh t ⟨ht.1,htT⟩ + · apply hδb + refine ⟨?_,ht.1⟩ + rw [mem_ball,Real.dist_eq,abs_of_pos (sub_pos.mpr (lt_of_not_ge htT))] + have hts := ht.2 + dsimp only [S] at hts + linarith + obtain ⟨t,ht,htmin⟩ := isCompact_Icc.exists_isMinOn (nonempty_Icc.mpr hS) + (Complex.continuous_im.comp_continuousOn (hc.mono Icc_subset_Ici_self)) + refine ⟨S,hTS,j,hj,(hpoint t ht).trans_le ?_⟩ + apply le_csInf (Set.Nonempty.image _ (nonempty_Icc.mpr hS)) + rintro y ⟨s,hs,rfl⟩ + exact htmin hs + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetSurvivalLifetime.lean b/lean/OAI/Probability/SLE/Loewner/TargetSurvivalLifetime.lean index 0ec979fc807fe0ef96d13419efcdce6b3d969a9e..ca984a73068ac9540b11118a4dd8953f1f0eb754 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetSurvivalLifetime.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetSurvivalLifetime.lean @@ -1,33 +1,33 @@ -import OAI.Probability.SLE.Loewner.TargetSurvivalExtension -import OAI.Probability.SLE.Loewner.TargetInverseProperties -import OAI.Probability.SLE.Loewner.RawSurvivingMonotone -import OAI.Probability.SLE.Loewner.OpenSurvivalLifetime - -namespace OAI - -/-! Every raw point lost by a finite time has a positive finite lifetime, -and survives precisely the earlier nonnegative times. -/ -noncomputable section -open Set Filter Complex -open scoped Topology - -namespace SLEExactGauge.Target - -lemma surviving_zero (κ : ℝ) (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) : - Surviving κ b z 0 := by - obtain ⟨j,hj,hjz⟩ := Real.exists_nat_pos_inv_lt (show 0 < z.im from hz) - refine ⟨j,hj,?_⟩ - simpa only [Icc_self,image_singleton,cutoffFlow_zero κ b j hj,csInf_singleton] using hjz - -theorem exists_survival_lifetime (κ : ℝ) (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) - {T : ℝ} (hT : 0 ≤ T) (hdead : ¬Surviving κ b z T) : - ∃ τ : ℝ, 0 < τ ∧ τ ≤ T ∧ ∀ t, 0 ≤ t → (Surviving κ b z t ↔ t < τ) := by - exact exists_open_survival_lifetime (Surviving κ b z) (surviving_zero κ b z hz) - (fun s t hs hst h => surviving_of_le_time κ b z hs hst h) - (fun _ ht h => exists_later_survival κ b z ht h) hT hdead - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetSurvivalExtension +import OAI.Probability.SLE.Loewner.TargetInverseProperties +import OAI.Probability.SLE.Loewner.RawSurvivingMonotone +import OAI.Probability.SLE.Loewner.OpenSurvivalLifetime + +namespace OAI + +/-! Every raw point lost by a finite time has a positive finite lifetime, +and survives precisely the earlier nonnegative times. -/ +noncomputable section +open Set Filter Complex +open scoped Topology + +namespace SLEExactGauge.Target + +lemma surviving_zero (κ : ℝ) (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) : + Surviving κ b z 0 := by + obtain ⟨j,hj,hjz⟩ := Real.exists_nat_pos_inv_lt (show 0 < z.im from hz) + refine ⟨j,hj,?_⟩ + simpa only [Icc_self,image_singleton,cutoffFlow_zero κ b j hj,csInf_singleton] using hjz + +theorem exists_survival_lifetime (κ : ℝ) (b : Driver) (z : ℂ) (hz : z ∈ upperHalfPlane) + {T : ℝ} (hT : 0 ≤ T) (hdead : ¬Surviving κ b z T) : + ∃ τ : ℝ, 0 < τ ∧ τ ≤ T ∧ ∀ t, 0 ≤ t → (Surviving κ b z t ↔ t < τ) := by + exact exists_open_survival_lifetime (Surviving κ b z) (surviving_zero κ b z hz) + (fun s t hs hst h => surviving_of_le_time κ b z hs hst h) + (fun _ ht h => exists_later_survival κ b z ht h) hT hdead + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TargetTraceCandidate.lean b/lean/OAI/Probability/SLE/Loewner/TargetTraceCandidate.lean index 55eba50eb1cd4b4a2514f824dffc314519c39385..3d3ce4809e50432afee8f39736ccbe70b737b92c 100644 --- a/lean/OAI/Probability/SLE/Loewner/TargetTraceCandidate.lean +++ b/lean/OAI/Probability/SLE/Loewner/TargetTraceCandidate.lean @@ -1,85 +1,85 @@ -import OAI.Probability.SLE.Loewner.TargetInverseMeasurability -import OAI.Probability.SLE.Angle.TargetFlowIdentification - -namespace OAI - -/-! A measurable trace candidate from the prescribed inverse flow. -It equals the actual trace whenever a capacity witness is present. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -def rawTipApprox (κ : ℝ) (n : ℕ) (T : ℝ≥0) (b : Driver) : ℂ := - inverseFlow κ b (κ.toNNReal*T) - ((b (κ.toNNReal*T):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I) - -def traceCandidate (κ : ℝ) (T : ℝ≥0) (b : Driver) : ℂ := - limUnder atTop (fun n => rawTipApprox κ n T b) - -lemma measurable_rawTipApprox (κ : ℝ) (n : ℕ) : - Measurable (fun q : ℝ≥0 × Driver => rawTipApprox κ n q.1 q.2) := by - have ht : Measurable (fun q : ℝ≥0 × Driver => κ.toNNReal*q.1) := measurable_const.mul measurable_fst - have hw : Measurable (fun q : ℝ≥0 × Driver => - (q.2 (κ.toNNReal*q.1):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I) := - ((continuous_eval.measurable.comp (measurable_snd.prodMk ht)).complex_ofReal).add_const _ - have hq : Measurable (fun q : ℝ≥0 × Driver => - (κ.toNNReal*q.1,(q.2,(q.2 (κ.toNNReal*q.1):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I))) := - ht.prodMk (measurable_snd.prodMk hw) - exact @Measurable.comp (ℝ≥0 × Driver) (ℝ≥0 × (Driver × ℂ)) ℂ - inferInstance inferInstance inferInstance - (fun q => inverseFlow κ q.2.1 q.1 q.2.2) - (fun q => (κ.toNNReal*q.1,(q.2,(q.2 (κ.toNNReal*q.1):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I))) - (measurable_inverseFlow κ) hq - -lemma measurable_traceCandidate (κ : ℝ) : - Measurable (fun q : ℝ≥0 × Driver => traceCandidate κ q.1 q.2) := by - exact (StronglyMeasurable.limUnder - (fun n => (measurable_rawTipApprox κ n).stronglyMeasurable)).measurable - -lemma rawTipApprox_eq_witness - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) - (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) - (hw : IsCapacityTwoWitness U γ G F) - (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) - (T : ℝ≥0) (n : ℕ) : - rawTipApprox κ n T b=F T ((U T:ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I) := by - have heTime : (κ*(T:ℝ)).toNNReal=κ.toNNReal*T := by - rw [Real.toNNReal_mul hκ.le,Real.toNNReal_coe] - have hb : b (κ.toNNReal*T)=U T := by - have h := hU T T.coe_nonneg - simpa only [Real.toNNReal_coe,driverValue,heTime] using h.symm - unfold rawTipApprox - rw [Real.coe_toNNReal _ hκ.le,hb] - exact inverseFlow_eq_witness κ hκ b U γ G F hw hU T _ (by - change 0 < ((U T:ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I).im - simp only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re, - Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero] - positivity) - -lemma rawTipApprox_tendsto_witness - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) - (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) - (hw : IsCapacityTwoWitness U γ G F) - (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) - (T : ℝ≥0) : Tendsto (fun n => rawTipApprox κ n T b) atTop (𝓝 (γ T)) := by - have hseq : Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝[>] (0:ℝ)) := - tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, - Eventually.of_forall (fun n => by change 0 < (1:ℝ)/((n:ℝ)+1); positivity)⟩ - simpa only [rawTipApprox_eq_witness κ hκ b U γ G F hw hU T,Function.comp_def] using - (hw.2.2.2.2.2.2.2.2.2.2.2.2 T).comp hseq - -lemma traceCandidate_eq_witness - (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) - (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) - (hw : IsCapacityTwoWitness U γ G F) - (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) - (T : ℝ≥0) : traceCandidate κ T b=γ T := - (rawTipApprox_tendsto_witness κ hκ b U γ G F hw hU T).limUnder_eq - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Loewner.TargetInverseMeasurability +import OAI.Probability.SLE.Angle.TargetFlowIdentification + +namespace OAI + +/-! A measurable trace candidate from the prescribed inverse flow. +It equals the actual trace whenever a capacity witness is present. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +def rawTipApprox (κ : ℝ) (n : ℕ) (T : ℝ≥0) (b : Driver) : ℂ := + inverseFlow κ b (κ.toNNReal*T) + ((b (κ.toNNReal*T):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I) + +def traceCandidate (κ : ℝ) (T : ℝ≥0) (b : Driver) : ℂ := + limUnder atTop (fun n => rawTipApprox κ n T b) + +lemma measurable_rawTipApprox (κ : ℝ) (n : ℕ) : + Measurable (fun q : ℝ≥0 × Driver => rawTipApprox κ n q.1 q.2) := by + have ht : Measurable (fun q : ℝ≥0 × Driver => κ.toNNReal*q.1) := measurable_const.mul measurable_fst + have hw : Measurable (fun q : ℝ≥0 × Driver => + (q.2 (κ.toNNReal*q.1):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I) := + ((continuous_eval.measurable.comp (measurable_snd.prodMk ht)).complex_ofReal).add_const _ + have hq : Measurable (fun q : ℝ≥0 × Driver => + (κ.toNNReal*q.1,(q.2,(q.2 (κ.toNNReal*q.1):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I))) := + ht.prodMk (measurable_snd.prodMk hw) + exact @Measurable.comp (ℝ≥0 × Driver) (ℝ≥0 × (Driver × ℂ)) ℂ + inferInstance inferInstance inferInstance + (fun q => inverseFlow κ q.2.1 q.1 q.2.2) + (fun q => (κ.toNNReal*q.1,(q.2,(q.2 (κ.toNNReal*q.1):ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I))) + (measurable_inverseFlow κ) hq + +lemma measurable_traceCandidate (κ : ℝ) : + Measurable (fun q : ℝ≥0 × Driver => traceCandidate κ q.1 q.2) := by + exact (StronglyMeasurable.limUnder + (fun n => (measurable_rawTipApprox κ n).stronglyMeasurable)).measurable + +lemma rawTipApprox_eq_witness + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) + (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) + (hw : IsCapacityTwoWitness U γ G F) + (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) + (T : ℝ≥0) (n : ℕ) : + rawTipApprox κ n T b=F T ((U T:ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I) := by + have heTime : (κ*(T:ℝ)).toNNReal=κ.toNNReal*T := by + rw [Real.toNNReal_mul hκ.le,Real.toNNReal_coe] + have hb : b (κ.toNNReal*T)=U T := by + have h := hU T T.coe_nonneg + simpa only [Real.toNNReal_coe,driverValue,heTime] using h.symm + unfold rawTipApprox + rw [Real.coe_toNNReal _ hκ.le,hb] + exact inverseFlow_eq_witness κ hκ b U γ G F hw hU T _ (by + change 0 < ((U T:ℂ)+((1/((n:ℝ)+1):ℝ):ℂ)*Complex.I).im + simp only [Complex.add_im,Complex.ofReal_im,Complex.mul_im,Complex.ofReal_re, + Complex.I_im,Complex.I_re,mul_one,mul_zero,zero_add,add_zero] + positivity) + +lemma rawTipApprox_tendsto_witness + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) + (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) + (hw : IsCapacityTwoWitness U γ G F) + (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) + (T : ℝ≥0) : Tendsto (fun n => rawTipApprox κ n T b) atTop (𝓝 (γ T)) := by + have hseq : Tendsto (fun n : ℕ => 1/((n:ℝ)+1)) atTop (𝓝[>] (0:ℝ)) := + tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, + Eventually.of_forall (fun n => by change 0 < (1:ℝ)/((n:ℝ)+1); positivity)⟩ + simpa only [rawTipApprox_eq_witness κ hκ b U γ G F hw hU T,Function.comp_def] using + (hw.2.2.2.2.2.2.2.2.2.2.2.2 T).comp hseq + +lemma traceCandidate_eq_witness + (κ : ℝ) (hκ : 0 < κ) (b : Driver) (U : ℝ≥0 → ℝ) + (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) + (hw : IsCapacityTwoWitness U γ G F) + (hU : ∀ s : ℝ, 0 ≤ s → U s.toNNReal=driverValue b (κ*s)) + (T : ℝ≥0) : traceCandidate κ T b=γ T := + (rawTipApprox_tendsto_witness κ hκ b U γ G F hw hU T).limUnder_eq + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TraceBoundaryBarrier.lean b/lean/OAI/Probability/SLE/Loewner/TraceBoundaryBarrier.lean index 5cee05d23f43035564bb318c7c7182e59ea4dd55..bea30bf498ecefa752b51cd99de0070a18c0b206 100644 --- a/lean/OAI/Probability/SLE/Loewner/TraceBoundaryBarrier.lean +++ b/lean/OAI/Probability/SLE/Loewner/TraceBoundaryBarrier.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Geometry.CompactPathNeighborhoods -import OAI.Probability.SLE.Loewner.TraceDomainBoundary - -namespace OAI - -/-! A finite trace together with a bounded real interval is a compact -locally path-connected barrier outside the surviving Loewner domain. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -def traceBoundaryBarrier (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (R : ℝ) : Set ℂ := - (γ '' Icc 0 T) ∪ (Complex.ofReal '' Icc (-R) R) - -lemma traceBoundaryBarrier_compact {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) - (T : ℝ≥0) (R : ℝ) : IsCompact (traceBoundaryBarrier γ T R) := - (finiteTrace_isCompact hγ T).union (isCompact_Icc.image Complex.continuous_ofReal) - -lemma traceBoundaryBarrier_locallyPathConnected {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) - (T : ℝ≥0) (R : ℝ) (hR : 0≤R) : - LocallyPathConnectedSpace (traceBoundaryBarrier γ T R) := by - have : LocallyPathConnectedSpace (γ '' Icc 0 T) := finiteTrace_locallyPathConnected hγ T - have : LocallyPathConnectedSpace (Icc (-R) R) := - (isQuotientMap_projIcc (a := -R) (b := R) (h := by linarith)).locallyPathConnectedSpace - let p : Icc (-R) R → Complex.ofReal '' Icc (-R) R := fun x => ⟨x,⟨x,x.2,rfl⟩⟩ - have hp : Continuous p := (Complex.continuous_ofReal.comp continuous_subtype_val).subtype_mk _ - have hs : Surjective p := by - rintro ⟨_,x,hx,rfl⟩ - exact ⟨⟨x,hx⟩,rfl⟩ - have : LocallyPathConnectedSpace (Complex.ofReal '' Icc (-R) R) := - (hp.isClosedMap.isQuotientMap hp hs).locallyPathConnectedSpace - exact compact_union_locallyPathConnected (finiteTrace_isCompact hγ T) - (isCompact_Icc.image Complex.continuous_ofReal) - -lemma traceBoundaryBarrier_disjoint_domain (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (R : ℝ) : - Disjoint (traceBoundaryBarrier γ T R) (survivingDomain γ T) := by - apply disjoint_left.mpr - intro z hz hD - rcases hz with hz | ⟨x,_,rfl⟩ - · exact hD.1.2 hz - · have him := hD.1.1 - change 0<(x:ℂ).im at him - simp at him - -lemma frontier_mem_traceBoundaryBarrier {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) - {T : ℝ≥0} {R : ℝ} {z : ℂ} (hz : z∈frontier (survivingDomain γ T)) - (hR : ‖z‖≤R) : z∈traceBoundaryBarrier γ T R := by - rcases frontier_survivingDomain_subset hγ T hz with ht | hr - · exact Or.inl ht - · right - change z.im=0 at hr - refine ⟨z.re,?_,?_⟩ - · have h := (Complex.abs_re_le_norm z).trans hR - exact ⟨by linarith [(neg_abs_le z.re)],(le_abs_self z.re).trans h⟩ - · apply Complex.ext <;> simp [hr] - -lemma traceBoundaryBarrier_uniform_paths {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) - (T : ℝ≥0) (R : ℝ) (hR : 0≤R) (ε : ℝ) (hε : 0<ε) : - ∃ δ : ℝ, 0<δ ∧ ∀ x∈traceBoundaryBarrier γ T R, ∀ y∈traceBoundaryBarrier γ T R, - dist x y<δ → ∃ p : Path x y, (∀ s, p s∈traceBoundaryBarrier γ T R) ∧ - ∀ s, dist (p s) x<ε := by - have : CompactSpace (traceBoundaryBarrier γ T R) := - isCompact_iff_compactSpace.mp (traceBoundaryBarrier_compact hγ T R) - have : LocallyPathConnectedSpace (traceBoundaryBarrier γ T R) := - traceBoundaryBarrier_locallyPathConnected hγ T R hR - obtain ⟨δ,hδ,hpath⟩ := compact_locallyPathConnected_uniform_path - (A := traceBoundaryBarrier γ T R) ε hε - refine ⟨δ,hδ,fun x hx y hy hxy => ?_⟩ - obtain ⟨p,hp⟩ := hpath ⟨x,hx⟩ ⟨y,hy⟩ hxy - exact ⟨p.map continuous_subtype_val,fun s => (p s).property,hp⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.CompactPathNeighborhoods +import OAI.Probability.SLE.Loewner.TraceDomainBoundary + +namespace OAI + +/-! A finite trace together with a bounded real interval is a compact +locally path-connected barrier outside the surviving Loewner domain. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +def traceBoundaryBarrier (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (R : ℝ) : Set ℂ := + (γ '' Icc 0 T) ∪ (Complex.ofReal '' Icc (-R) R) + +lemma traceBoundaryBarrier_compact {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) + (T : ℝ≥0) (R : ℝ) : IsCompact (traceBoundaryBarrier γ T R) := + (finiteTrace_isCompact hγ T).union (isCompact_Icc.image Complex.continuous_ofReal) + +lemma traceBoundaryBarrier_locallyPathConnected {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) + (T : ℝ≥0) (R : ℝ) (hR : 0≤R) : + LocallyPathConnectedSpace (traceBoundaryBarrier γ T R) := by + have : LocallyPathConnectedSpace (γ '' Icc 0 T) := finiteTrace_locallyPathConnected hγ T + have : LocallyPathConnectedSpace (Icc (-R) R) := + (isQuotientMap_projIcc (a := -R) (b := R) (h := by linarith)).locallyPathConnectedSpace + let p : Icc (-R) R → Complex.ofReal '' Icc (-R) R := fun x => ⟨x,⟨x,x.2,rfl⟩⟩ + have hp : Continuous p := (Complex.continuous_ofReal.comp continuous_subtype_val).subtype_mk _ + have hs : Surjective p := by + rintro ⟨_,x,hx,rfl⟩ + exact ⟨⟨x,hx⟩,rfl⟩ + have : LocallyPathConnectedSpace (Complex.ofReal '' Icc (-R) R) := + (hp.isClosedMap.isQuotientMap hp hs).locallyPathConnectedSpace + exact compact_union_locallyPathConnected (finiteTrace_isCompact hγ T) + (isCompact_Icc.image Complex.continuous_ofReal) + +lemma traceBoundaryBarrier_disjoint_domain (γ : ℝ≥0 → ℂ) (T : ℝ≥0) (R : ℝ) : + Disjoint (traceBoundaryBarrier γ T R) (survivingDomain γ T) := by + apply disjoint_left.mpr + intro z hz hD + rcases hz with hz | ⟨x,_,rfl⟩ + · exact hD.1.2 hz + · have him := hD.1.1 + change 0<(x:ℂ).im at him + simp at him + +lemma frontier_mem_traceBoundaryBarrier {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) + {T : ℝ≥0} {R : ℝ} {z : ℂ} (hz : z∈frontier (survivingDomain γ T)) + (hR : ‖z‖≤R) : z∈traceBoundaryBarrier γ T R := by + rcases frontier_survivingDomain_subset hγ T hz with ht | hr + · exact Or.inl ht + · right + change z.im=0 at hr + refine ⟨z.re,?_,?_⟩ + · have h := (Complex.abs_re_le_norm z).trans hR + exact ⟨by linarith [(neg_abs_le z.re)],(le_abs_self z.re).trans h⟩ + · apply Complex.ext <;> simp [hr] + +lemma traceBoundaryBarrier_uniform_paths {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) + (T : ℝ≥0) (R : ℝ) (hR : 0≤R) (ε : ℝ) (hε : 0<ε) : + ∃ δ : ℝ, 0<δ ∧ ∀ x∈traceBoundaryBarrier γ T R, ∀ y∈traceBoundaryBarrier γ T R, + dist x y<δ → ∃ p : Path x y, (∀ s, p s∈traceBoundaryBarrier γ T R) ∧ + ∀ s, dist (p s) x<ε := by + have : CompactSpace (traceBoundaryBarrier γ T R) := + isCompact_iff_compactSpace.mp (traceBoundaryBarrier_compact hγ T R) + have : LocallyPathConnectedSpace (traceBoundaryBarrier γ T R) := + traceBoundaryBarrier_locallyPathConnected hγ T R hR + obtain ⟨δ,hδ,hpath⟩ := compact_locallyPathConnected_uniform_path + (A := traceBoundaryBarrier γ T R) ε hε + refine ⟨δ,hδ,fun x hx y hy hxy => ?_⟩ + obtain ⟨p,hp⟩ := hpath ⟨x,hx⟩ ⟨y,hy⟩ hxy + exact ⟨p.map continuous_subtype_val,fun s => (p s).property,hp⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TraceDomainBoundary.lean b/lean/OAI/Probability/SLE/Loewner/TraceDomainBoundary.lean index 675c521e5f9d5c2e97f187d574802c28650143f8..52ec746bd93c3c6d8e7e2d2cc1905b79318d815d 100644 --- a/lean/OAI/Probability/SLE/Loewner/TraceDomainBoundary.lean +++ b/lean/OAI/Probability/SLE/Loewner/TraceDomainBoundary.lean @@ -1,46 +1,46 @@ -import OAI.Probability.SLE.Geometry.FiniteTraceTopology - -namespace OAI - -/-! Every finite Loewner-domain boundary point is on the finite trace or -on the real axis. This identifies the possible inverse cluster points. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma frontier_survivingDomain_subset {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) (T : ℝ≥0) : - frontier (survivingDomain γ T) ⊆ (γ '' Icc 0 T) ∪ {z : ℂ | z.im=0} := by - intro z hz - by_cases htrace : z∈γ '' Icc 0 T - · exact Or.inl htrace - right - have hclosed : IsClosed {z : ℂ | 0≤z.im} := isClosed_le continuous_const Complex.continuous_im - have hsub : survivingDomain γ T ⊆ {w : ℂ | 0≤w.im} := by - intro w hw - change 0≤w.im - exact hw.1.1.le - have hznonneg : 0≤z.im := (closure_minimal hsub hclosed) hz.1 - by_contra hz0 - have hzH : z∈upperHalfPlane := lt_of_le_of_ne hznonneg (Ne.symm hz0) - let S := upperHalfPlane \ (γ '' Icc 0 T) - have hS : IsOpen S := isOpen_upperHalfPlane.sdiff (finiteTrace_isCompact hγ T).isClosed - have hzS : z∈S := ⟨hzH,htrace⟩ - have hznot : z∉survivingDomain γ T := by - simpa only [(isOpen_survivingDomain hγ T).interior_eq] using hz.2 - have hb : Bornology.IsBounded (connectedComponentIn S z) := by - by_contra hn - exact hznot ⟨hzS,hn⟩ - have hCn : connectedComponentIn S z∈𝓝 z := - (hS.connectedComponentIn).mem_nhds (mem_connectedComponentIn hzS) - obtain ⟨w,hwC,hwD⟩ := mem_closure_iff_nhds.mp hz.1 _ hCn - have hwBound : Bornology.IsBounded (connectedComponentIn S w) := by - rwa [← connectedComponentIn_eq hwC] - exact hwD.2 hwBound - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.FiniteTraceTopology + +namespace OAI + +/-! Every finite Loewner-domain boundary point is on the finite trace or +on the real axis. This identifies the possible inverse cluster points. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma frontier_survivingDomain_subset {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) (T : ℝ≥0) : + frontier (survivingDomain γ T) ⊆ (γ '' Icc 0 T) ∪ {z : ℂ | z.im=0} := by + intro z hz + by_cases htrace : z∈γ '' Icc 0 T + · exact Or.inl htrace + right + have hclosed : IsClosed {z : ℂ | 0≤z.im} := isClosed_le continuous_const Complex.continuous_im + have hsub : survivingDomain γ T ⊆ {w : ℂ | 0≤w.im} := by + intro w hw + change 0≤w.im + exact hw.1.1.le + have hznonneg : 0≤z.im := (closure_minimal hsub hclosed) hz.1 + by_contra hz0 + have hzH : z∈upperHalfPlane := lt_of_le_of_ne hznonneg (Ne.symm hz0) + let S := upperHalfPlane \ (γ '' Icc 0 T) + have hS : IsOpen S := isOpen_upperHalfPlane.sdiff (finiteTrace_isCompact hγ T).isClosed + have hzS : z∈S := ⟨hzH,htrace⟩ + have hznot : z∉survivingDomain γ T := by + simpa only [(isOpen_survivingDomain hγ T).interior_eq] using hz.2 + have hb : Bornology.IsBounded (connectedComponentIn S z) := by + by_contra hn + exact hznot ⟨hzS,hn⟩ + have hCn : connectedComponentIn S z∈𝓝 z := + (hS.connectedComponentIn).mem_nhds (mem_connectedComponentIn hzS) + obtain ⟨w,hwC,hwD⟩ := mem_closure_iff_nhds.mp hz.1 _ hCn + have hwBound : Bornology.IsBounded (connectedComponentIn S w) := by + rwa [← connectedComponentIn_eq hwC] + exact hwD.2 hwBound + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TraceRestrictionContinuity.lean b/lean/OAI/Probability/SLE/Loewner/TraceRestrictionContinuity.lean index 71d0fd8b82a6fd34ae4006c8fcb1adaace86d28a..704f7b8a21612d94a9eb88c9d98e5dcb567021df 100644 --- a/lean/OAI/Probability/SLE/Loewner/TraceRestrictionContinuity.lean +++ b/lean/OAI/Probability/SLE/Loewner/TraceRestrictionContinuity.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Mass.CanonicalMassContinuity - -namespace OAI - -/-! Rational first-visit identities extend simultaneously to all finite -times by local total-variation continuity. -/ - -noncomputable section -open Set Filter MeasureTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem trace_restriction_of_rational_times - (γ : ℝ≥0 → ℂ) (hγ : Continuous γ) - (μ : ℚ → Measure upperHalfPlane) (ν : Measure upperHalfPlane) - [ν.Regular] (N : ℝ≥0 → Measure upperHalfPlane) - (hp : ContinuousMassPath γ μ ν N) - (hpast : ∀ t, N t {z : upperHalfPlane | (z:ℂ) ∉ γ '' Icc 0 t} = 0) - (hq : ∀ q : ℚ, 0 ≤ q → N (q:ℝ).toNNReal = - ν.restrict {z : upperHalfPlane | (z:ℂ) ∈ γ '' Icc 0 (q:ℝ).toNNReal}) : - ∀ t, N t = ν.restrict {z : upperHalfPlane | (z:ℂ) ∈ γ '' Icc 0 t} := by - intro t - let A : Set upperHalfPlane := {z | (z:ℂ) ∈ γ '' Icc 0 t} - have hA : MeasurableSet A := - ((isCompact_Icc.image hγ).isClosed.preimage continuous_subtype_val).measurableSet - have hNpast : (N t).restrict A = N t := by - apply Measure.restrict_eq_self_of_ae_mem - filter_upwards [measure_eq_zero_iff_ae_notMem.mp (hpast t)] with z hz - exact not_not.mp hz - let D := localTestFamily upperHalfPlane - have hcover : ⋃ n, D.compactSet n = univ := by - apply eq_univ_of_forall - intro z - obtain ⟨n,hn⟩ := D.exhaustive {z} isCompact_singleton - exact mem_iUnion.mpr ⟨n,hn (mem_singleton z)⟩ - apply Measure.ext_of_iUnion_eq_univ hcover - intro n - apply Measure.ext - intro E hE - rw [Measure.restrict_apply hE,Measure.restrict_apply hE] - have hEK : MeasurableSet (E ∩ D.compactSet n) := hE.inter (D.isCompact n).measurableSet - rw [Measure.restrict_apply hEK] - have hcut : N t (E ∩ D.compactSet n) = N t ((E ∩ D.compactSet n) ∩ A) := by - conv_lhs => rw [← hNpast] - rw [Measure.restrict_apply hEK] - rw [hcut] - let L := (E ∩ D.compactSet n) ∩ A - have hL : MeasurableSet L := hEK.inter hA - have hLK : L ⊆ D.compactSet n := fun _ hz => hz.1.2 - have hνfin : ν L ≠ ∞ := (lt_of_le_of_lt (measure_mono hLK) (D.isCompact n).measure_lt_top).ne - have hNfin : N t L ≠ ∞ := (lt_of_le_of_lt (hp.dominated t L) (lt_top_iff_ne_top.mpr hνfin)).ne - have heReal : (N t L).toReal = (ν L).toReal := by - apply eq_of_forall_dist_le - intro ε hε - obtain ⟨δ,hδ,hvar⟩ := hp.local_variation (D.compactSet n) (D.isCompact n) t ε hε - obtain ⟨q,htq,hqt⟩ := exists_rat_btwn (lt_add_of_pos_right (t:ℝ) hδ) - have hq0 : 0 ≤ q := Rat.cast_nonneg.mp ((NNReal.coe_nonneg t).trans htq.le) - have hqr : ((q:ℝ).toNNReal:ℝ) = q := Real.coe_toNNReal _ (Rat.cast_nonneg.mpr hq0) - have htqNN : t ≤ (q:ℝ).toNNReal := by - apply NNReal.coe_le_coe.mp - rw [hqr] - exact htq.le - have hnear : dist (((q:ℝ).toNNReal):ℝ) t < δ := by - rw [hqr,Real.dist_eq,abs_of_pos (sub_pos.mpr htq)] - linarith - have hvalues : N (q:ℝ).toNNReal L = ν L := by - rw [hq q hq0,Measure.restrict_apply hL] - congr 1 - apply inter_eq_left.mpr - intro z hz - exact image_mono (Icc_subset_Icc le_rfl htqNN) hz.2 - have hv := hvar t (q:ℝ).toNNReal (by simpa using hδ) hnear L hL hLK - rw [hvalues] at hv - exact le_of_lt (by simpa only [Real.dist_eq] using hv) - calc - N t L = ENNReal.ofReal (N t L).toReal := (ENNReal.ofReal_toReal hNfin).symm - _ = ENNReal.ofReal (ν L).toReal := congrArg ENNReal.ofReal heReal - _ = ν L := ENNReal.ofReal_toReal hνfin - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassContinuity + +namespace OAI + +/-! Rational first-visit identities extend simultaneously to all finite +times by local total-variation continuity. -/ + +noncomputable section +open Set Filter MeasureTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem trace_restriction_of_rational_times + (γ : ℝ≥0 → ℂ) (hγ : Continuous γ) + (μ : ℚ → Measure upperHalfPlane) (ν : Measure upperHalfPlane) + [ν.Regular] (N : ℝ≥0 → Measure upperHalfPlane) + (hp : ContinuousMassPath γ μ ν N) + (hpast : ∀ t, N t {z : upperHalfPlane | (z:ℂ) ∉ γ '' Icc 0 t} = 0) + (hq : ∀ q : ℚ, 0 ≤ q → N (q:ℝ).toNNReal = + ν.restrict {z : upperHalfPlane | (z:ℂ) ∈ γ '' Icc 0 (q:ℝ).toNNReal}) : + ∀ t, N t = ν.restrict {z : upperHalfPlane | (z:ℂ) ∈ γ '' Icc 0 t} := by + intro t + let A : Set upperHalfPlane := {z | (z:ℂ) ∈ γ '' Icc 0 t} + have hA : MeasurableSet A := + ((isCompact_Icc.image hγ).isClosed.preimage continuous_subtype_val).measurableSet + have hNpast : (N t).restrict A = N t := by + apply Measure.restrict_eq_self_of_ae_mem + filter_upwards [measure_eq_zero_iff_ae_notMem.mp (hpast t)] with z hz + exact not_not.mp hz + let D := localTestFamily upperHalfPlane + have hcover : ⋃ n, D.compactSet n = univ := by + apply eq_univ_of_forall + intro z + obtain ⟨n,hn⟩ := D.exhaustive {z} isCompact_singleton + exact mem_iUnion.mpr ⟨n,hn (mem_singleton z)⟩ + apply Measure.ext_of_iUnion_eq_univ hcover + intro n + apply Measure.ext + intro E hE + rw [Measure.restrict_apply hE,Measure.restrict_apply hE] + have hEK : MeasurableSet (E ∩ D.compactSet n) := hE.inter (D.isCompact n).measurableSet + rw [Measure.restrict_apply hEK] + have hcut : N t (E ∩ D.compactSet n) = N t ((E ∩ D.compactSet n) ∩ A) := by + conv_lhs => rw [← hNpast] + rw [Measure.restrict_apply hEK] + rw [hcut] + let L := (E ∩ D.compactSet n) ∩ A + have hL : MeasurableSet L := hEK.inter hA + have hLK : L ⊆ D.compactSet n := fun _ hz => hz.1.2 + have hνfin : ν L ≠ ∞ := (lt_of_le_of_lt (measure_mono hLK) (D.isCompact n).measure_lt_top).ne + have hNfin : N t L ≠ ∞ := (lt_of_le_of_lt (hp.dominated t L) (lt_top_iff_ne_top.mpr hνfin)).ne + have heReal : (N t L).toReal = (ν L).toReal := by + apply eq_of_forall_dist_le + intro ε hε + obtain ⟨δ,hδ,hvar⟩ := hp.local_variation (D.compactSet n) (D.isCompact n) t ε hε + obtain ⟨q,htq,hqt⟩ := exists_rat_btwn (lt_add_of_pos_right (t:ℝ) hδ) + have hq0 : 0 ≤ q := Rat.cast_nonneg.mp ((NNReal.coe_nonneg t).trans htq.le) + have hqr : ((q:ℝ).toNNReal:ℝ) = q := Real.coe_toNNReal _ (Rat.cast_nonneg.mpr hq0) + have htqNN : t ≤ (q:ℝ).toNNReal := by + apply NNReal.coe_le_coe.mp + rw [hqr] + exact htq.le + have hnear : dist (((q:ℝ).toNNReal):ℝ) t < δ := by + rw [hqr,Real.dist_eq,abs_of_pos (sub_pos.mpr htq)] + linarith + have hvalues : N (q:ℝ).toNNReal L = ν L := by + rw [hq q hq0,Measure.restrict_apply hL] + congr 1 + apply inter_eq_left.mpr + intro z hz + exact image_mono (Icc_subset_Icc le_rfl htqNN) hz.2 + have hv := hvar t (q:ℝ).toNNReal (by simpa using hδ) hnear L hL hLK + rw [hvalues] at hv + exact le_of_lt (by simpa only [Real.dist_eq] using hv) + calc + N t L = ENNReal.ofReal (N t L).toReal := (ENNReal.ofReal_toReal hNfin).symm + _ = ENNReal.ofReal (ν L).toReal := congrArg ENNReal.ofReal heReal + _ = ν L := ENNReal.ofReal_toReal hνfin + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/TraceWitnessUniqueness.lean b/lean/OAI/Probability/SLE/Loewner/TraceWitnessUniqueness.lean index 95bfeee57e710dc6bfed410b1211f1b7971ece00..c04c37f66d895a5b24ce5b980d66c127ada03c52 100644 --- a/lean/OAI/Probability/SLE/Loewner/TraceWitnessUniqueness.lean +++ b/lean/OAI/Probability/SLE/Loewner/TraceWitnessUniqueness.lean @@ -1,57 +1,57 @@ -import OAI.Probability.SLE.StoppedMeasures - -namespace OAI - -/-! The reverse Loewner equation determines the inverse maps and hence -the continuous trace. This is deterministic uniqueness for a fixed driver. -/ - -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -theorem IsCapacityTwoWitness.inverse_unique - {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F H J : ℝ≥0 → ℂ → ℂ} - (hγ : IsCapacityTwoWitness U γ G F) (hβ : IsCapacityTwoWitness U β H J) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ upperHalfPlane) : F T z = J T z := by - let Q : ℝ → ℂ := fun t => G ((T : ℝ) - t).toNNReal (F T z) - have hQ0 : Q 0 = z := by - simp only [Q, sub_zero, Real.toNNReal_coe] - exact hγ.2.2.2.2.2.2.2.2.1 T z hz - have hQt : Q T = F T z := by - simp only [Q, sub_self, Real.toNNReal_zero] - exact hγ.initial_value _ (hγ.2.2.2.2.2.2.1 T hz).1.1 - have he := hβ.reverse_flow_identifies T z hz le_rfl Q hQ0 - (fun t ht => hγ.reverse_path_derivative T z hz le_rfl t ht) - rwa [hQt] at he - -theorem IsCapacityTwoWitness.trace_unique - {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F H J : ℝ≥0 → ℂ → ℂ} - (hγ : IsCapacityTwoWitness U γ G F) (hβ : IsCapacityTwoWitness U β H J) : γ = β := by - funext T - let y : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1) - have hy : Tendsto y atTop (𝓝[>] (0 : ℝ)) := - tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, - Eventually.of_forall (fun n => by change 0 < y n; dsimp only [y]; positivity)⟩ - have he (n : ℕ) : F T ((U T : ℂ) + (y n : ℂ) * Complex.I) = - J T ((U T : ℂ) + (y n : ℂ) * Complex.I) := by - apply hγ.inverse_unique hβ - change 0 < ((U T : ℂ) + (y n : ℂ) * Complex.I).im - simpa only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, - Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero, - zero_add] using (show 0 < y n by dsimp only [y]; positivity) - exact tendsto_nhds_unique (hγ.2.2.2.2.2.2.2.2.2.2.2.2 T |>.comp hy) - ((hβ.2.2.2.2.2.2.2.2.2.2.2.2 T |>.comp hy).congr (fun n => (he n).symm)) - -theorem IsCapacityTwoTrace.unique {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} - (hγ : IsCapacityTwoTrace U γ) (hβ : IsCapacityTwoTrace U β) : γ = β := by - obtain ⟨G, F, hγ⟩ := hγ - obtain ⟨H, J, hβ⟩ := hβ - exact hγ.trace_unique hβ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.StoppedMeasures + +namespace OAI + +/-! The reverse Loewner equation determines the inverse maps and hence +the continuous trace. This is deterministic uniqueness for a fixed driver. -/ + +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +theorem IsCapacityTwoWitness.inverse_unique + {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F H J : ℝ≥0 → ℂ → ℂ} + (hγ : IsCapacityTwoWitness U γ G F) (hβ : IsCapacityTwoWitness U β H J) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ upperHalfPlane) : F T z = J T z := by + let Q : ℝ → ℂ := fun t => G ((T : ℝ) - t).toNNReal (F T z) + have hQ0 : Q 0 = z := by + simp only [Q, sub_zero, Real.toNNReal_coe] + exact hγ.2.2.2.2.2.2.2.2.1 T z hz + have hQt : Q T = F T z := by + simp only [Q, sub_self, Real.toNNReal_zero] + exact hγ.initial_value _ (hγ.2.2.2.2.2.2.1 T hz).1.1 + have he := hβ.reverse_flow_identifies T z hz le_rfl Q hQ0 + (fun t ht => hγ.reverse_path_derivative T z hz le_rfl t ht) + rwa [hQt] at he + +theorem IsCapacityTwoWitness.trace_unique + {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} {G F H J : ℝ≥0 → ℂ → ℂ} + (hγ : IsCapacityTwoWitness U γ G F) (hβ : IsCapacityTwoWitness U β H J) : γ = β := by + funext T + let y : ℕ → ℝ := fun n => 1 / ((n : ℝ) + 1) + have hy : Tendsto y atTop (𝓝[>] (0 : ℝ)) := + tendsto_nhdsWithin_iff.mpr ⟨tendsto_one_div_add_atTop_nhds_zero_nat, + Eventually.of_forall (fun n => by change 0 < y n; dsimp only [y]; positivity)⟩ + have he (n : ℕ) : F T ((U T : ℂ) + (y n : ℂ) * Complex.I) = + J T ((U T : ℂ) + (y n : ℂ) * Complex.I) := by + apply hγ.inverse_unique hβ + change 0 < ((U T : ℂ) + (y n : ℂ) * Complex.I).im + simpa only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, + Complex.ofReal_re, Complex.I_im, Complex.I_re, mul_one, mul_zero, add_zero, + zero_add] using (show 0 < y n by dsimp only [y]; positivity) + exact tendsto_nhds_unique (hγ.2.2.2.2.2.2.2.2.2.2.2.2 T |>.comp hy) + ((hβ.2.2.2.2.2.2.2.2.2.2.2.2 T |>.comp hy).congr (fun n => (he n).symm)) + +theorem IsCapacityTwoTrace.unique {U : ℝ≥0 → ℝ} {γ β : ℝ≥0 → ℂ} + (hγ : IsCapacityTwoTrace U γ) (hβ : IsCapacityTwoTrace U β) : γ = β := by + obtain ⟨G, F, hγ⟩ := hγ + obtain ⟨H, J, hβ⟩ := hβ + exact hγ.trace_unique hβ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentArcRayCrossing.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentArcRayCrossing.lean index 5a5cdf5f981cac158b689a65b59b59b061aceef8..a195b65288faf6904e57bc4cb5f27056fed0e661 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentArcRayCrossing.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentArcRayCrossing.lean @@ -1,80 +1,80 @@ -import OAI.Probability.SLE.Geometry.TransverseCurveDifference -import OAI.Probability.SLE.Loewner.UnivalentInverseAnalytic -import OAI.Probability.SLE.DyadicStrips - -namespace OAI - -/-! Radial and angular image curves cross transversely under an actual -univalent half-plane map. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma univalent_polar_point_mem (x r θ : ℝ) (hr : 0 F ((x:ℂ)+(r:ℂ)*Complex.exp ((u:ℂ)*Complex.I))) - (deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) * - ((r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)*Complex.I)) θ := by - have hd := (hF.analyticAt (isOpen_upperHalfPlane.mem_nhds - (univalent_polar_point_mem x r θ hr hθ))).hasStrictDerivAt - have hi : HasStrictDerivAt (fun u : ℝ => (u:ℂ)*Complex.I) Complex.I θ := by - simpa using (Complex.ofRealCLM.hasStrictDerivAt (x := θ)).mul_const Complex.I - have he := ((Complex.hasStrictDerivAt_exp ((θ:ℂ)*Complex.I)).hasStrictFDerivAt.restrictScalars ℝ).comp_hasStrictDerivAt θ hi - have ha := (he.const_mul (r:ℂ)).const_add (x:ℂ) - have hf := (hd.hasStrictFDerivAt.restrictScalars ℝ).comp_hasStrictDerivAt θ ha - simpa [ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul,mul_assoc,mul_comm,mul_left_comm, - Function.comp_def] using hf - -lemma holomorphic_polar_ray_strict {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (x r θ : ℝ) (hr : 0 F ((x:ℂ)+(u:ℂ)*Complex.exp ((θ:ℂ)*Complex.I))) - (deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) * - Complex.exp ((θ:ℂ)*Complex.I)) r := by - have hd := (hF.analyticAt (isOpen_upperHalfPlane.mem_nhds - (univalent_polar_point_mem x r θ hr hθ))).hasStrictDerivAt - have ha : HasStrictDerivAt - (fun u : ℝ => (x:ℂ)+(u:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) - (Complex.exp ((θ:ℂ)*Complex.I)) r := by - simpa using ((Complex.ofRealCLM.hasStrictDerivAt (x := r)).mul_const - (Complex.exp ((θ:ℂ)*Complex.I))).const_add (x:ℂ) - have hf := (hd.hasStrictFDerivAt.restrictScalars ℝ).comp_hasStrictDerivAt r ha - simpa [ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul,mul_comm,Function.comp_def] using hf - -lemma univalent_arc_ray_locally_injective {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (hi : InjOn F upperHalfPlane) - (x r θ : ℝ) (hr : 0 - F ((x:ℂ)+(r:ℂ)*Complex.exp ((z.im:ℂ)*Complex.I))- - F ((x:ℂ)+(z.re:ℂ)*Complex.exp ((θ:ℂ)*Complex.I))) - (closedBall ((r:ℂ)+(θ:ℂ)*Complex.I) δ) := by - apply transverse_difference_locally_injective - (holomorphic_polar_arc_strict hF x r θ hr hθ) - (holomorphic_polar_ray_strict hF x r θ hr hθ) - let q := deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I))*Complex.exp ((θ:ℂ)*Complex.I) - have hq : q≠0 := mul_ne_zero - (univalent_deriv_ne_zero isOpen_upperHalfPlane hF hi (univalent_polar_point_mem x r θ hr hθ)) - (Complex.exp_ne_zero _) - have hα : deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) * - ((r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)*Complex.I)=(r:ℂ)*Complex.I*q := by - dsimp [q] - ring - rw [hα] - simpa only [Complex.ofReal_one,one_mul] using - transverse_radial_angular r 1 q hr.ne' one_ne_zero hq - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.TransverseCurveDifference +import OAI.Probability.SLE.Loewner.UnivalentInverseAnalytic +import OAI.Probability.SLE.DyadicStrips + +namespace OAI + +/-! Radial and angular image curves cross transversely under an actual +univalent half-plane map. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma univalent_polar_point_mem (x r θ : ℝ) (hr : 0 F ((x:ℂ)+(r:ℂ)*Complex.exp ((u:ℂ)*Complex.I))) + (deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) * + ((r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)*Complex.I)) θ := by + have hd := (hF.analyticAt (isOpen_upperHalfPlane.mem_nhds + (univalent_polar_point_mem x r θ hr hθ))).hasStrictDerivAt + have hi : HasStrictDerivAt (fun u : ℝ => (u:ℂ)*Complex.I) Complex.I θ := by + simpa using (Complex.ofRealCLM.hasStrictDerivAt (x := θ)).mul_const Complex.I + have he := ((Complex.hasStrictDerivAt_exp ((θ:ℂ)*Complex.I)).hasStrictFDerivAt.restrictScalars ℝ).comp_hasStrictDerivAt θ hi + have ha := (he.const_mul (r:ℂ)).const_add (x:ℂ) + have hf := (hd.hasStrictFDerivAt.restrictScalars ℝ).comp_hasStrictDerivAt θ ha + simpa [ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul,mul_assoc,mul_comm,mul_left_comm, + Function.comp_def] using hf + +lemma holomorphic_polar_ray_strict {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (x r θ : ℝ) (hr : 0 F ((x:ℂ)+(u:ℂ)*Complex.exp ((θ:ℂ)*Complex.I))) + (deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) * + Complex.exp ((θ:ℂ)*Complex.I)) r := by + have hd := (hF.analyticAt (isOpen_upperHalfPlane.mem_nhds + (univalent_polar_point_mem x r θ hr hθ))).hasStrictDerivAt + have ha : HasStrictDerivAt + (fun u : ℝ => (x:ℂ)+(u:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) + (Complex.exp ((θ:ℂ)*Complex.I)) r := by + simpa using ((Complex.ofRealCLM.hasStrictDerivAt (x := r)).mul_const + (Complex.exp ((θ:ℂ)*Complex.I))).const_add (x:ℂ) + have hf := (hd.hasStrictFDerivAt.restrictScalars ℝ).comp_hasStrictDerivAt r ha + simpa [ContinuousLinearMap.toSpanSingleton_apply,smul_eq_mul,mul_comm,Function.comp_def] using hf + +lemma univalent_arc_ray_locally_injective {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (hi : InjOn F upperHalfPlane) + (x r θ : ℝ) (hr : 0 + F ((x:ℂ)+(r:ℂ)*Complex.exp ((z.im:ℂ)*Complex.I))- + F ((x:ℂ)+(z.re:ℂ)*Complex.exp ((θ:ℂ)*Complex.I))) + (closedBall ((r:ℂ)+(θ:ℂ)*Complex.I) δ) := by + apply transverse_difference_locally_injective + (holomorphic_polar_arc_strict hF x r θ hr hθ) + (holomorphic_polar_ray_strict hF x r θ hr hθ) + let q := deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I))*Complex.exp ((θ:ℂ)*Complex.I) + have hq : q≠0 := mul_ne_zero + (univalent_deriv_ne_zero isOpen_upperHalfPlane hF hi (univalent_polar_point_mem x r θ hr hθ)) + (Complex.exp_ne_zero _) + have hα : deriv F ((x:ℂ)+(r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)) * + ((r:ℂ)*Complex.exp ((θ:ℂ)*Complex.I)*Complex.I)=(r:ℂ)*Complex.I*q := by + dsimp [q] + ring + rw [hα] + simpa only [Complex.ofReal_one,one_mul] using + transverse_radial_angular r 1 q hr.ne' one_ne_zero hq + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentFamilyPropagation.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentFamilyPropagation.lean index b08447232abe6604b379b13cdb8ebd2d3d34edb3..122dcc3c3747510fd57f27469e07ad0b01122c28 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentFamilyPropagation.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentFamilyPropagation.lean @@ -1,128 +1,128 @@ -import OAI.Probability.SLE.Loewner.UnivalentLocalBound -import OAI.Probability.SLE.Geometry.HolomorphicFamilyCompactness - -namespace OAI - -/-! A local bound for an injective holomorphic family propagates throughout - the half-plane. Only the coarse, proved Bloch bound is used. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -def familyBoundedLocus (f : ℕ → ℂ → ℂ) : Set ℂ := - {z | ∃ r : ℝ, 0 hC n v ?_⟩ - have h := dist_triangle v w z - rw [mem_ball] at hw hv ⊢ - linarith - -lemma univalent_family_enlarged_bound - (f : ℕ → ℂ → ℂ) (hf : ∀ n, DifferentiableOn ℂ (f n) upperHalfPlane) - (hi : ∀ n, InjOn (f n) upperHalfPlane) {z : ℂ} (hz : z∈upperHalfPlane) - (hb : z∈familyBoundedLocus f) : - ∃ B : ℝ, ∀ n, ∀ w∈ball z (z.im/(2*(128*Real.pi)^2)), ‖f n w‖≤B := by - obtain ⟨r,hr,C,hC⟩ := hb - let t := min (r/2) (z.im/2) - have ht : 0 ?_⟩ - have hsmall : w∈ball z (R/(128*Real.pi)^2) := by - have he : R/(128*Real.pi)^2=z.im/(2*(128*Real.pi)^2) := by dsimp [R]; ring - rw [he] - exact hw - have h := univalent_coarse_local_bound hR ((hf n).mono hRH) ((hi n).mono hRH) hsmall - calc - ‖f n w‖ ≤ ‖f n w-f n z‖+‖f n z‖ := norm_le_norm_sub_add _ _ - _ ≤ R*‖deriv (f n) z‖/(128*Real.pi)+C := - add_le_add h.le (hC n z (mem_ball_self hr)) - _ ≤ R*(C/t)/(128*Real.pi)+C := by gcongr; exact hder n - -lemma univalent_family_bound_propagates - (f : ℕ → ℂ → ℂ) (hf : ∀ n, DifferentiableOn ℂ (f n) upperHalfPlane) - (hi : ∀ n, InjOn (f n) upperHalfPlane) - (hseed : (upperHalfPlane ∩ familyBoundedLocus f).Nonempty) : - upperHalfPlane ⊆ familyBoundedLocus f := by - have hconn : IsPreconnected upperHalfPlane := - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected - apply hconn.subset_of_closure_inter_subset (isOpen_familyBoundedLocus f) hseed - rintro x ⟨hx,hxH⟩ - let Q : ℝ := (128*Real.pi)^2 - have hQ : 1≤Q := by dsimp [Q]; nlinarith [Real.pi_gt_three] - have hQpos : 0 hB n w ?_⟩ - rw [mem_ball] at hw ⊢ - have htri := dist_triangle w x y - change dist w y hb z.property - obtain ⟨t,ht⟩ := hK.elim_finite_subcover (fun z : K => ball z.1 (r z)) - (fun _ => isOpen_ball) (fun z hz => mem_iUnion.mpr ⟨⟨z,hz⟩,mem_ball_self (hr ⟨z,hz⟩)⟩) - refine ⟨∑ z∈t, max (C z) 0,fun n z hz => ?_⟩ - obtain ⟨a,ha,hza⟩ := mem_iUnion₂.mp (ht hz) - exact (hC a n z hza).trans ((le_max_left _ _).trans - (Finset.single_le_sum (fun a _ => le_max_right (C a) 0) ha)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentLocalBound +import OAI.Probability.SLE.Geometry.HolomorphicFamilyCompactness + +namespace OAI + +/-! A local bound for an injective holomorphic family propagates throughout + the half-plane. Only the coarse, proved Bloch bound is used. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +def familyBoundedLocus (f : ℕ → ℂ → ℂ) : Set ℂ := + {z | ∃ r : ℝ, 0 hC n v ?_⟩ + have h := dist_triangle v w z + rw [mem_ball] at hw hv ⊢ + linarith + +lemma univalent_family_enlarged_bound + (f : ℕ → ℂ → ℂ) (hf : ∀ n, DifferentiableOn ℂ (f n) upperHalfPlane) + (hi : ∀ n, InjOn (f n) upperHalfPlane) {z : ℂ} (hz : z∈upperHalfPlane) + (hb : z∈familyBoundedLocus f) : + ∃ B : ℝ, ∀ n, ∀ w∈ball z (z.im/(2*(128*Real.pi)^2)), ‖f n w‖≤B := by + obtain ⟨r,hr,C,hC⟩ := hb + let t := min (r/2) (z.im/2) + have ht : 0 ?_⟩ + have hsmall : w∈ball z (R/(128*Real.pi)^2) := by + have he : R/(128*Real.pi)^2=z.im/(2*(128*Real.pi)^2) := by dsimp [R]; ring + rw [he] + exact hw + have h := univalent_coarse_local_bound hR ((hf n).mono hRH) ((hi n).mono hRH) hsmall + calc + ‖f n w‖ ≤ ‖f n w-f n z‖+‖f n z‖ := norm_le_norm_sub_add _ _ + _ ≤ R*‖deriv (f n) z‖/(128*Real.pi)+C := + add_le_add h.le (hC n z (mem_ball_self hr)) + _ ≤ R*(C/t)/(128*Real.pi)+C := by gcongr; exact hder n + +lemma univalent_family_bound_propagates + (f : ℕ → ℂ → ℂ) (hf : ∀ n, DifferentiableOn ℂ (f n) upperHalfPlane) + (hi : ∀ n, InjOn (f n) upperHalfPlane) + (hseed : (upperHalfPlane ∩ familyBoundedLocus f).Nonempty) : + upperHalfPlane ⊆ familyBoundedLocus f := by + have hconn : IsPreconnected upperHalfPlane := + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected + apply hconn.subset_of_closure_inter_subset (isOpen_familyBoundedLocus f) hseed + rintro x ⟨hx,hxH⟩ + let Q : ℝ := (128*Real.pi)^2 + have hQ : 1≤Q := by dsimp [Q]; nlinarith [Real.pi_gt_three] + have hQpos : 0 hB n w ?_⟩ + rw [mem_ball] at hw ⊢ + have htri := dist_triangle w x y + change dist w y hb z.property + obtain ⟨t,ht⟩ := hK.elim_finite_subcover (fun z : K => ball z.1 (r z)) + (fun _ => isOpen_ball) (fun z hz => mem_iUnion.mpr ⟨⟨z,hz⟩,mem_ball_self (hr ⟨z,hz⟩)⟩) + refine ⟨∑ z∈t, max (C z) 0,fun n z hz => ?_⟩ + obtain ⟨a,ha,hza⟩ := mem_iUnion₂.mp (ht hz) + exact (hC a n z hza).trans ((le_max_left _ _).trans + (Finset.single_le_sum (fun a _ => le_max_right (C a) 0) ha)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentInverseAnalytic.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentInverseAnalytic.lean index 1a65ac0487c6c8be533bffa3fd1bbc70244879e9..0b9b2236f028669767e1cb84581ecc5239fccf57 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentInverseAnalytic.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentInverseAnalytic.lean @@ -1,106 +1,106 @@ -import Mathlib - -namespace OAI - -/-! The local inverse of an injective holomorphic map is holomorphic. -The proof uses the open mapping theorem and removes the isolated possible -singularity of the inverse, without a conformal mapping assumption. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma univalent_not_eventually_constant {D : Set ℂ} {f : ℂ → ℂ} - (hD : IsOpen D) (hi : InjOn f D) {a : ℂ} (ha : a ∈ D) : - ¬∀ᶠ z in 𝓝 a, f z=f a := by - intro h - have he : ∀ᶠ z in 𝓝[≠] a, z=a := by - filter_upwards [h.filter_mono nhdsWithin_le_nhds, - (show ∀ᶠ z in 𝓝 a, z∈D from hD.mem_nhds ha).filter_mono - nhdsWithin_le_nhds] with z hz hDz - exact hi hDz ha hz - have hn : ∀ᶠ z in 𝓝[≠] a, z≠a := self_mem_nhdsWithin - obtain ⟨z,hz,hnz⟩ := (he.and hn).exists - exact hnz hz - -lemma univalent_nhds_le_map {D : Set ℂ} {f : ℂ → ℂ} - (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - {a : ℂ} (ha : a ∈ D) : 𝓝 (f a) ≤ map f (𝓝 a) := - ((hf.analyticAt (hD.mem_nhds ha)).eventually_constant_or_nhds_le_map_nhds).resolve_left - (univalent_not_eventually_constant hD hi ha) - -lemma univalent_inverse_continuousAt {D : Set ℂ} {f : ℂ → ℂ} - (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - {a : ℂ} (ha : a ∈ D) : ContinuousAt (invFunOn f D) (f a) := by - have hleft : (invFunOn f D ∘ f) =ᶠ[𝓝 a] id := by - filter_upwards [hD.mem_nhds ha] with z hz - exact hi.leftInvOn_invFunOn hz - change map (invFunOn f D) (𝓝 (f a)) ≤ 𝓝 (invFunOn f D (f a)) - rw [hi.leftInvOn_invFunOn ha] - calc - map (invFunOn f D) (𝓝 (f a)) ≤ map (invFunOn f D) (map f (𝓝 a)) := - Filter.map_mono (univalent_nhds_le_map hD hf hi ha) - _ = 𝓝 a := by rw [Filter.map_map,Filter.map_congr hleft,Filter.map_id] - -lemma univalent_deriv_eventually_ne {D : Set ℂ} {f : ℂ → ℂ} - (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - {a : ℂ} (ha : a ∈ D) : ∀ᶠ z in 𝓝[≠] a, deriv f z≠0 := by - have han := hf.analyticAt (hD.mem_nhds ha) - apply (han.deriv.eventually_eq_or_eventually_ne analyticAt_const).resolve_left - intro hz - obtain ⟨r,hr,hball⟩ := Metric.nhds_basis_ball.mem_iff.mp (hz.and (hD.mem_nhds ha)) - have hbD : ball a r ⊆ D := fun z hz => (hball hz).2 - have he : ∀ᶠ z in 𝓝 a, f z=f a := by - filter_upwards [ball_mem_nhds a hr] with z hz - exact isOpen_ball.is_const_of_deriv_eq_zero (convex_ball a r).isPreconnected - (hf.mono hbD) (fun w hw => (hball hw).1) hz (mem_ball_self hr) - exact univalent_not_eventually_constant hD hi ha he - -lemma univalent_inverse_analyticAt {D : Set ℂ} {f : ℂ → ℂ} - (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - {a : ℂ} (ha : a ∈ D) : AnalyticAt ℂ (invFunOn f D) (f a) := by - have hc := univalent_inverse_continuousAt hD hf hi ha - apply Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt _ hc - have hne : ∀ᶠ z in 𝓝 a, z≠a → deriv f z≠0 := - eventually_nhdsWithin_iff.mp (univalent_deriv_eventually_ne hD hf hi ha) - have hpre : ∀ᶠ w in 𝓝 (f a), invFunOn f D w ≠ a → deriv f (invFunOn f D w) ≠ 0 := by - have hg : Tendsto (invFunOn f D) (𝓝 (f a)) (𝓝 a) := by - simpa only [hi.leftInvOn_invFunOn ha] using hc.tendsto - exact hg.eventually hne - have himage : f '' D ∈ 𝓝 (f a) := - univalent_nhds_le_map hD hf hi ha (image_mem_map (hD.mem_nhds ha)) - filter_upwards [hpre.filter_mono nhdsWithin_le_nhds, - (show ∀ᶠ w in 𝓝 (f a), w ∈ f '' D from himage).filter_mono nhdsWithin_le_nhds, - (show ∀ᶠ w in 𝓝[≠] (f a), w≠f a from self_mem_nhdsWithin)] with w hn hw hwa - have hzD := (surjOn_image f D).mapsTo_invFunOn hw - have hright := (surjOn_image f D).rightInvOn_invFunOn hw - have hz : invFunOn f D w≠a := by - intro h - exact hwa (hright.symm.trans (congrArg f h)) - have hleft : ∀ᶠ z in 𝓝 (invFunOn f D w), invFunOn f D (f z)=z := by - filter_upwards [hD.mem_nhds hzD] with z hz - exact hi.leftInvOn_invFunOn hz - have hd := (hf.analyticAt (hD.mem_nhds hzD)).hasStrictDerivAt.to_local_left_inverse - (g := invFunOn f D) (hn hz) hleft - rw [hright] at hd - exact hd.hasDerivAt.differentiableAt - -lemma univalent_deriv_ne_zero {D : Set ℂ} {f : ℂ → ℂ} - (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) - {a : ℂ} (ha : a ∈ D) : deriv f a≠0 := by - have hg := (univalent_inverse_analyticAt hD hf hi ha).differentiableAt.hasDerivAt - have h := hg.comp a (hf.differentiableAt (hD.mem_nhds ha)).hasDerivAt - have he : (invFunOn f D ∘ f) =ᶠ[𝓝 a] id := by - filter_upwards [hD.mem_nhds ha] with z hz - exact hi.leftInvOn_invFunOn hz - have hid := (he.hasDerivAt_iff).mp h - have hone := hid.unique (hasDerivAt_id a) - intro hz - exact zero_ne_one (by simpa only [hz,mul_zero] using hone) - -end SLEExactGauge - -end - -end OAI +import Mathlib + +namespace OAI + +/-! The local inverse of an injective holomorphic map is holomorphic. +The proof uses the open mapping theorem and removes the isolated possible +singularity of the inverse, without a conformal mapping assumption. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma univalent_not_eventually_constant {D : Set ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hi : InjOn f D) {a : ℂ} (ha : a ∈ D) : + ¬∀ᶠ z in 𝓝 a, f z=f a := by + intro h + have he : ∀ᶠ z in 𝓝[≠] a, z=a := by + filter_upwards [h.filter_mono nhdsWithin_le_nhds, + (show ∀ᶠ z in 𝓝 a, z∈D from hD.mem_nhds ha).filter_mono + nhdsWithin_le_nhds] with z hz hDz + exact hi hDz ha hz + have hn : ∀ᶠ z in 𝓝[≠] a, z≠a := self_mem_nhdsWithin + obtain ⟨z,hz,hnz⟩ := (he.and hn).exists + exact hnz hz + +lemma univalent_nhds_le_map {D : Set ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + {a : ℂ} (ha : a ∈ D) : 𝓝 (f a) ≤ map f (𝓝 a) := + ((hf.analyticAt (hD.mem_nhds ha)).eventually_constant_or_nhds_le_map_nhds).resolve_left + (univalent_not_eventually_constant hD hi ha) + +lemma univalent_inverse_continuousAt {D : Set ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + {a : ℂ} (ha : a ∈ D) : ContinuousAt (invFunOn f D) (f a) := by + have hleft : (invFunOn f D ∘ f) =ᶠ[𝓝 a] id := by + filter_upwards [hD.mem_nhds ha] with z hz + exact hi.leftInvOn_invFunOn hz + change map (invFunOn f D) (𝓝 (f a)) ≤ 𝓝 (invFunOn f D (f a)) + rw [hi.leftInvOn_invFunOn ha] + calc + map (invFunOn f D) (𝓝 (f a)) ≤ map (invFunOn f D) (map f (𝓝 a)) := + Filter.map_mono (univalent_nhds_le_map hD hf hi ha) + _ = 𝓝 a := by rw [Filter.map_map,Filter.map_congr hleft,Filter.map_id] + +lemma univalent_deriv_eventually_ne {D : Set ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + {a : ℂ} (ha : a ∈ D) : ∀ᶠ z in 𝓝[≠] a, deriv f z≠0 := by + have han := hf.analyticAt (hD.mem_nhds ha) + apply (han.deriv.eventually_eq_or_eventually_ne analyticAt_const).resolve_left + intro hz + obtain ⟨r,hr,hball⟩ := Metric.nhds_basis_ball.mem_iff.mp (hz.and (hD.mem_nhds ha)) + have hbD : ball a r ⊆ D := fun z hz => (hball hz).2 + have he : ∀ᶠ z in 𝓝 a, f z=f a := by + filter_upwards [ball_mem_nhds a hr] with z hz + exact isOpen_ball.is_const_of_deriv_eq_zero (convex_ball a r).isPreconnected + (hf.mono hbD) (fun w hw => (hball hw).1) hz (mem_ball_self hr) + exact univalent_not_eventually_constant hD hi ha he + +lemma univalent_inverse_analyticAt {D : Set ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + {a : ℂ} (ha : a ∈ D) : AnalyticAt ℂ (invFunOn f D) (f a) := by + have hc := univalent_inverse_continuousAt hD hf hi ha + apply Complex.analyticAt_of_differentiable_on_punctured_nhds_of_continuousAt _ hc + have hne : ∀ᶠ z in 𝓝 a, z≠a → deriv f z≠0 := + eventually_nhdsWithin_iff.mp (univalent_deriv_eventually_ne hD hf hi ha) + have hpre : ∀ᶠ w in 𝓝 (f a), invFunOn f D w ≠ a → deriv f (invFunOn f D w) ≠ 0 := by + have hg : Tendsto (invFunOn f D) (𝓝 (f a)) (𝓝 a) := by + simpa only [hi.leftInvOn_invFunOn ha] using hc.tendsto + exact hg.eventually hne + have himage : f '' D ∈ 𝓝 (f a) := + univalent_nhds_le_map hD hf hi ha (image_mem_map (hD.mem_nhds ha)) + filter_upwards [hpre.filter_mono nhdsWithin_le_nhds, + (show ∀ᶠ w in 𝓝 (f a), w ∈ f '' D from himage).filter_mono nhdsWithin_le_nhds, + (show ∀ᶠ w in 𝓝[≠] (f a), w≠f a from self_mem_nhdsWithin)] with w hn hw hwa + have hzD := (surjOn_image f D).mapsTo_invFunOn hw + have hright := (surjOn_image f D).rightInvOn_invFunOn hw + have hz : invFunOn f D w≠a := by + intro h + exact hwa (hright.symm.trans (congrArg f h)) + have hleft : ∀ᶠ z in 𝓝 (invFunOn f D w), invFunOn f D (f z)=z := by + filter_upwards [hD.mem_nhds hzD] with z hz + exact hi.leftInvOn_invFunOn hz + have hd := (hf.analyticAt (hD.mem_nhds hzD)).hasStrictDerivAt.to_local_left_inverse + (g := invFunOn f D) (hn hz) hleft + rw [hright] at hd + exact hd.hasDerivAt.differentiableAt + +lemma univalent_deriv_ne_zero {D : Set ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hf : DifferentiableOn ℂ f D) (hi : InjOn f D) + {a : ℂ} (ha : a ∈ D) : deriv f a≠0 := by + have hg := (univalent_inverse_analyticAt hD hf hi ha).differentiableAt.hasDerivAt + have h := hg.comp a (hf.differentiableAt (hD.mem_nhds ha)).hasDerivAt + have he : (invFunOn f D ∘ f) =ᶠ[𝓝 a] id := by + filter_upwards [hD.mem_nhds ha] with z hz + exact hi.leftInvOn_invFunOn hz + have hid := (he.hasDerivAt_iff).mp h + have hone := hid.unique (hasDerivAt_id a) + intro hz + exact zero_ne_one (by simpa only [hz,mul_zero] using hone) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentKoebeTransform.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentKoebeTransform.lean index 4070126913ad19989ab70d8e3794ed19842da41d..bd36fe089bd23cec752d45bd05185901cb4a477d 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentKoebeTransform.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentKoebeTransform.lean @@ -1,89 +1,89 @@ -import OAI.Probability.SLE.Geometry.DiskRecentering -import OAI.Probability.SLE.Loewner.UnivalentOddRoot - -namespace OAI - -/-! The exact second-derivative calculation for disk recentering. -A bound on the second coefficient of normalized univalent functions -therefore controls their logarithmic derivative throughout the disk. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma diskRecenter_second_deriv_zero {a : ℂ} (ha : ‖a‖<1) : - HasDerivAt (deriv (diskRecenter a)) (-2*star a*(1-star a*a)) 0 := by - let D (z : ℂ) := (1-star a*a)/(1+star a*z)^2 - have hd : HasDerivAt D (-2*star a*(1-star a*a)) 0 := by - have hh := (hasDerivAt_const (0:ℂ) (1-star a*a)).div - ((((hasDerivAt_id (0:ℂ)).const_mul (star a)).const_add 1).pow 2) - (by simp : (1+star a*(0:ℂ))^2≠0) - change HasDerivAt D _ 0 at hh - convert hh using 1 - simp - ring - apply hd.congr_of_eventuallyEq - filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz - exact (diskRecenter_hasDerivAt ha (mem_ball_zero_iff.mp hz)).deriv - -lemma deriv_comp_diskRecenter_zero {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) {a : ℂ} (ha : ‖a‖<1) : - deriv (f ∘ diskRecenter a) 0=deriv f a*(1-star a*a) := by - have hfa := hf.differentiableAt (isOpen_ball.mem_nhds (mem_ball_zero_iff.mpr ha)) - have hfa' : HasDerivAt f (deriv f a) (diskRecenter a 0) := by simpa using hfa.hasDerivAt - have hd := hfa'.comp 0 (diskRecenter_hasDerivAt ha (by simp)) - simpa using hd.deriv - -lemma second_deriv_comp_diskRecenter_zero {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) {a : ℂ} (ha : ‖a‖<1) : - deriv (deriv (f ∘ diskRecenter a)) 0 = - deriv (deriv f) a*(1-star a*a)^2-2*star a*deriv f a*(1-star a*a) := by - have hfa := hf.analyticAt (isOpen_ball.mem_nhds (mem_ball_zero_iff.mpr ha)) - have hDfa : HasDerivAt (deriv f) (deriv (deriv f) a) (diskRecenter a 0) := by - simpa using hfa.deriv.differentiableAt.hasDerivAt - have hφ := diskRecenter_hasDerivAt ha (by simp : ‖(0:ℂ)‖<1) - have hleft : HasDerivAt (deriv f ∘ diskRecenter a) - (deriv (deriv f) a*(1-star a*a)) 0 := by - simpa using hDfa.comp 0 hφ - have hprod := hleft.mul (diskRecenter_second_deriv_zero ha) - have hg : HasDerivAt (deriv (f ∘ diskRecenter a)) - (deriv (deriv f) a*(1-star a*a)*deriv (diskRecenter a) 0+ - deriv f (diskRecenter a 0)*(-2*star a*(1-star a*a))) 0 := by - apply hprod.congr_of_eventuallyEq - filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz - exact (deriv_comp z (hf.differentiableAt (isOpen_ball.mem_nhds (diskRecenter_maps ha hz))) - (diskRecenter_hasDerivAt ha (mem_ball_zero_iff.mp hz)).differentiableAt) - rw [hg.deriv] - rw [(diskRecenter_hasDerivAt ha (by simp : ‖(0:ℂ)‖<1)).deriv] - simp only [diskRecenter_zero,mul_zero,add_zero,one_pow,div_one] - ring - -lemma univalent_preSchwarzian_of_second_derivative - (hsecond : ∀ f : ℂ → ℂ, DifferentiableOn ℂ f (ball 0 1) → - InjOn f (ball 0 1) → ‖deriv (deriv f) 0‖≤4*‖deriv f 0‖) - {f : ℂ → ℂ} (hf : DifferentiableOn ℂ f (ball 0 1)) - (hi : InjOn f (ball 0 1)) {a : ℂ} (ha : ‖a‖<1) : - ‖((1-‖a‖^2:ℝ):ℂ)*deriv (deriv f) a/deriv f a-2*star a‖≤4 := by - let d : ℝ := 1-‖a‖^2 - have hd : 00, closedBall a r ⊆ D ∧ b∉closedBall a r ∧ - ∀ z∈sphere a r, f z≠f a := by - have hn : ¬∀ᶠ z in 𝓝 a, f z=f a := by - intro h - exact hnon ⟨f a,fun z hz => (hf.analyticOnNhd hD).eqOn_of_preconnected_of_eventuallyEq - analyticOnNhd_const hconn ha h hz⟩ - have hne := ((hf.analyticAt (hD.mem_nhds ha)).eventually_eq_or_eventually_ne - analyticAt_const).resolve_left hn - have hh : ∀ᶠ z in 𝓝 a, z∈D ∧ z≠b ∧ (z≠a → f z≠f a) := by - filter_upwards [hD.mem_nhds ha,isOpen_ne.mem_nhds hab, - eventually_nhdsWithin_iff.mp hne] with z hz hzb hza - exact ⟨hz,hzb,hza⟩ - obtain ⟨r,hr,hball⟩ := Metric.nhds_basis_closedBall.mem_iff.mp hh - refine ⟨r,hr,fun z hz => (hball hz).1,?_,?_⟩ - · intro hb - exact (hball hb).2.1 rfl - · intro z hz - exact (hball (sphere_subset_closedBall hz)).2.2 (ne_of_mem_sphere hz hr.ne') - -lemma tendstoLocallyUniformlyOn_injOn - {D : Set ℂ} {F : ℕ → ℂ → ℂ} {f : ℂ → ℂ} - (hD : IsOpen D) (hconn : IsPreconnected D) - (hF : ∀ n, DifferentiableOn ℂ (F n) D) (hi : ∀ n, InjOn (F n) D) - (hlim : TendstoLocallyUniformlyOn F f atTop D) - (hnon : ¬∃ c, ∀ z∈D, f z=c) : InjOn f D := by - have hf := hlim.differentiableOn (Eventually.of_forall hF) hD - intro a ha b hb he - by_contra hab - obtain ⟨r,hr,hKD,hbK,hsep⟩ := isolated_level_closedBall hD hconn hf hnon ha hab - have hc : ContinuousOn (fun z => ‖f z-f a‖) (sphere a r) := - ((hf.continuousOn.mono (sphere_subset_closedBall.trans hKD)).sub continuousOn_const).norm - obtain ⟨w,hw,hmin⟩ := (isCompact_sphere a r).exists_isMinOn - (NormedSpace.sphere_nonempty.mpr hr.le) hc - let δ : ℝ := ‖f w-f a‖ - have hδ : 0<δ := norm_pos_iff.mpr (sub_ne_zero.mpr (hsep w hw)) - have hlower (z : ℂ) (hz : z∈sphere a r) : δ≤‖f z-f a‖ := hmin hz - have hu := (tendstoLocallyUniformlyOn_iff_forall_isCompact hD).mp hlim - (closedBall a r) hKD (isCompact_closedBall a r) - have hε : 0<δ/8 := by positivity - have hk := (Metric.tendstoUniformlyOn_iff.mp hu) (δ/8) hε - have hbb := (Metric.tendsto_nhds.mp (hlim.tendsto_at hb)) (δ/8) hε - obtain ⟨n,hkn,hbn⟩ := (hk.and hbb).exists - have hnear (z : ℂ) (hz : z∈closedBall a r) : ‖F n z-f z‖<δ/8 := by - simpa only [dist_eq_norm,norm_sub_rev] using hkn z hz - have hnearB : ‖F n b-f b‖<δ/8 := by - simpa only [dist_eq_norm,norm_sub_rev] using hbn - have hnearA := hnear a (mem_closedBall_self hr.le) - have hboundary (z : ℂ) (hz : z∈sphere a r) : δ/2≤‖F n z-F n a‖ := by - have htri : ‖f z-f a‖ ≤ ‖f z-F n z‖+‖F n z-F n a‖+‖F n a-f a‖ := by - calc - _ = ‖(f z-F n z)+(F n z-F n a)+(F n a-f a)‖ := by congr 1; ring - _ ≤ _ := (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) - rw [norm_sub_rev (f z) (F n z)] at htri - have hznear := hnear z (sphere_subset_closedBall hz) - linarith [hlower z hz] - have hsmall : F n b∈ball (F n a) ((δ/2)/2) := by - rw [mem_ball,dist_eq_norm] - have htri : ‖F n b-F n a‖≤‖F n b-f b‖+‖F n a-f a‖ := by - rw [he] - have h := norm_sub_le_norm_sub_add_norm_sub (F n b) (f b) (F n a) - simpa only [norm_sub_rev (f b) (F n a)] using h - linarith - have hdc : DiffContOnCl ℂ (F n) (ball a r) := - ⟨(hF n).mono (ball_subset_closedBall.trans hKD),by - rw [closure_ball a hr.ne'] - exact (hF n).continuousOn.mono hKD⟩ - have hnot : ∃ᶠ z in 𝓝 a, F n z≠F n a := - not_eventually.mp (univalent_not_eventually_constant hD (hi n) ha) - obtain ⟨z,hz,heq⟩ := hdc.ball_subset_image_closedBall hr hboundary hnot hsmall - have hzb : z=b := hi n (hKD hz) hb heq - exact hbK (hzb ▸ hz) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentInverseAnalytic + +namespace OAI + +/-! A nonconstant locally uniform limit of injective holomorphic maps is +injective. Boundary separation and the minimum principle provide the +local preimages needed for the proof. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma isolated_level_closedBall {D : Set ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hconn : IsPreconnected D) (hf : DifferentiableOn ℂ f D) + (hnon : ¬∃ c, ∀ z∈D, f z=c) {a b : ℂ} (ha : a∈D) (hab : a≠b) : + ∃ r>0, closedBall a r ⊆ D ∧ b∉closedBall a r ∧ + ∀ z∈sphere a r, f z≠f a := by + have hn : ¬∀ᶠ z in 𝓝 a, f z=f a := by + intro h + exact hnon ⟨f a,fun z hz => (hf.analyticOnNhd hD).eqOn_of_preconnected_of_eventuallyEq + analyticOnNhd_const hconn ha h hz⟩ + have hne := ((hf.analyticAt (hD.mem_nhds ha)).eventually_eq_or_eventually_ne + analyticAt_const).resolve_left hn + have hh : ∀ᶠ z in 𝓝 a, z∈D ∧ z≠b ∧ (z≠a → f z≠f a) := by + filter_upwards [hD.mem_nhds ha,isOpen_ne.mem_nhds hab, + eventually_nhdsWithin_iff.mp hne] with z hz hzb hza + exact ⟨hz,hzb,hza⟩ + obtain ⟨r,hr,hball⟩ := Metric.nhds_basis_closedBall.mem_iff.mp hh + refine ⟨r,hr,fun z hz => (hball hz).1,?_,?_⟩ + · intro hb + exact (hball hb).2.1 rfl + · intro z hz + exact (hball (sphere_subset_closedBall hz)).2.2 (ne_of_mem_sphere hz hr.ne') + +lemma tendstoLocallyUniformlyOn_injOn + {D : Set ℂ} {F : ℕ → ℂ → ℂ} {f : ℂ → ℂ} + (hD : IsOpen D) (hconn : IsPreconnected D) + (hF : ∀ n, DifferentiableOn ℂ (F n) D) (hi : ∀ n, InjOn (F n) D) + (hlim : TendstoLocallyUniformlyOn F f atTop D) + (hnon : ¬∃ c, ∀ z∈D, f z=c) : InjOn f D := by + have hf := hlim.differentiableOn (Eventually.of_forall hF) hD + intro a ha b hb he + by_contra hab + obtain ⟨r,hr,hKD,hbK,hsep⟩ := isolated_level_closedBall hD hconn hf hnon ha hab + have hc : ContinuousOn (fun z => ‖f z-f a‖) (sphere a r) := + ((hf.continuousOn.mono (sphere_subset_closedBall.trans hKD)).sub continuousOn_const).norm + obtain ⟨w,hw,hmin⟩ := (isCompact_sphere a r).exists_isMinOn + (NormedSpace.sphere_nonempty.mpr hr.le) hc + let δ : ℝ := ‖f w-f a‖ + have hδ : 0<δ := norm_pos_iff.mpr (sub_ne_zero.mpr (hsep w hw)) + have hlower (z : ℂ) (hz : z∈sphere a r) : δ≤‖f z-f a‖ := hmin hz + have hu := (tendstoLocallyUniformlyOn_iff_forall_isCompact hD).mp hlim + (closedBall a r) hKD (isCompact_closedBall a r) + have hε : 0<δ/8 := by positivity + have hk := (Metric.tendstoUniformlyOn_iff.mp hu) (δ/8) hε + have hbb := (Metric.tendsto_nhds.mp (hlim.tendsto_at hb)) (δ/8) hε + obtain ⟨n,hkn,hbn⟩ := (hk.and hbb).exists + have hnear (z : ℂ) (hz : z∈closedBall a r) : ‖F n z-f z‖<δ/8 := by + simpa only [dist_eq_norm,norm_sub_rev] using hkn z hz + have hnearB : ‖F n b-f b‖<δ/8 := by + simpa only [dist_eq_norm,norm_sub_rev] using hbn + have hnearA := hnear a (mem_closedBall_self hr.le) + have hboundary (z : ℂ) (hz : z∈sphere a r) : δ/2≤‖F n z-F n a‖ := by + have htri : ‖f z-f a‖ ≤ ‖f z-F n z‖+‖F n z-F n a‖+‖F n a-f a‖ := by + calc + _ = ‖(f z-F n z)+(F n z-F n a)+(F n a-f a)‖ := by congr 1; ring + _ ≤ _ := (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) + rw [norm_sub_rev (f z) (F n z)] at htri + have hznear := hnear z (sphere_subset_closedBall hz) + linarith [hlower z hz] + have hsmall : F n b∈ball (F n a) ((δ/2)/2) := by + rw [mem_ball,dist_eq_norm] + have htri : ‖F n b-F n a‖≤‖F n b-f b‖+‖F n a-f a‖ := by + rw [he] + have h := norm_sub_le_norm_sub_add_norm_sub (F n b) (f b) (F n a) + simpa only [norm_sub_rev (f b) (F n a)] using h + linarith + have hdc : DiffContOnCl ℂ (F n) (ball a r) := + ⟨(hF n).mono (ball_subset_closedBall.trans hKD),by + rw [closure_ball a hr.ne'] + exact (hF n).continuousOn.mono hKD⟩ + have hnot : ∃ᶠ z in 𝓝 a, F n z≠F n a := + not_eventually.mp (univalent_not_eventually_constant hD (hi n) ha) + obtain ⟨z,hz,heq⟩ := hdc.ball_subset_image_closedBall hr hboundary hnot hsmall + have hzb : z=b := hi n (hKD hz) hb heq + exact hbK (hzb ▸ hz) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentMovingNormalization.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentMovingNormalization.lean index e6f4aacb68947a73110af661e242efb00e25a951..eb214120b881895978fffea5f5192ea23c7d3f3a 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentMovingNormalization.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentMovingNormalization.lean @@ -1,72 +1,72 @@ -import OAI.Probability.SLE.Loewner.UnivalentFamilyPropagation - -namespace OAI - -/-! Moving normalizations in a compact subset of the half-plane give a - locally bounded subsequence, using the coarse Bloch estimate. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma univalent_moving_normalization_subsequence - (f : ℕ → ℂ → ℂ) (z : ℕ → ℂ) (s M : ℝ) (hs : 0 hs.trans_le hw.2 - have hzK (n : ℕ) : z n∈K := by - exact ⟨by simpa only [mem_sphere,dist_zero_right] using (hz n).1,(hz n).2⟩ - obtain ⟨a,ha,φ,hφ,hlim⟩ := hK.tendsto_subseq hzK - let R := s/2 - let ρ := R/(128*Real.pi)^2 - have hR : 0 φ (n+N) - have hσ : StrictMono σ := hφ.comp (strictMono_id.add_const N) - have hball (n : ℕ) : ball (z (σ n)) R ⊆ upperHalfPlane := by - intro w hw - have him := Complex.abs_im_le_norm (w-z (σ n)) - have hdw : ‖w-z (σ n)‖ f (σ n))).Nonempty := by - refine ⟨a,hKH ha,ρ/2,by positivity,R*M/(128*Real.pi)+M,?_⟩ - intro n w hw - have hcenter : dist a (z (σ n))<ρ/2 := by - simpa only [Function.comp_apply,dist_comm] using hN (n+N) (Nat.le_add_left N n) - have hnear : w∈ball (z (σ n)) ρ := by - rw [mem_ball] at hw ⊢ - have := dist_triangle w a (z (σ n)) - linarith - have hg := univalent_coarse_local_bound hR ((hf (σ n)).mono (hball n)) - ((hi (σ n)).mono (hball n)) hnear - calc - ‖f (σ n) w‖ ≤ ‖f (σ n) w-f (σ n) (z (σ n))‖+‖f (σ n) (z (σ n))‖ := - norm_le_norm_sub_add _ _ - _ ≤ R*‖deriv (f (σ n)) (z (σ n))‖/(128*Real.pi)+M := add_le_add hg.le (hv (σ n)) - _ ≤ R*M/(128*Real.pi)+M := by gcongr; exact hd (σ n) - have hb := univalent_family_bound_propagates (fun n => f (σ n)) - (fun n => hf (σ n)) (fun n => hi (σ n)) hseed - refine ⟨σ,hσ,fun L hL hLH => ?_⟩ - exact familyBoundedLocus_compact_bound _ hL (hLH.trans hb) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentFamilyPropagation + +namespace OAI + +/-! Moving normalizations in a compact subset of the half-plane give a + locally bounded subsequence, using the coarse Bloch estimate. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma univalent_moving_normalization_subsequence + (f : ℕ → ℂ → ℂ) (z : ℕ → ℂ) (s M : ℝ) (hs : 0 hs.trans_le hw.2 + have hzK (n : ℕ) : z n∈K := by + exact ⟨by simpa only [mem_sphere,dist_zero_right] using (hz n).1,(hz n).2⟩ + obtain ⟨a,ha,φ,hφ,hlim⟩ := hK.tendsto_subseq hzK + let R := s/2 + let ρ := R/(128*Real.pi)^2 + have hR : 0 φ (n+N) + have hσ : StrictMono σ := hφ.comp (strictMono_id.add_const N) + have hball (n : ℕ) : ball (z (σ n)) R ⊆ upperHalfPlane := by + intro w hw + have him := Complex.abs_im_le_norm (w-z (σ n)) + have hdw : ‖w-z (σ n)‖ f (σ n))).Nonempty := by + refine ⟨a,hKH ha,ρ/2,by positivity,R*M/(128*Real.pi)+M,?_⟩ + intro n w hw + have hcenter : dist a (z (σ n))<ρ/2 := by + simpa only [Function.comp_apply,dist_comm] using hN (n+N) (Nat.le_add_left N n) + have hnear : w∈ball (z (σ n)) ρ := by + rw [mem_ball] at hw ⊢ + have := dist_triangle w a (z (σ n)) + linarith + have hg := univalent_coarse_local_bound hR ((hf (σ n)).mono (hball n)) + ((hi (σ n)).mono (hball n)) hnear + calc + ‖f (σ n) w‖ ≤ ‖f (σ n) w-f (σ n) (z (σ n))‖+‖f (σ n) (z (σ n))‖ := + norm_le_norm_sub_add _ _ + _ ≤ R*‖deriv (f (σ n)) (z (σ n))‖/(128*Real.pi)+M := add_le_add hg.le (hv (σ n)) + _ ≤ R*M/(128*Real.pi)+M := by gcongr; exact hd (σ n) + have hb := univalent_family_bound_propagates (fun n => f (σ n)) + (fun n => hf (σ n)) (fun n => hi (σ n)) hseed + refine ⟨σ,hσ,fun L hL hLH => ?_⟩ + exact familyBoundedLocus_compact_bound _ hL (hLH.trans hb) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentOddRoot.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentOddRoot.lean index b989f729adaad9be2f6b2dbcffcd1f211b5b772d..d6f27fbb5ff25f189a9d7361e29cbd6a4651ab96 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentOddRoot.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentOddRoot.lean @@ -1,121 +1,121 @@ -import OAI.Probability.SLE.Loewner.UnivalentInverseAnalytic -import Mathlib.Analysis.Complex.BranchLogRoot -import Mathlib.Analysis.Convex.Contractible - -namespace OAI - -/-! The normalized odd square-root transform used in the sharp disk -growth argument. The square root is constructed, not assumed. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma hasDerivAt_of_continuous_square_root {g q : ℂ → ℂ} {a d : ℂ} - (hg : HasDerivAt g d a) (hq : ContinuousAt q a) (hqa : q a≠0) - (he : ∀ᶠ z in 𝓝 a, q z^2=g z) : - HasDerivAt q (d/(2*q a)) a := by - have hea : q a^2=g a := he.self_of_nhds - have hn : q a+q a≠0 := by simpa only [←two_mul] using mul_ne_zero (by norm_num) hqa - have hsum : Tendsto (fun z => q z+q a) (𝓝[≠] a) (𝓝 (q a+q a)) := - (hq.add_const (q a)).tendsto.mono_left nhdsWithin_le_nhds - have ht := hg.tendsto_slope.div hsum hn - have heq : (fun z => slope g a z/(q z+q a)) =ᶠ[𝓝[≠] a] slope q a := by - filter_upwards [he.filter_mono nhdsWithin_le_nhds, - ((hq.add_const (q a)).eventually_ne hn).filter_mono nhdsWithin_le_nhds, - self_mem_nhdsWithin] with z hz hsum hza - simp only [slope_def_field,←hz,←hea] - field_simp - ring - apply hasDerivAt_iff_tendsto_slope.mpr - simpa only [←two_mul] using ht.congr' heq - -lemma differentiableOn_continuous_square_root {D : Set ℂ} (hD : IsOpen D) - {g q : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) (hq : ContinuousOn q D) - (he : ∀ z∈D, q z^2=g z) (hne : ∀ z∈D, g z≠0) : - DifferentiableOn ℂ q D := by - intro a ha - have hqa : q a≠0 := by intro h; apply hne a ha; rw [←he a ha,h,zero_pow (by norm_num)] - exact (hasDerivAt_of_continuous_square_root (hg.differentiableAt (hD.mem_nhds ha)).hasDerivAt - ((hq a ha).continuousAt (hD.mem_nhds ha)) hqa - (by filter_upwards [hD.mem_nhds ha] with z hz; exact he z hz)).differentiableAt.differentiableWithinAt - -lemma exists_normalized_analytic_square_root {g : ℂ → ℂ} - (hg : DifferentiableOn ℂ g (ball 0 1)) (hne : ∀ z∈ball (0:ℂ) 1, g z≠0) - (h0 : g 0=1) : - ∃ q : ℂ → ℂ, DifferentiableOn ℂ q (ball 0 1) ∧ q 0=1 ∧ - ∀ z∈ball (0:ℂ) 1, q z^2=g z := by - have hb : (ball (0:ℂ) 1).Nonempty := ⟨0,mem_ball_self zero_lt_one⟩ - have hcontract : ContractibleSpace ↥(ball (0:ℂ) 1) := - (convex_ball (0:ℂ) 1).contractibleSpace hb - have hsimply : IsSimplyConnected (ball (0:ℂ) 1) := by - change SimplyConnectedSpace ↥(ball (0:ℂ) 1) - infer_instance - obtain ⟨s,hsc,hs⟩ := Complex.exists_continuousOn_pow_eq hsimply isOpen_ball hg.continuousOn - (by rintro ⟨z,hz,hze⟩; exact hne z hz hze) (n := 2) (by norm_num) - have hsD := differentiableOn_continuous_square_root isOpen_ball hg hsc (fun z _ => hs z) hne - have hs0 : s 0^2=1 := (hs 0).trans h0 - refine ⟨fun z => s z*s 0,hsD.mul_const _,by simpa only [pow_two] using hs0,?_⟩ - intro z hz - rw [mul_pow,hs z,hs0,mul_one] - -/-- Every normalized disk-univalent map has a normalized odd univalent -square-root transform on the same disk. -/ -theorem univalent_odd_square_root {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) - (hf0 : f 0=0) (hd0 : deriv f 0=1) : - ∃ h : ℂ → ℂ, DifferentiableOn ℂ h (ball 0 1) ∧ InjOn h (ball 0 1) ∧ - h 0=0 ∧ deriv h 0=1 ∧ - (∀ z∈ball (0:ℂ) 1, h z^2=f (z^2)) ∧ (∀ z, h (-z)= -h z) := by - have hzero : (0:ℂ)∈ball 0 1 := mem_ball_self zero_lt_one - let g := dslope f 0 - have hg : DifferentiableOn ℂ g (ball 0 1) := - (Complex.differentiableOn_dslope (ball_mem_nhds (0:ℂ) zero_lt_one)).mpr hf - have hg0 : g 0=1 := by simpa [g] using hd0 - have hgne (z : ℂ) (hz : z∈ball (0:ℂ) 1) : g z≠0 := by - by_cases hz0 : z=0 - · simpa only [hz0,hg0] using one_ne_zero - · dsimp only [g] - rw [dslope_of_ne _ hz0,slope_def_field,hf0,sub_zero,sub_zero] - exact div_ne_zero (fun h => hz0 (hi hz hzero (h.trans hf0.symm))) hz0 - obtain ⟨q,hq,hq0,hq2⟩ := exists_normalized_analytic_square_root hg hgne hg0 - let h (z : ℂ) := z*q (z^2) - have hsq (z : ℂ) (hz : z∈ball (0:ℂ) 1) : z^2∈ball (0:ℂ) 1 := by - rw [mem_ball,dist_zero_right,norm_pow] - exact pow_lt_one₀ (norm_nonneg z) (by simpa using hz) (by norm_num) - have hh : DifferentiableOn ℂ h (ball 0 1) := - differentiableOn_id.mul (hq.comp (differentiable_id.pow 2).differentiableOn hsq) - have hh0 : h 0=0 := by simp [h] - have hhD : deriv h 0=1 := by - have hdq := ((hq.differentiableAt (isOpen_ball.mem_nhds (hsq 0 hzero))).hasDerivAt).comp 0 - (hasDerivAt_pow 2 (0:ℂ)) - have hd := (hasDerivAt_id (0:ℂ)).mul hdq - change HasDerivAt h _ 0 at hd - simpa [hq0] using hd.deriv - have hh2 (z : ℂ) (hz : z∈ball (0:ℂ) 1) : h z^2=f (z^2) := by - dsimp only [h] - rw [mul_pow,hq2 _ (hsq z hz)] - have he := sub_smul_dslope f 0 (z^2) - simpa only [sub_zero,hf0,smul_eq_mul] using he - have hodd (z : ℂ) : h (-z)= -h z := by simp [h] - refine ⟨h,hh,?_,hh0,hhD,hh2,hodd⟩ - intro z hz w hw he - have he2 : z^2=w^2 := hi (hsq z hz) (hsq w hw) - ((hh2 z hz).symm.trans ((congrArg (fun v : ℂ => v^2) he).trans (hh2 w hw))) - rcases (sq_eq_sq_iff_eq_or_eq_neg).mp he2 with hzw | hzw - · exact hzw - · have hhz : h z=0 := by - have he' : h z= -h z := he.trans (by rw [show w= -z by rw [hzw,neg_neg],hodd]) - linear_combination (1/2:ℂ)*he' - have hz2 : z^2=0 := hi (hsq z hz) hzero (by rw [←hh2 z hz,hhz,hf0]; simp) - have hz0 : z=0 := (pow_eq_zero_iff (by norm_num : (2:ℕ)≠0)).mp hz2 - rw [hz0] at hzw - rw [hz0] - simpa [eq_comm] using hzw - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentInverseAnalytic +import Mathlib.Analysis.Complex.BranchLogRoot +import Mathlib.Analysis.Convex.Contractible + +namespace OAI + +/-! The normalized odd square-root transform used in the sharp disk +growth argument. The square root is constructed, not assumed. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma hasDerivAt_of_continuous_square_root {g q : ℂ → ℂ} {a d : ℂ} + (hg : HasDerivAt g d a) (hq : ContinuousAt q a) (hqa : q a≠0) + (he : ∀ᶠ z in 𝓝 a, q z^2=g z) : + HasDerivAt q (d/(2*q a)) a := by + have hea : q a^2=g a := he.self_of_nhds + have hn : q a+q a≠0 := by simpa only [←two_mul] using mul_ne_zero (by norm_num) hqa + have hsum : Tendsto (fun z => q z+q a) (𝓝[≠] a) (𝓝 (q a+q a)) := + (hq.add_const (q a)).tendsto.mono_left nhdsWithin_le_nhds + have ht := hg.tendsto_slope.div hsum hn + have heq : (fun z => slope g a z/(q z+q a)) =ᶠ[𝓝[≠] a] slope q a := by + filter_upwards [he.filter_mono nhdsWithin_le_nhds, + ((hq.add_const (q a)).eventually_ne hn).filter_mono nhdsWithin_le_nhds, + self_mem_nhdsWithin] with z hz hsum hza + simp only [slope_def_field,←hz,←hea] + field_simp + ring + apply hasDerivAt_iff_tendsto_slope.mpr + simpa only [←two_mul] using ht.congr' heq + +lemma differentiableOn_continuous_square_root {D : Set ℂ} (hD : IsOpen D) + {g q : ℂ → ℂ} (hg : DifferentiableOn ℂ g D) (hq : ContinuousOn q D) + (he : ∀ z∈D, q z^2=g z) (hne : ∀ z∈D, g z≠0) : + DifferentiableOn ℂ q D := by + intro a ha + have hqa : q a≠0 := by intro h; apply hne a ha; rw [←he a ha,h,zero_pow (by norm_num)] + exact (hasDerivAt_of_continuous_square_root (hg.differentiableAt (hD.mem_nhds ha)).hasDerivAt + ((hq a ha).continuousAt (hD.mem_nhds ha)) hqa + (by filter_upwards [hD.mem_nhds ha] with z hz; exact he z hz)).differentiableAt.differentiableWithinAt + +lemma exists_normalized_analytic_square_root {g : ℂ → ℂ} + (hg : DifferentiableOn ℂ g (ball 0 1)) (hne : ∀ z∈ball (0:ℂ) 1, g z≠0) + (h0 : g 0=1) : + ∃ q : ℂ → ℂ, DifferentiableOn ℂ q (ball 0 1) ∧ q 0=1 ∧ + ∀ z∈ball (0:ℂ) 1, q z^2=g z := by + have hb : (ball (0:ℂ) 1).Nonempty := ⟨0,mem_ball_self zero_lt_one⟩ + have hcontract : ContractibleSpace ↥(ball (0:ℂ) 1) := + (convex_ball (0:ℂ) 1).contractibleSpace hb + have hsimply : IsSimplyConnected (ball (0:ℂ) 1) := by + change SimplyConnectedSpace ↥(ball (0:ℂ) 1) + infer_instance + obtain ⟨s,hsc,hs⟩ := Complex.exists_continuousOn_pow_eq hsimply isOpen_ball hg.continuousOn + (by rintro ⟨z,hz,hze⟩; exact hne z hz hze) (n := 2) (by norm_num) + have hsD := differentiableOn_continuous_square_root isOpen_ball hg hsc (fun z _ => hs z) hne + have hs0 : s 0^2=1 := (hs 0).trans h0 + refine ⟨fun z => s z*s 0,hsD.mul_const _,by simpa only [pow_two] using hs0,?_⟩ + intro z hz + rw [mul_pow,hs z,hs0,mul_one] + +/-- Every normalized disk-univalent map has a normalized odd univalent +square-root transform on the same disk. -/ +theorem univalent_odd_square_root {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) + (hf0 : f 0=0) (hd0 : deriv f 0=1) : + ∃ h : ℂ → ℂ, DifferentiableOn ℂ h (ball 0 1) ∧ InjOn h (ball 0 1) ∧ + h 0=0 ∧ deriv h 0=1 ∧ + (∀ z∈ball (0:ℂ) 1, h z^2=f (z^2)) ∧ (∀ z, h (-z)= -h z) := by + have hzero : (0:ℂ)∈ball 0 1 := mem_ball_self zero_lt_one + let g := dslope f 0 + have hg : DifferentiableOn ℂ g (ball 0 1) := + (Complex.differentiableOn_dslope (ball_mem_nhds (0:ℂ) zero_lt_one)).mpr hf + have hg0 : g 0=1 := by simpa [g] using hd0 + have hgne (z : ℂ) (hz : z∈ball (0:ℂ) 1) : g z≠0 := by + by_cases hz0 : z=0 + · simpa only [hz0,hg0] using one_ne_zero + · dsimp only [g] + rw [dslope_of_ne _ hz0,slope_def_field,hf0,sub_zero,sub_zero] + exact div_ne_zero (fun h => hz0 (hi hz hzero (h.trans hf0.symm))) hz0 + obtain ⟨q,hq,hq0,hq2⟩ := exists_normalized_analytic_square_root hg hgne hg0 + let h (z : ℂ) := z*q (z^2) + have hsq (z : ℂ) (hz : z∈ball (0:ℂ) 1) : z^2∈ball (0:ℂ) 1 := by + rw [mem_ball,dist_zero_right,norm_pow] + exact pow_lt_one₀ (norm_nonneg z) (by simpa using hz) (by norm_num) + have hh : DifferentiableOn ℂ h (ball 0 1) := + differentiableOn_id.mul (hq.comp (differentiable_id.pow 2).differentiableOn hsq) + have hh0 : h 0=0 := by simp [h] + have hhD : deriv h 0=1 := by + have hdq := ((hq.differentiableAt (isOpen_ball.mem_nhds (hsq 0 hzero))).hasDerivAt).comp 0 + (hasDerivAt_pow 2 (0:ℂ)) + have hd := (hasDerivAt_id (0:ℂ)).mul hdq + change HasDerivAt h _ 0 at hd + simpa [hq0] using hd.deriv + have hh2 (z : ℂ) (hz : z∈ball (0:ℂ) 1) : h z^2=f (z^2) := by + dsimp only [h] + rw [mul_pow,hq2 _ (hsq z hz)] + have he := sub_smul_dslope f 0 (z^2) + simpa only [sub_zero,hf0,smul_eq_mul] using he + have hodd (z : ℂ) : h (-z)= -h z := by simp [h] + refine ⟨h,hh,?_,hh0,hhD,hh2,hodd⟩ + intro z hz w hw he + have he2 : z^2=w^2 := hi (hsq z hz) (hsq w hw) + ((hh2 z hz).symm.trans ((congrArg (fun v : ℂ => v^2) he).trans (hh2 w hw))) + rcases (sq_eq_sq_iff_eq_or_eq_neg).mp he2 with hzw | hzw + · exact hzw + · have hhz : h z=0 := by + have he' : h z= -h z := he.trans (by rw [show w= -z by rw [hzw,neg_neg],hodd]) + linear_combination (1/2:ℂ)*he' + have hz2 : z^2=0 := hi (hsq z hz) hzero (by rw [←hh2 z hz,hhz,hf0]; simp) + have hz0 : z=0 := (pow_eq_zero_iff (by norm_num : (2:ℕ)≠0)).mp hz2 + rw [hz0] at hzw + rw [hz0] + simpa [eq_comm] using hzw + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentRadialDerivative.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentRadialDerivative.lean index 054f4ed45cd9430ab5e44c4c1d853047b9fcb4ac..55c23501b36d82625e31a11d5fa6f1aaee5a9168 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentRadialDerivative.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentRadialDerivative.lean @@ -1,127 +1,127 @@ -import OAI.Probability.SLE.Loewner.UnivalentKoebeTransform -import OAI.Probability.SLE.LoewnerCharts - -namespace OAI - -/-! Integrating the exact logarithmic-derivative inequality gives the -sharp radial derivative majorant. -/ -noncomputable section -open Set Filter Metric Function MeasureTheory -open scoped Topology - -namespace SLEExactGauge - -lemma disk_logarithmic_derivative_re_bound {f : ℂ → ℂ} - (hpre : ∀ a : ℂ, ‖a‖<1 → - ‖((1-‖a‖^2:ℝ):ℂ)*deriv (deriv f) a/deriv f a-2*star a‖≤4) - (u : ℂ) (hu : ‖u‖=1) (t : ℝ) (ht : t∈Ico 0 1) : - (u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)).re≤ - 3/(1-t)+1/(1+t) := by - have ha : ‖(t:ℂ)*u‖=t := by simp [hu,abs_of_nonneg ht.1] - have hd : 0<1-t^2 := by nlinarith [ht.1,ht.2] - have hunit : star u*u=1 := by - rw [Complex.star_def,Complex.conj_mul',hu] - norm_num - have hstarcast : star (t:ℂ)=(t:ℂ) := by apply Complex.ext <;> simp - have he : - (((1-t^2:ℝ):ℂ)*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)-2*star ((t:ℂ)*u))*u = - ((1-t^2:ℝ):ℂ)*(u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u))-((2*t:ℝ):ℂ) := by - calc - _ = ((1-t^2:ℝ):ℂ)*(u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u))- - 2*(t:ℂ)*(star u*u) := by simp only [star_mul,hstarcast]; ring - _ = _ := by rw [hunit]; push_cast; ring - have hn := hpre ((t:ℂ)*u) (by rw [ha]; exact ht.2) - rw [ha] at hn - have hRe := (Complex.re_le_norm _).trans (show - ‖((((1-t^2:ℝ):ℂ)*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)-2*star ((t:ℂ)*u))*u)‖≤4 by - rw [norm_mul,hu,mul_one]; exact hn) - rw [he] at hRe - simp only [Complex.sub_re,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] at hRe - have hb : (u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)).re≤(4+2*t)/(1-t^2) := - (le_div_iff₀ hd).mpr (by linarith) - have heq : (4+2*t)/(1-t^2)=3/(1-t)+1/(1+t) := by - have h1 : 1-t≠0 := by linarith [ht.2] - have h2 : 1+t≠0 := by linarith [ht.1] - field_simp - ring - exact hb.trans_eq heq - -lemma univalent_radial_derivative_bound {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) - (hpre : ∀ a : ℂ, ‖a‖<1 → - ‖((1-‖a‖^2:ℝ):ℂ)*deriv (deriv f) a/deriv f a-2*star a‖≤4) - (u : ℂ) (hu : ‖u‖=1) (r : ℝ) (hr : r∈Ico 0 1) : - ‖deriv f ((r:ℂ)*u)‖≤‖deriv f 0‖*(1+r)/(1-r)^3 := by - let y (t : ℝ) := deriv f ((t:ℂ)*u) - let c (t : ℝ) := u*deriv (deriv f) ((t:ℂ)*u)/y t - let q (t : ℝ) := 3/(1-t)+1/(1+t) - let Q (t : ℝ) := Real.log (1+t)-3*Real.log (1-t) - have hat (t : ℝ) (ht : t∈Icc 0 r) : (t:ℂ)*u∈ball 0 1 := by - rw [mem_ball_zero_iff,norm_mul,hu,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] - exact ht.2.trans_lt hr.2 - have hyt (t : ℝ) (ht : t∈Icc 0 r) : y t≠0 := - univalent_deriv_ne_zero isOpen_ball hf hi (hat t ht) - have hyd (t : ℝ) (ht : t∈Icc 0 r) : - HasDerivAt y (u*deriv (deriv f) ((t:ℂ)*u)) t := by - have ha := (hf.analyticAt (isOpen_ball.mem_nhds (hat t ht))).deriv - have hd := (ha.differentiableAt.hasDerivAt.hasFDerivAt.restrictScalars ℝ).comp_hasDerivAt t - ((Complex.ofRealCLM.hasDerivAt (x := t)).mul_const u) - simpa only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply, - smul_eq_mul,Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul,Function.comp_def,y] using hd - have hyc : ContinuousOn y (Icc 0 r) := fun t ht => (hyd t ht).continuousAt.continuousWithinAt - have hcc : ContinuousOn c (Icc 0 r) := by - apply ContinuousOn.div ?_ hyc hyt - apply continuousOn_const.mul - intro t ht - have houter : ContinuousAt (deriv (deriv f)) ((t:ℂ)*u) := - (hf.analyticAt (isOpen_ball.mem_nhds (hat t ht))).deriv.deriv.continuousAt - exact (ContinuousAt.comp (g := deriv (deriv f)) (f := fun s : ℝ => (s:ℂ)*u) - houter (by fun_prop)).continuousWithinAt - have hyode (t : ℝ) (ht : t∈Icc 0 r) : HasDerivWithinAt y (c t*y t) (Icc 0 r) t := by - have he : c t*y t=u*deriv (deriv f) ((t:ℂ)*u) := by dsimp only [c]; field_simp [hyt t ht] - rw [he] - exact (hyd t ht).hasDerivWithinAt - have hformula := complex_linear_ode_exp r hr.1 y c hcc hyode - have hqc : ContinuousOn q (Icc 0 r) := by - apply ContinuousOn.add - · exact continuousOn_const.div (by fun_prop) (fun t ht => by linarith [ht.2,hr.2]) - · exact continuousOn_const.div (by fun_prop) (fun t ht => by linarith [ht.1]) - have hQ (t : ℝ) (ht : t∈Icc 0 r) : HasDerivAt Q (q t) t := by - have hp : 1+t≠0 := by linarith [ht.1] - have hm : 1-t≠0 := by linarith [ht.2,hr.2] - have h := (((hasDerivAt_id t).const_add 1).log hp).sub - ((((hasDerivAt_id t).const_sub 1).log hm).const_mul 3) - change HasDerivAt Q _ t at h - convert h using 1 - dsimp [q] - ring - have hQI : ∫ t in (0:ℝ)..r, q t=Q r := by - have h := intervalIntegral.integral_eq_sub_of_hasDerivAt (a := 0) (b := r) - (fun t ht => hQ t (by simpa only [uIcc_of_le hr.1] using ht)) (hqc.intervalIntegrable_of_Icc hr.1) - simpa [Q] using h - have hbound : (∫ t in (0:ℝ)..r, c t).re≤Q r := by - have hre : (∫ t in (0:ℝ)..r, c t).re=∫ t in (0:ℝ)..r, (c t).re := - (Complex.reCLM.intervalIntegral_comp_comm (hcc.intervalIntegrable_of_Icc hr.1)).symm - rw [←hQI,hre] - apply intervalIntegral.integral_mono_on hr.1 - ((Complex.continuous_re.comp_continuousOn hcc).intervalIntegrable_of_Icc hr.1) (hqc.intervalIntegrable_of_Icc hr.1) - intro t ht - exact disk_logarithmic_derivative_re_bound hpre u hu t ⟨ht.1,ht.2.trans_lt hr.2⟩ - have hexp : Real.exp (Q r)=(1+r)/(1-r)^3 := by - have hp : 0<1+r := by linarith [hr.1] - have hm : 0<1-r := by linarith [hr.2] - dsimp [Q] - rw [Real.exp_sub,Real.exp_log hp,show 3*Real.log (1-r)=(3:ℕ)*Real.log (1-r) by norm_num, - Real.exp_nat_mul,Real.exp_log hm] - calc - ‖deriv f ((r:ℂ)*u)‖ = ‖y 0‖*Real.exp ((∫ t in (0:ℝ)..r, c t).re) := by - change ‖y r‖=_ - rw [hformula,norm_mul,Complex.norm_exp] - _ ≤ ‖y 0‖*Real.exp (Q r) := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hbound) (norm_nonneg _) - _ = ‖deriv f 0‖*(1+r)/(1-r)^3 := by simp only [y,Complex.ofReal_zero,zero_mul]; rw [hexp]; ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentKoebeTransform +import OAI.Probability.SLE.LoewnerCharts + +namespace OAI + +/-! Integrating the exact logarithmic-derivative inequality gives the +sharp radial derivative majorant. -/ +noncomputable section +open Set Filter Metric Function MeasureTheory +open scoped Topology + +namespace SLEExactGauge + +lemma disk_logarithmic_derivative_re_bound {f : ℂ → ℂ} + (hpre : ∀ a : ℂ, ‖a‖<1 → + ‖((1-‖a‖^2:ℝ):ℂ)*deriv (deriv f) a/deriv f a-2*star a‖≤4) + (u : ℂ) (hu : ‖u‖=1) (t : ℝ) (ht : t∈Ico 0 1) : + (u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)).re≤ + 3/(1-t)+1/(1+t) := by + have ha : ‖(t:ℂ)*u‖=t := by simp [hu,abs_of_nonneg ht.1] + have hd : 0<1-t^2 := by nlinarith [ht.1,ht.2] + have hunit : star u*u=1 := by + rw [Complex.star_def,Complex.conj_mul',hu] + norm_num + have hstarcast : star (t:ℂ)=(t:ℂ) := by apply Complex.ext <;> simp + have he : + (((1-t^2:ℝ):ℂ)*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)-2*star ((t:ℂ)*u))*u = + ((1-t^2:ℝ):ℂ)*(u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u))-((2*t:ℝ):ℂ) := by + calc + _ = ((1-t^2:ℝ):ℂ)*(u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u))- + 2*(t:ℂ)*(star u*u) := by simp only [star_mul,hstarcast]; ring + _ = _ := by rw [hunit]; push_cast; ring + have hn := hpre ((t:ℂ)*u) (by rw [ha]; exact ht.2) + rw [ha] at hn + have hRe := (Complex.re_le_norm _).trans (show + ‖((((1-t^2:ℝ):ℂ)*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)-2*star ((t:ℂ)*u))*u)‖≤4 by + rw [norm_mul,hu,mul_one]; exact hn) + rw [he] at hRe + simp only [Complex.sub_re,Complex.mul_re,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero] at hRe + have hb : (u*deriv (deriv f) ((t:ℂ)*u)/deriv f ((t:ℂ)*u)).re≤(4+2*t)/(1-t^2) := + (le_div_iff₀ hd).mpr (by linarith) + have heq : (4+2*t)/(1-t^2)=3/(1-t)+1/(1+t) := by + have h1 : 1-t≠0 := by linarith [ht.2] + have h2 : 1+t≠0 := by linarith [ht.1] + field_simp + ring + exact hb.trans_eq heq + +lemma univalent_radial_derivative_bound {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) + (hpre : ∀ a : ℂ, ‖a‖<1 → + ‖((1-‖a‖^2:ℝ):ℂ)*deriv (deriv f) a/deriv f a-2*star a‖≤4) + (u : ℂ) (hu : ‖u‖=1) (r : ℝ) (hr : r∈Ico 0 1) : + ‖deriv f ((r:ℂ)*u)‖≤‖deriv f 0‖*(1+r)/(1-r)^3 := by + let y (t : ℝ) := deriv f ((t:ℂ)*u) + let c (t : ℝ) := u*deriv (deriv f) ((t:ℂ)*u)/y t + let q (t : ℝ) := 3/(1-t)+1/(1+t) + let Q (t : ℝ) := Real.log (1+t)-3*Real.log (1-t) + have hat (t : ℝ) (ht : t∈Icc 0 r) : (t:ℂ)*u∈ball 0 1 := by + rw [mem_ball_zero_iff,norm_mul,hu,mul_one,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg ht.1] + exact ht.2.trans_lt hr.2 + have hyt (t : ℝ) (ht : t∈Icc 0 r) : y t≠0 := + univalent_deriv_ne_zero isOpen_ball hf hi (hat t ht) + have hyd (t : ℝ) (ht : t∈Icc 0 r) : + HasDerivAt y (u*deriv (deriv f) ((t:ℂ)*u)) t := by + have ha := (hf.analyticAt (isOpen_ball.mem_nhds (hat t ht))).deriv + have hd := (ha.differentiableAt.hasDerivAt.hasFDerivAt.restrictScalars ℝ).comp_hasDerivAt t + ((Complex.ofRealCLM.hasDerivAt (x := t)).mul_const u) + simpa only [ContinuousLinearMap.coe_restrictScalars',ContinuousLinearMap.toSpanSingleton_apply, + smul_eq_mul,Complex.ofRealCLM_apply,Complex.ofReal_one,one_mul,Function.comp_def,y] using hd + have hyc : ContinuousOn y (Icc 0 r) := fun t ht => (hyd t ht).continuousAt.continuousWithinAt + have hcc : ContinuousOn c (Icc 0 r) := by + apply ContinuousOn.div ?_ hyc hyt + apply continuousOn_const.mul + intro t ht + have houter : ContinuousAt (deriv (deriv f)) ((t:ℂ)*u) := + (hf.analyticAt (isOpen_ball.mem_nhds (hat t ht))).deriv.deriv.continuousAt + exact (ContinuousAt.comp (g := deriv (deriv f)) (f := fun s : ℝ => (s:ℂ)*u) + houter (by fun_prop)).continuousWithinAt + have hyode (t : ℝ) (ht : t∈Icc 0 r) : HasDerivWithinAt y (c t*y t) (Icc 0 r) t := by + have he : c t*y t=u*deriv (deriv f) ((t:ℂ)*u) := by dsimp only [c]; field_simp [hyt t ht] + rw [he] + exact (hyd t ht).hasDerivWithinAt + have hformula := complex_linear_ode_exp r hr.1 y c hcc hyode + have hqc : ContinuousOn q (Icc 0 r) := by + apply ContinuousOn.add + · exact continuousOn_const.div (by fun_prop) (fun t ht => by linarith [ht.2,hr.2]) + · exact continuousOn_const.div (by fun_prop) (fun t ht => by linarith [ht.1]) + have hQ (t : ℝ) (ht : t∈Icc 0 r) : HasDerivAt Q (q t) t := by + have hp : 1+t≠0 := by linarith [ht.1] + have hm : 1-t≠0 := by linarith [ht.2,hr.2] + have h := (((hasDerivAt_id t).const_add 1).log hp).sub + ((((hasDerivAt_id t).const_sub 1).log hm).const_mul 3) + change HasDerivAt Q _ t at h + convert h using 1 + dsimp [q] + ring + have hQI : ∫ t in (0:ℝ)..r, q t=Q r := by + have h := intervalIntegral.integral_eq_sub_of_hasDerivAt (a := 0) (b := r) + (fun t ht => hQ t (by simpa only [uIcc_of_le hr.1] using ht)) (hqc.intervalIntegrable_of_Icc hr.1) + simpa [Q] using h + have hbound : (∫ t in (0:ℝ)..r, c t).re≤Q r := by + have hre : (∫ t in (0:ℝ)..r, c t).re=∫ t in (0:ℝ)..r, (c t).re := + (Complex.reCLM.intervalIntegral_comp_comm (hcc.intervalIntegrable_of_Icc hr.1)).symm + rw [←hQI,hre] + apply intervalIntegral.integral_mono_on hr.1 + ((Complex.continuous_re.comp_continuousOn hcc).intervalIntegrable_of_Icc hr.1) (hqc.intervalIntegrable_of_Icc hr.1) + intro t ht + exact disk_logarithmic_derivative_re_bound hpre u hu t ⟨ht.1,ht.2.trans_lt hr.2⟩ + have hexp : Real.exp (Q r)=(1+r)/(1-r)^3 := by + have hp : 0<1+r := by linarith [hr.1] + have hm : 0<1-r := by linarith [hr.2] + dsimp [Q] + rw [Real.exp_sub,Real.exp_log hp,show 3*Real.log (1-r)=(3:ℕ)*Real.log (1-r) by norm_num, + Real.exp_nat_mul,Real.exp_log hm] + calc + ‖deriv f ((r:ℂ)*u)‖ = ‖y 0‖*Real.exp ((∫ t in (0:ℝ)..r, c t).re) := by + change ‖y r‖=_ + rw [hformula,norm_mul,Complex.norm_exp] + _ ≤ ‖y 0‖*Real.exp (Q r) := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hbound) (norm_nonneg _) + _ = ‖deriv f 0‖*(1+r)/(1-r)^3 := by simp only [y,Complex.ofReal_zero,zero_mul]; rw [hexp]; ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentRootFactor.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentRootFactor.lean index 8c190e239c9ee5bd94967a0cb5ede6e45ab97c50..7310967e91556303323b63e207f009eb42f55f90 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentRootFactor.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentRootFactor.lean @@ -1,66 +1,66 @@ -import OAI.Probability.SLE.Loewner.UnivalentOddRoot - -namespace OAI - -/-! The normalized odd-root construction with its analytic factor kept -explicit for coefficient extraction. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma univalent_root_factor {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) - (hf0 : f 0=0) (hd0 : deriv f 0=1) : - ∃ q : ℂ → ℂ, DifferentiableOn ℂ q (ball 0 1) ∧ q 0=1 ∧ - (∀ z∈ball (0:ℂ) 1, q z^2=dslope f 0 z) ∧ - (∀ z∈ball (0:ℂ) 1, q z≠0) ∧ - InjOn (fun z : ℂ => z*q (z^2)) (ball 0 1) := by - have hzero : (0:ℂ)∈ball 0 1 := mem_ball_self zero_lt_one - let g := dslope f 0 - have hg : DifferentiableOn ℂ g (ball 0 1) := - (Complex.differentiableOn_dslope (ball_mem_nhds (0:ℂ) zero_lt_one)).mpr hf - have hg0 : g 0=1 := by simpa [g] using hd0 - have hgne (z : ℂ) (hz : z∈ball (0:ℂ) 1) : g z≠0 := by - by_cases hz0 : z=0 - · simpa only [hz0,hg0] using one_ne_zero - · dsimp only [g] - rw [dslope_of_ne _ hz0,slope_def_field,hf0,sub_zero,sub_zero] - exact div_ne_zero (fun h => hz0 (hi hz hzero (h.trans hf0.symm))) hz0 - obtain ⟨q,hq,hq0,hq2⟩ := exists_normalized_analytic_square_root hg hgne hg0 - have hqne (z : ℂ) (hz : z∈ball (0:ℂ) 1) : q z≠0 := by - intro he - apply hgne z hz - rw [←hq2 z hz,he,zero_pow (by norm_num : (2:ℕ)≠0)] - refine ⟨q,hq,hq0,hq2,hqne,?_⟩ - let h (z : ℂ) := z*q (z^2) - have hs (z : ℂ) (hz : z∈ball (0:ℂ) 1) : z^2∈ball (0:ℂ) 1 := by - rw [mem_ball_zero_iff,norm_pow] - exact pow_lt_one₀ (norm_nonneg _) (mem_ball_zero_iff.mp hz) (by norm_num) - have hh2 (z : ℂ) (hz : z∈ball (0:ℂ) 1) : h z^2=f (z^2) := by - dsimp only [h] - rw [mul_pow,hq2 _ (hs z hz)] - have he := sub_smul_dslope f 0 (z^2) - simpa only [sub_zero,hf0,smul_eq_mul] using he - have hodd (z : ℂ) : h (-z)= -h z := by simp [h] - intro z hz w hw he - change h z=h w at he - have he2 : z^2=w^2 := hi (hs z hz) (hs w hw) - ((hh2 z hz).symm.trans ((congrArg (fun v : ℂ => v^2) he).trans (hh2 w hw))) - rcases (sq_eq_sq_iff_eq_or_eq_neg).mp he2 with hzw | hzw - · exact hzw - · have hhz : h z=0 := by - have he' : h z= -h z := he.trans (by rw [show w= -z by rw [hzw,neg_neg],hodd]) - linear_combination (1/2:ℂ)*he' - have hz2 : z^2=0 := hi (hs z hz) hzero (by rw [←hh2 z hz,hhz,hf0]; simp) - have hz0 : z=0 := (pow_eq_zero_iff (by norm_num : (2:ℕ)≠0)).mp hz2 - rw [hz0] at hzw - rw [hz0] - simpa [eq_comm] using hzw - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentOddRoot + +namespace OAI + +/-! The normalized odd-root construction with its analytic factor kept +explicit for coefficient extraction. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma univalent_root_factor {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) + (hf0 : f 0=0) (hd0 : deriv f 0=1) : + ∃ q : ℂ → ℂ, DifferentiableOn ℂ q (ball 0 1) ∧ q 0=1 ∧ + (∀ z∈ball (0:ℂ) 1, q z^2=dslope f 0 z) ∧ + (∀ z∈ball (0:ℂ) 1, q z≠0) ∧ + InjOn (fun z : ℂ => z*q (z^2)) (ball 0 1) := by + have hzero : (0:ℂ)∈ball 0 1 := mem_ball_self zero_lt_one + let g := dslope f 0 + have hg : DifferentiableOn ℂ g (ball 0 1) := + (Complex.differentiableOn_dslope (ball_mem_nhds (0:ℂ) zero_lt_one)).mpr hf + have hg0 : g 0=1 := by simpa [g] using hd0 + have hgne (z : ℂ) (hz : z∈ball (0:ℂ) 1) : g z≠0 := by + by_cases hz0 : z=0 + · simpa only [hz0,hg0] using one_ne_zero + · dsimp only [g] + rw [dslope_of_ne _ hz0,slope_def_field,hf0,sub_zero,sub_zero] + exact div_ne_zero (fun h => hz0 (hi hz hzero (h.trans hf0.symm))) hz0 + obtain ⟨q,hq,hq0,hq2⟩ := exists_normalized_analytic_square_root hg hgne hg0 + have hqne (z : ℂ) (hz : z∈ball (0:ℂ) 1) : q z≠0 := by + intro he + apply hgne z hz + rw [←hq2 z hz,he,zero_pow (by norm_num : (2:ℕ)≠0)] + refine ⟨q,hq,hq0,hq2,hqne,?_⟩ + let h (z : ℂ) := z*q (z^2) + have hs (z : ℂ) (hz : z∈ball (0:ℂ) 1) : z^2∈ball (0:ℂ) 1 := by + rw [mem_ball_zero_iff,norm_pow] + exact pow_lt_one₀ (norm_nonneg _) (mem_ball_zero_iff.mp hz) (by norm_num) + have hh2 (z : ℂ) (hz : z∈ball (0:ℂ) 1) : h z^2=f (z^2) := by + dsimp only [h] + rw [mul_pow,hq2 _ (hs z hz)] + have he := sub_smul_dslope f 0 (z^2) + simpa only [sub_zero,hf0,smul_eq_mul] using he + have hodd (z : ℂ) : h (-z)= -h z := by simp [h] + intro z hz w hw he + change h z=h w at he + have he2 : z^2=w^2 := hi (hs z hz) (hs w hw) + ((hh2 z hz).symm.trans ((congrArg (fun v : ℂ => v^2) he).trans (hh2 w hw))) + rcases (sq_eq_sq_iff_eq_or_eq_neg).mp he2 with hzw | hzw + · exact hzw + · have hhz : h z=0 := by + have he' : h z= -h z := he.trans (by rw [show w= -z by rw [hzw,neg_neg],hodd]) + linear_combination (1/2:ℂ)*he' + have hz2 : z^2=0 := hi (hs z hz) hzero (by rw [←hh2 z hz,hhz,hf0]; simp) + have hz0 : z=0 := (pow_eq_zero_iff (by norm_num : (2:ℕ)≠0)).mp hz2 + rw [hz0] at hzw + rw [hz0] + simpa [eq_comm] using hzw + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/UnivalentSecondCoefficient.lean b/lean/OAI/Probability/SLE/Loewner/UnivalentSecondCoefficient.lean index a6c50bc31ac4b121ff9987221d0b2a17c8a40ec5..8d98f890712efdcf68188f9f321615641f6f398c 100644 --- a/lean/OAI/Probability/SLE/Loewner/UnivalentSecondCoefficient.lean +++ b/lean/OAI/Probability/SLE/Loewner/UnivalentSecondCoefficient.lean @@ -1,83 +1,83 @@ -import OAI.Probability.SLE.Loewner.UnivalentRootFactor -import OAI.Probability.SLE.Geometry.RootCoefficientBound - -namespace OAI - -/-! The sharp second-coefficient estimate for disk-univalent functions, -obtained from the actual odd-root exterior map and its area inequality. -/ -noncomputable section -open Set Filter Metric Function -open scoped Topology - -namespace SLEExactGauge - -lemma second_derivative_eq_twice_dslope {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) (hf0 : f 0=0) : - deriv (deriv f) 0=2*deriv (dslope f 0) 0 := by - let g := dslope f 0 - have hg : DifferentiableOn ℂ g (ball 0 1) := - (Complex.differentiableOn_dslope (ball_mem_nhds (0:ℂ) zero_lt_one)).mpr hf - have he (z : ℂ) : f z=z*g z := by - have h := sub_smul_dslope f 0 z - simpa only [sub_zero,hf0,smul_eq_mul] using h.symm - have hD : deriv f =ᶠ[𝓝 (0:ℂ)] (fun z => g z+z*deriv g z) := by - filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz - have hd := (hasDerivAt_id z).mul ((hg.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt) - have hd' := hd.congr_of_eventuallyEq (Eventually.of_forall he) - simpa only [id_eq,one_mul] using hd'.deriv - have hga := hg.analyticAt (ball_mem_nhds (0:ℂ) zero_lt_one) - have hd := hga.differentiableAt.hasDerivAt.add - ((hasDerivAt_id (0:ℂ)).mul hga.deriv.differentiableAt.hasDerivAt) - have hh := (hd.congr_of_eventuallyEq hD).deriv - simpa only [id_eq,one_mul,zero_mul,add_zero,two_mul,g] using hh - -lemma normalized_univalent_second_derivative {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) - (hf0 : f 0=0) (hd0 : deriv f 0=1) : - ‖deriv (deriv f) 0‖≤4 := by - obtain ⟨q,hq,hq0,hq2,hqne,hqi⟩ := univalent_root_factor hf hi hf0 hd0 - have hqb := root_factor_derivative_le_one hq hq0 hqne hqi - have hqD := (hq.differentiableAt (ball_mem_nhds (0:ℂ) zero_lt_one)).hasDerivAt - have he : dslope f 0 =ᶠ[𝓝 (0:ℂ)] (fun z => q z^2) := by - filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz - exact (hq2 z hz).symm - have hgD := ((hqD.pow 2).congr_of_eventuallyEq he).deriv - have hg : deriv (dslope f 0) 0=2*deriv q 0 := by simpa [hq0] using hgD - have h4 : deriv (deriv f) 0=4*deriv q 0 := by - rw [second_derivative_eq_twice_dslope hf hf0,hg] - ring - rw [h4,norm_mul] - norm_num - linarith - -theorem univalent_second_derivative_bound {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) : - ‖deriv (deriv f) 0‖≤4*‖deriv f 0‖ := by - have hzero : (0:ℂ)∈ball 0 1 := mem_ball_self zero_lt_one - have hd : deriv f 0≠0 := univalent_deriv_ne_zero isOpen_ball hf hi hzero - let P (z : ℂ) := (f z-f 0)/deriv f 0 - have hP : DifferentiableOn ℂ P (ball 0 1) := (hf.sub_const _).div_const _ - have hPi : InjOn P (ball 0 1) := by - intro z hz w hw he - apply hi hz hw - have hh := (div_left_inj' hd).mp he - linear_combination hh - have hP0 : P 0=0 := by simp [P] - have hPD (z : ℂ) (hz : z∈ball (0:ℂ) 1) : deriv P z=deriv f z/deriv f 0 := by - exact (((hf.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.sub_const _).div_const _).deriv - have hP1 : deriv P 0=1 := by rw [hPD 0 hzero,div_self hd] - have hfA := hf.analyticAt (isOpen_ball.mem_nhds hzero) - have hP2 : deriv (deriv P) 0=deriv (deriv f) 0/deriv f 0 := by - have hh := hfA.deriv.differentiableAt.hasDerivAt.div_const (deriv f 0) - exact (hh.congr_of_eventuallyEq (by - filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz - exact hPD z hz)).deriv - have hh := normalized_univalent_second_derivative hP hPi hP0 hP1 - rw [hP2,norm_div] at hh - exact (div_le_iff₀ (norm_pos_iff.mpr hd)).mp hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.UnivalentRootFactor +import OAI.Probability.SLE.Geometry.RootCoefficientBound + +namespace OAI + +/-! The sharp second-coefficient estimate for disk-univalent functions, +obtained from the actual odd-root exterior map and its area inequality. -/ +noncomputable section +open Set Filter Metric Function +open scoped Topology + +namespace SLEExactGauge + +lemma second_derivative_eq_twice_dslope {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f (ball 0 1)) (hf0 : f 0=0) : + deriv (deriv f) 0=2*deriv (dslope f 0) 0 := by + let g := dslope f 0 + have hg : DifferentiableOn ℂ g (ball 0 1) := + (Complex.differentiableOn_dslope (ball_mem_nhds (0:ℂ) zero_lt_one)).mpr hf + have he (z : ℂ) : f z=z*g z := by + have h := sub_smul_dslope f 0 z + simpa only [sub_zero,hf0,smul_eq_mul] using h.symm + have hD : deriv f =ᶠ[𝓝 (0:ℂ)] (fun z => g z+z*deriv g z) := by + filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz + have hd := (hasDerivAt_id z).mul ((hg.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt) + have hd' := hd.congr_of_eventuallyEq (Eventually.of_forall he) + simpa only [id_eq,one_mul] using hd'.deriv + have hga := hg.analyticAt (ball_mem_nhds (0:ℂ) zero_lt_one) + have hd := hga.differentiableAt.hasDerivAt.add + ((hasDerivAt_id (0:ℂ)).mul hga.deriv.differentiableAt.hasDerivAt) + have hh := (hd.congr_of_eventuallyEq hD).deriv + simpa only [id_eq,one_mul,zero_mul,add_zero,two_mul,g] using hh + +lemma normalized_univalent_second_derivative {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) + (hf0 : f 0=0) (hd0 : deriv f 0=1) : + ‖deriv (deriv f) 0‖≤4 := by + obtain ⟨q,hq,hq0,hq2,hqne,hqi⟩ := univalent_root_factor hf hi hf0 hd0 + have hqb := root_factor_derivative_le_one hq hq0 hqne hqi + have hqD := (hq.differentiableAt (ball_mem_nhds (0:ℂ) zero_lt_one)).hasDerivAt + have he : dslope f 0 =ᶠ[𝓝 (0:ℂ)] (fun z => q z^2) := by + filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz + exact (hq2 z hz).symm + have hgD := ((hqD.pow 2).congr_of_eventuallyEq he).deriv + have hg : deriv (dslope f 0) 0=2*deriv q 0 := by simpa [hq0] using hgD + have h4 : deriv (deriv f) 0=4*deriv q 0 := by + rw [second_derivative_eq_twice_dslope hf hf0,hg] + ring + rw [h4,norm_mul] + norm_num + linarith + +theorem univalent_second_derivative_bound {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) : + ‖deriv (deriv f) 0‖≤4*‖deriv f 0‖ := by + have hzero : (0:ℂ)∈ball 0 1 := mem_ball_self zero_lt_one + have hd : deriv f 0≠0 := univalent_deriv_ne_zero isOpen_ball hf hi hzero + let P (z : ℂ) := (f z-f 0)/deriv f 0 + have hP : DifferentiableOn ℂ P (ball 0 1) := (hf.sub_const _).div_const _ + have hPi : InjOn P (ball 0 1) := by + intro z hz w hw he + apply hi hz hw + have hh := (div_left_inj' hd).mp he + linear_combination hh + have hP0 : P 0=0 := by simp [P] + have hPD (z : ℂ) (hz : z∈ball (0:ℂ) 1) : deriv P z=deriv f z/deriv f 0 := by + exact (((hf.differentiableAt (isOpen_ball.mem_nhds hz)).hasDerivAt.sub_const _).div_const _).deriv + have hP1 : deriv P 0=1 := by rw [hPD 0 hzero,div_self hd] + have hfA := hf.analyticAt (isOpen_ball.mem_nhds hzero) + have hP2 : deriv (deriv P) 0=deriv (deriv f) 0/deriv f 0 := by + have hh := hfA.deriv.differentiableAt.hasDerivAt.div_const (deriv f 0) + exact (hh.congr_of_eventuallyEq (by + filter_upwards [ball_mem_nhds (0:ℂ) zero_lt_one] with z hz + exact hPD z hz)).deriv + have hh := normalized_univalent_second_derivative hP hPi hP0 hP1 + rw [hP2,norm_div] at hh + exact (div_le_iff₀ (norm_pos_iff.mpr hd)).mp hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealBounds.lean b/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealBounds.lean index 570d488889c8d2878af5aa92c2b8b5dc5a1d9d06..26d77156d3dbb57a7e4e6d67849c4fdbc6d6098f 100644 --- a/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealBounds.lean +++ b/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealBounds.lean @@ -1,104 +1,104 @@ -import OAI.Probability.SLE.Geometry.OutwardRealBarrier -import OAI.Probability.SLE.Stochastic.ScalarExponentialLower - -namespace OAI - -/-! Vanishing forward height forces the real coordinate to remain -between the future driver bounds. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge - -lemma forward_height_lower_of_real_gap {a b ε : ℝ} (hε : 0 < ε) - (X y u : ℝ → ℝ) (hc : ContinuousOn y (Icc a b)) - (hd : ∀ t ∈ Ico a b, HasDerivWithinAt y - (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) - (hy : ∀ t ∈ Icc a b, 0 ≤ y t) - (hgap : ∀ t ∈ Ico a b, ε ≤ |X t-u t|) : - ∀ t ∈ Icc a b, y a*Real.exp (-(2/ε^2)*(t-a)) ≤ y t := by - apply scalar_exponential_lower y _ hc hd - intro t ht - have hsq := (sq_le_sq₀ hε.le (abs_nonneg (X t-u t))).mpr (hgap t ht) - rw [sq_abs] at hsq - have hden : ε^2 ≤ (X t-u t)^2+(y t)^2 := by nlinarith [sq_nonneg (y t)] - have hdv := div_le_div_of_nonneg_left - (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (hy t (Ico_subset_Icc_self ht))) - (sq_pos_of_pos hε) hden - calc - -(2/ε^2)*y t = -(2*y t/ε^2) := by ring - _ ≤ -(2*y t/((X t-u t)^2+(y t)^2)) := neg_le_neg hdv - _ = -2*y t/((X t-u t)^2+(y t)^2) := by ring - -theorem real_le_future_driver_bound_of_height_zero {a τ c : ℝ} (haτ : a < τ) - (X y u : ℝ → ℝ) - (hX : ContinuousOn X (Ico a τ)) (hyc : ContinuousOn y (Ico a τ)) - (hXd : ∀ t ∈ Ico a τ, HasDerivWithinAt X - (2*(X t-u t)/((X t-u t)^2+(y t)^2)) (Ici t) t) - (hyd : ∀ t ∈ Ico a τ, HasDerivWithinAt y - (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) - (hyp : ∀ t ∈ Ico a τ, 0 < y t) - (hu : ∀ t ∈ Ico a τ, u t ≤ c) - (hlim : Tendsto y (𝓝[<] τ) (𝓝 0)) : X a ≤ c := by - by_contra hn - have hca : c < X a := lt_of_not_ge hn - let ε : ℝ := X a-c - have hε : 0 < ε := sub_pos.mpr hca - have hya : 0 < y a := hyp a ⟨le_rfl,haτ⟩ - have hpoint (t : ℝ) (ht : t ∈ Ico a τ) : - y a*Real.exp (-(2/ε^2)*(τ-a)) ≤ y t := by - have hsub : Icc a t ⊆ Ico a τ := fun s hs => ⟨hs.1,hs.2.trans_lt ht.2⟩ - have hsub' : Ico a t ⊆ Ico a τ := fun s hs => ⟨hs.1,hs.2.trans ht.2⟩ - have hD (s : ℝ) (hs : s ∈ Ico a t) : 0 < (X s-u s)^2+(y s)^2 := by - have hp := hyp s (hsub' hs) - positivity - have hbar := outward_real_barrier X u (fun s => (X s-u s)^2+(y s)^2) - (hX.mono hsub) (fun s hs => hXd s (hsub' hs)) hD - (fun s hs => hu s (hsub' hs)) hca - have hlower := forward_height_lower_of_real_gap hε X y u (hyc.mono hsub) - (fun s hs => hyd s (hsub' hs)) (fun s hs => (hyp s (hsub hs)).le) - (fun s hs => ?_) t ⟨ht.1,le_rfl⟩ - · have hC : 0 ≤ 2/ε^2 := by positivity - have hexp : Real.exp (-(2/ε^2)*(τ-a)) ≤ Real.exp (-(2/ε^2)*(t-a)) := by - apply Real.exp_le_exp.mpr - nlinarith [mul_le_mul_of_nonneg_left ht.2.le hC] - exact (mul_le_mul_of_nonneg_left hexp hya.le).trans hlower - · have hb := hbar s (Ico_subset_Icc_self hs) - have hu' := hu s (hsub' hs) - exact (by dsimp [ε]; linarith : ε ≤ X s-u s).trans (le_abs_self _) - have hzero : y a*Real.exp (-(2/ε^2)*(τ-a)) ≤ 0 := by - apply ge_of_tendsto hlim - filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds haτ)] with t ht hta - exact hpoint t ⟨hta.le,ht⟩ - exact not_le_of_gt (mul_pos hya (Real.exp_pos _)) hzero - -theorem future_driver_lower_le_real_of_height_zero {a τ c : ℝ} (haτ : a < τ) - (X y u : ℝ → ℝ) - (hX : ContinuousOn X (Ico a τ)) (hyc : ContinuousOn y (Ico a τ)) - (hXd : ∀ t ∈ Ico a τ, HasDerivWithinAt X - (2*(X t-u t)/((X t-u t)^2+(y t)^2)) (Ici t) t) - (hyd : ∀ t ∈ Ico a τ, HasDerivWithinAt y - (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) - (hyp : ∀ t ∈ Ico a τ, 0 < y t) - (hu : ∀ t ∈ Ico a τ, c ≤ u t) - (hlim : Tendsto y (𝓝[<] τ) (𝓝 0)) : c ≤ X a := by - have hXd' (t : ℝ) (ht : t ∈ Ico a τ) : HasDerivWithinAt (fun s => -X s) - (2*((-X t)-(-u t))/(((-X t)-(-u t))^2+(y t)^2)) (Ici t) t := by - convert! (hXd t ht).neg using 1 - ring - have hyd' (t : ℝ) (ht : t ∈ Ico a τ) : HasDerivWithinAt y - (-2*y t/(((-X t)-(-u t))^2+(y t)^2)) (Ici t) t := by - convert! hyd t ht using 1 - congr 1 - ring - have hh := real_le_future_driver_bound_of_height_zero haτ - (fun t => -X t) y (fun t => -u t) hX.neg hyc hXd' hyd' hyp - (fun t ht => neg_le_neg (hu t ht)) hlim - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.OutwardRealBarrier +import OAI.Probability.SLE.Stochastic.ScalarExponentialLower + +namespace OAI + +/-! Vanishing forward height forces the real coordinate to remain +between the future driver bounds. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge + +lemma forward_height_lower_of_real_gap {a b ε : ℝ} (hε : 0 < ε) + (X y u : ℝ → ℝ) (hc : ContinuousOn y (Icc a b)) + (hd : ∀ t ∈ Ico a b, HasDerivWithinAt y + (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) + (hy : ∀ t ∈ Icc a b, 0 ≤ y t) + (hgap : ∀ t ∈ Ico a b, ε ≤ |X t-u t|) : + ∀ t ∈ Icc a b, y a*Real.exp (-(2/ε^2)*(t-a)) ≤ y t := by + apply scalar_exponential_lower y _ hc hd + intro t ht + have hsq := (sq_le_sq₀ hε.le (abs_nonneg (X t-u t))).mpr (hgap t ht) + rw [sq_abs] at hsq + have hden : ε^2 ≤ (X t-u t)^2+(y t)^2 := by nlinarith [sq_nonneg (y t)] + have hdv := div_le_div_of_nonneg_left + (mul_nonneg (by norm_num : (0:ℝ) ≤ 2) (hy t (Ico_subset_Icc_self ht))) + (sq_pos_of_pos hε) hden + calc + -(2/ε^2)*y t = -(2*y t/ε^2) := by ring + _ ≤ -(2*y t/((X t-u t)^2+(y t)^2)) := neg_le_neg hdv + _ = -2*y t/((X t-u t)^2+(y t)^2) := by ring + +theorem real_le_future_driver_bound_of_height_zero {a τ c : ℝ} (haτ : a < τ) + (X y u : ℝ → ℝ) + (hX : ContinuousOn X (Ico a τ)) (hyc : ContinuousOn y (Ico a τ)) + (hXd : ∀ t ∈ Ico a τ, HasDerivWithinAt X + (2*(X t-u t)/((X t-u t)^2+(y t)^2)) (Ici t) t) + (hyd : ∀ t ∈ Ico a τ, HasDerivWithinAt y + (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) + (hyp : ∀ t ∈ Ico a τ, 0 < y t) + (hu : ∀ t ∈ Ico a τ, u t ≤ c) + (hlim : Tendsto y (𝓝[<] τ) (𝓝 0)) : X a ≤ c := by + by_contra hn + have hca : c < X a := lt_of_not_ge hn + let ε : ℝ := X a-c + have hε : 0 < ε := sub_pos.mpr hca + have hya : 0 < y a := hyp a ⟨le_rfl,haτ⟩ + have hpoint (t : ℝ) (ht : t ∈ Ico a τ) : + y a*Real.exp (-(2/ε^2)*(τ-a)) ≤ y t := by + have hsub : Icc a t ⊆ Ico a τ := fun s hs => ⟨hs.1,hs.2.trans_lt ht.2⟩ + have hsub' : Ico a t ⊆ Ico a τ := fun s hs => ⟨hs.1,hs.2.trans ht.2⟩ + have hD (s : ℝ) (hs : s ∈ Ico a t) : 0 < (X s-u s)^2+(y s)^2 := by + have hp := hyp s (hsub' hs) + positivity + have hbar := outward_real_barrier X u (fun s => (X s-u s)^2+(y s)^2) + (hX.mono hsub) (fun s hs => hXd s (hsub' hs)) hD + (fun s hs => hu s (hsub' hs)) hca + have hlower := forward_height_lower_of_real_gap hε X y u (hyc.mono hsub) + (fun s hs => hyd s (hsub' hs)) (fun s hs => (hyp s (hsub hs)).le) + (fun s hs => ?_) t ⟨ht.1,le_rfl⟩ + · have hC : 0 ≤ 2/ε^2 := by positivity + have hexp : Real.exp (-(2/ε^2)*(τ-a)) ≤ Real.exp (-(2/ε^2)*(t-a)) := by + apply Real.exp_le_exp.mpr + nlinarith [mul_le_mul_of_nonneg_left ht.2.le hC] + exact (mul_le_mul_of_nonneg_left hexp hya.le).trans hlower + · have hb := hbar s (Ico_subset_Icc_self hs) + have hu' := hu s (hsub' hs) + exact (by dsimp [ε]; linarith : ε ≤ X s-u s).trans (le_abs_self _) + have hzero : y a*Real.exp (-(2/ε^2)*(τ-a)) ≤ 0 := by + apply ge_of_tendsto hlim + filter_upwards [self_mem_nhdsWithin,mem_nhdsWithin_of_mem_nhds (Ioi_mem_nhds haτ)] with t ht hta + exact hpoint t ⟨hta.le,ht⟩ + exact not_le_of_gt (mul_pos hya (Real.exp_pos _)) hzero + +theorem future_driver_lower_le_real_of_height_zero {a τ c : ℝ} (haτ : a < τ) + (X y u : ℝ → ℝ) + (hX : ContinuousOn X (Ico a τ)) (hyc : ContinuousOn y (Ico a τ)) + (hXd : ∀ t ∈ Ico a τ, HasDerivWithinAt X + (2*(X t-u t)/((X t-u t)^2+(y t)^2)) (Ici t) t) + (hyd : ∀ t ∈ Ico a τ, HasDerivWithinAt y + (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) + (hyp : ∀ t ∈ Ico a τ, 0 < y t) + (hu : ∀ t ∈ Ico a τ, c ≤ u t) + (hlim : Tendsto y (𝓝[<] τ) (𝓝 0)) : c ≤ X a := by + have hXd' (t : ℝ) (ht : t ∈ Ico a τ) : HasDerivWithinAt (fun s => -X s) + (2*((-X t)-(-u t))/(((-X t)-(-u t))^2+(y t)^2)) (Ici t) t := by + convert! (hXd t ht).neg using 1 + ring + have hyd' (t : ℝ) (ht : t ∈ Ico a τ) : HasDerivWithinAt y + (-2*y t/(((-X t)-(-u t))^2+(y t)^2)) (Ici t) t := by + convert! hyd t ht using 1 + congr 1 + ring + have hh := real_le_future_driver_bound_of_height_zero haτ + (fun t => -X t) y (fun t => -u t) hX.neg hyc hXd' hyd' hyp + (fun t ht => neg_le_neg (hu t ht)) hlim + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealLimit.lean b/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealLimit.lean index 7518dfd7a1d9631d10419523538ea3f323de6b5c..b4b7707b13f5eff454c795ab0c029549c6c996e4 100644 --- a/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/VanishingHeightRealLimit.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Loewner.VanishingHeightRealBounds - -namespace OAI - -/-! At a finite lifetime, vanishing height and continuity of the driver -force the forward real coordinate to approach the driver. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge - -theorem forward_real_tendsto_driver_of_height_zero {τ : ℝ} (hτ : 0 < τ) - (X y u : ℝ → ℝ) - (hX : ContinuousOn X (Ico 0 τ)) (hyc : ContinuousOn y (Ico 0 τ)) - (hXd : ∀ t ∈ Ico 0 τ, HasDerivWithinAt X - (2*(X t-u t)/((X t-u t)^2+(y t)^2)) (Ici t) t) - (hyd : ∀ t ∈ Ico 0 τ, HasDerivWithinAt y - (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) - (hyp : ∀ t ∈ Ico 0 τ, 0 < y t) - (hu : ContinuousAt u τ) - (hlim : Tendsto y (𝓝[<] τ) (𝓝 0)) : - Tendsto X (𝓝[<] τ) (𝓝 (u τ)) := by - apply tendsto_order.mpr - constructor - · intro c hc - let d : ℝ := (c+u τ)/2 - have hcd : c < d := by dsimp [d]; linarith - have hdu : d < u τ := by dsimp [d]; linarith - have he : ∀ᶠ t in 𝓝[<] τ, d < u t := - (hu.tendsto.mono_left nhdsWithin_le_nhds).eventually (lt_mem_nhds hdu) - obtain ⟨l,hl,hle⟩ := mem_nhdsLT_iff_exists_Ioo_subset.mp he - filter_upwards [Ioo_mem_nhdsLT (max_lt hl hτ)] with a ha - have hasub : Ico a τ ⊆ Ico 0 τ := - fun t ht => ⟨(le_max_right l 0).trans (ha.1.le.trans ht.1),ht.2⟩ - have hh := future_driver_lower_le_real_of_height_zero ha.2 X y u - (hX.mono hasub) (hyc.mono hasub) - (fun t ht => hXd t (hasub ht)) (fun t ht => hyd t (hasub ht)) - (fun t ht => hyp t (hasub ht)) - (fun t ht => (hle ⟨(le_max_left l 0).trans_lt (ha.1.trans_le ht.1),ht.2⟩).le) hlim - exact hcd.trans_le hh - · intro c hc - let d : ℝ := (c+u τ)/2 - have hdc : d < c := by dsimp [d]; linarith - have hud : u τ < d := by dsimp [d]; linarith - have he : ∀ᶠ t in 𝓝[<] τ, u t < d := - (hu.tendsto.mono_left nhdsWithin_le_nhds).eventually (gt_mem_nhds hud) - obtain ⟨l,hl,hle⟩ := mem_nhdsLT_iff_exists_Ioo_subset.mp he - filter_upwards [Ioo_mem_nhdsLT (max_lt hl hτ)] with a ha - have hasub : Ico a τ ⊆ Ico 0 τ := - fun t ht => ⟨(le_max_right l 0).trans (ha.1.le.trans ht.1),ht.2⟩ - have hh := real_le_future_driver_bound_of_height_zero ha.2 X y u - (hX.mono hasub) (hyc.mono hasub) - (fun t ht => hXd t (hasub ht)) (fun t ht => hyd t (hasub ht)) - (fun t ht => hyp t (hasub ht)) - (fun t ht => (hle ⟨(le_max_left l 0).trans_lt (ha.1.trans_le ht.1),ht.2⟩).le) hlim - exact hh.trans_lt hdc - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.VanishingHeightRealBounds + +namespace OAI + +/-! At a finite lifetime, vanishing height and continuity of the driver +force the forward real coordinate to approach the driver. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge + +theorem forward_real_tendsto_driver_of_height_zero {τ : ℝ} (hτ : 0 < τ) + (X y u : ℝ → ℝ) + (hX : ContinuousOn X (Ico 0 τ)) (hyc : ContinuousOn y (Ico 0 τ)) + (hXd : ∀ t ∈ Ico 0 τ, HasDerivWithinAt X + (2*(X t-u t)/((X t-u t)^2+(y t)^2)) (Ici t) t) + (hyd : ∀ t ∈ Ico 0 τ, HasDerivWithinAt y + (-2*y t/((X t-u t)^2+(y t)^2)) (Ici t) t) + (hyp : ∀ t ∈ Ico 0 τ, 0 < y t) + (hu : ContinuousAt u τ) + (hlim : Tendsto y (𝓝[<] τ) (𝓝 0)) : + Tendsto X (𝓝[<] τ) (𝓝 (u τ)) := by + apply tendsto_order.mpr + constructor + · intro c hc + let d : ℝ := (c+u τ)/2 + have hcd : c < d := by dsimp [d]; linarith + have hdu : d < u τ := by dsimp [d]; linarith + have he : ∀ᶠ t in 𝓝[<] τ, d < u t := + (hu.tendsto.mono_left nhdsWithin_le_nhds).eventually (lt_mem_nhds hdu) + obtain ⟨l,hl,hle⟩ := mem_nhdsLT_iff_exists_Ioo_subset.mp he + filter_upwards [Ioo_mem_nhdsLT (max_lt hl hτ)] with a ha + have hasub : Ico a τ ⊆ Ico 0 τ := + fun t ht => ⟨(le_max_right l 0).trans (ha.1.le.trans ht.1),ht.2⟩ + have hh := future_driver_lower_le_real_of_height_zero ha.2 X y u + (hX.mono hasub) (hyc.mono hasub) + (fun t ht => hXd t (hasub ht)) (fun t ht => hyd t (hasub ht)) + (fun t ht => hyp t (hasub ht)) + (fun t ht => (hle ⟨(le_max_left l 0).trans_lt (ha.1.trans_le ht.1),ht.2⟩).le) hlim + exact hcd.trans_le hh + · intro c hc + let d : ℝ := (c+u τ)/2 + have hdc : d < c := by dsimp [d]; linarith + have hud : u τ < d := by dsimp [d]; linarith + have he : ∀ᶠ t in 𝓝[<] τ, u t < d := + (hu.tendsto.mono_left nhdsWithin_le_nhds).eventually (gt_mem_nhds hud) + obtain ⟨l,hl,hle⟩ := mem_nhdsLT_iff_exists_Ioo_subset.mp he + filter_upwards [Ioo_mem_nhdsLT (max_lt hl hτ)] with a ha + have hasub : Ico a τ ⊆ Ico 0 τ := + fun t ht => ⟨(le_max_right l 0).trans (ha.1.le.trans ht.1),ht.2⟩ + have hh := real_le_future_driver_bound_of_height_zero ha.2 X y u + (hX.mono hasub) (hyc.mono hasub) + (fun t ht => hXd t (hasub ht)) (fun t ht => hyd t (hasub ht)) + (fun t ht => hyp t (hasub ht)) + (fun t ht => (hle ⟨(le_max_left l 0).trans_lt (ha.1.trans_le ht.1),ht.2⟩).le) hlim + exact hh.trans_lt hdc + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/VerticalDisplacementLimit.lean b/lean/OAI/Probability/SLE/Loewner/VerticalDisplacementLimit.lean index a6adfcad63361ea28a70d71ec4c89f36334a4baf..a0b2c45f8556b3488957399d758c57a319a7308b 100644 --- a/lean/OAI/Probability/SLE/Loewner/VerticalDisplacementLimit.lean +++ b/lean/OAI/Probability/SLE/Loewner/VerticalDisplacementLimit.lean @@ -1,41 +1,41 @@ -import OAI.Probability.SLE.Loewner.VerticalHalfPlaneUniqueness - -namespace OAI - -/-! Vertical displacement bounds imply the normalization used in half-plane -automorphism uniqueness. -/ -noncomputable section -open Set Filter Function Complex -open scoped Topology - -namespace SLEExactGauge - -lemma vertical_displacement_zero_of_bound {F : ℂ → ℂ} {C : ℝ} - (h : ∀ᶠ n : ℕ in atTop, ‖F (verticalInfinity n)-verticalInfinity n‖≤C/((n:ℝ)+1)) : - Tendsto (fun n : ℕ => F (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0) := by - apply squeeze_zero_norm' h - simpa only [div_eq_mul_inv,mul_zero] using - (tendsto_const_nhds.mul (tendsto_inv_atTop_zero.comp verticalHeights_tendsto) : - Tendsto (fun n : ℕ => C*((n:ℝ)+1)⁻¹) atTop (𝓝 (C*0))) - -lemma vertical_normalization_of_displacement {F : ℂ → ℂ} - (h : Tendsto (fun n : ℕ => F (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0)) : - Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I) := by - have hi : Tendsto (fun n : ℕ => ((((n:ℝ)+1)⁻¹:ℝ):ℂ)) atTop (𝓝 0) := by - exact_mod_cast continuous_ofReal.continuousAt.tendsto.comp - (tendsto_inv_atTop_zero.comp verticalHeights_tendsto) - have hh := (h.mul hi).add_const I - simp only [zero_mul,zero_add] at hh - apply hh.congr - intro n - have hn : (((n:ℝ)+1:ℝ):ℂ)≠0 := ofReal_ne_zero.mpr (by positivity) - dsimp only [verticalInfinity] - push_cast - field_simp - ring - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.VerticalHalfPlaneUniqueness + +namespace OAI + +/-! Vertical displacement bounds imply the normalization used in half-plane +automorphism uniqueness. -/ +noncomputable section +open Set Filter Function Complex +open scoped Topology + +namespace SLEExactGauge + +lemma vertical_displacement_zero_of_bound {F : ℂ → ℂ} {C : ℝ} + (h : ∀ᶠ n : ℕ in atTop, ‖F (verticalInfinity n)-verticalInfinity n‖≤C/((n:ℝ)+1)) : + Tendsto (fun n : ℕ => F (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0) := by + apply squeeze_zero_norm' h + simpa only [div_eq_mul_inv,mul_zero] using + (tendsto_const_nhds.mul (tendsto_inv_atTop_zero.comp verticalHeights_tendsto) : + Tendsto (fun n : ℕ => C*((n:ℝ)+1)⁻¹) atTop (𝓝 (C*0))) + +lemma vertical_normalization_of_displacement {F : ℂ → ℂ} + (h : Tendsto (fun n : ℕ => F (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0)) : + Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I) := by + have hi : Tendsto (fun n : ℕ => ((((n:ℝ)+1)⁻¹:ℝ):ℂ)) atTop (𝓝 0) := by + exact_mod_cast continuous_ofReal.continuousAt.tendsto.comp + (tendsto_inv_atTop_zero.comp verticalHeights_tendsto) + have hh := (h.mul hi).add_const I + simp only [zero_mul,zero_add] at hh + apply hh.congr + intro n + have hn : (((n:ℝ)+1:ℝ):ℂ)≠0 := ofReal_ne_zero.mpr (by positivity) + dsimp only [verticalInfinity] + push_cast + field_simp + ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/VerticalHalfPlaneUniqueness.lean b/lean/OAI/Probability/SLE/Loewner/VerticalHalfPlaneUniqueness.lean index 328c9c1c97f2bd218bdf7c4f9c8bc552e9bf3e2f..e06ea12136e4f3bb9c55e36622dc59e66e51dabc 100644 --- a/lean/OAI/Probability/SLE/Loewner/VerticalHalfPlaneUniqueness.lean +++ b/lean/OAI/Probability/SLE/Loewner/VerticalHalfPlaneUniqueness.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.Geometry.HalfPlaneNormalizedUniqueness - -namespace OAI - -/-! Normalized half-plane uniqueness using only the vertical asymptotics -preserved by bounded-capacity limits. -/ -noncomputable section -open Set Filter Complex Function -open scoped Topology - -namespace SLEExactGauge - -lemma halfPlane_height_of_vertical_normalization {F : ℂ → ℂ} - (hF : DifferentiableOn ℂ F upperHalfPlane) (hm : MapsTo F upperHalfPlane upperHalfPlane) - (hn : Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) - atTop (𝓝 I)) {z : ℂ} (hz : z∈upperHalfPlane) : z.im≤(F z).im := by - apply halfPlane_julia_at_infinity hF hm (fun n : ℕ => (n:ℝ)+1) - (fun n => by positivity) _ hn hz - exact tendsto_inv_atTop_zero.comp verticalHeights_tendsto - -lemma vertical_halfPlane_left_inverse_eq_id {G F : ℂ → ℂ} - (hG : DifferentiableOn ℂ G upperHalfPlane) (hF : DifferentiableOn ℂ F upperHalfPlane) - (hGm : MapsTo G upperHalfPlane upperHalfPlane) (hFm : MapsTo F upperHalfPlane upperHalfPlane) - (hFG : ∀ z∈upperHalfPlane, F (G z)=z) - (hGn : Tendsto (fun n : ℕ => G (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I)) - (hFn : Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I)) - (hG0 : Tendsto (fun n : ℕ => G (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0)) : - EqOn G id upperHalfPlane := by - have him (z : ℂ) (hz : z∈upperHalfPlane) : (G z-z).im=0 := by - have h1 := halfPlane_height_of_vertical_normalization hG hGm hGn hz - have h2 := halfPlane_height_of_vertical_normalization hF hFm hFn (hGm hz) - rw [hFG z hz] at h2 - simp only [sub_im] - linarith - have hc : IsConnected upperHalfPlane := - ⟨⟨I,by simp [upperHalfPlane]⟩,(convex_halfSpace_im_gt (0:ℝ)).isPreconnected⟩ - have ha := (hG.sub differentiableOn_id).analyticOnNhd isOpen_upperHalfPlane - obtain ⟨c,hc⟩ := ha.eq_const_of_im_eq_const him isOpen_upperHalfPlane hc - have ht : Tendsto (fun _ : ℕ => c) atTop (𝓝 0) := by - apply hG0.congr - intro n - exact hc _ (verticalInfinity_mem n) - have hc0 : c=0 := tendsto_nhds_unique tendsto_const_nhds ht - intro z hz - exact sub_eq_zero.mp ((hc z hz).trans hc0) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.HalfPlaneNormalizedUniqueness + +namespace OAI + +/-! Normalized half-plane uniqueness using only the vertical asymptotics +preserved by bounded-capacity limits. -/ +noncomputable section +open Set Filter Complex Function +open scoped Topology + +namespace SLEExactGauge + +lemma halfPlane_height_of_vertical_normalization {F : ℂ → ℂ} + (hF : DifferentiableOn ℂ F upperHalfPlane) (hm : MapsTo F upperHalfPlane upperHalfPlane) + (hn : Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) + atTop (𝓝 I)) {z : ℂ} (hz : z∈upperHalfPlane) : z.im≤(F z).im := by + apply halfPlane_julia_at_infinity hF hm (fun n : ℕ => (n:ℝ)+1) + (fun n => by positivity) _ hn hz + exact tendsto_inv_atTop_zero.comp verticalHeights_tendsto + +lemma vertical_halfPlane_left_inverse_eq_id {G F : ℂ → ℂ} + (hG : DifferentiableOn ℂ G upperHalfPlane) (hF : DifferentiableOn ℂ F upperHalfPlane) + (hGm : MapsTo G upperHalfPlane upperHalfPlane) (hFm : MapsTo F upperHalfPlane upperHalfPlane) + (hFG : ∀ z∈upperHalfPlane, F (G z)=z) + (hGn : Tendsto (fun n : ℕ => G (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I)) + (hFn : Tendsto (fun n : ℕ => F (verticalInfinity n)/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 I)) + (hG0 : Tendsto (fun n : ℕ => G (verticalInfinity n)-verticalInfinity n) atTop (𝓝 0)) : + EqOn G id upperHalfPlane := by + have him (z : ℂ) (hz : z∈upperHalfPlane) : (G z-z).im=0 := by + have h1 := halfPlane_height_of_vertical_normalization hG hGm hGn hz + have h2 := halfPlane_height_of_vertical_normalization hF hFm hFn (hGm hz) + rw [hFG z hz] at h2 + simp only [sub_im] + linarith + have hc : IsConnected upperHalfPlane := + ⟨⟨I,by simp [upperHalfPlane]⟩,(convex_halfSpace_im_gt (0:ℝ)).isPreconnected⟩ + have ha := (hG.sub differentiableOn_id).analyticOnNhd isOpen_upperHalfPlane + obtain ⟨c,hc⟩ := ha.eq_const_of_im_eq_const him isOpen_upperHalfPlane hc + have ht : Tendsto (fun _ : ℕ => c) atTop (𝓝 0) := by + apply hG0.congr + intro n + exact hc _ (verticalInfinity_mem n) + have hc0 : c=0 := tendsto_nhds_unique tendsto_const_nhds ht + intro z hz + exact sub_eq_zero.mp ((hc z hz).trans hc0) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/VerticalIntervalEnergy.lean b/lean/OAI/Probability/SLE/Loewner/VerticalIntervalEnergy.lean index 0ebf7f2ec1c18ecf4f62e7ede1a897e1eaaca535..567583170f9ba9b8b776066bd60fb5f666f3e0eb 100644 --- a/lean/OAI/Probability/SLE/Loewner/VerticalIntervalEnergy.lean +++ b/lean/OAI/Probability/SLE/Loewner/VerticalIntervalEnergy.lean @@ -1,54 +1,54 @@ -import OAI.Probability.SLE.Geometry.ComplexAngularEnergy - -namespace OAI - -/-! A vertical segment's squared oscillation is bounded by its height -times the one-dimensional derivative energy. -/ -noncomputable section -open Set Filter Function Complex Metric MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -lemma integral_norm_le_interval_energy {h : ℝ} (hh : 0 (1:ℝ)) 2 (volume.restrict (Ioc 0 h)) := memLp_const 1 - have hb := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two - (ae_of_all _ (fun y => norm_nonneg (g y))) (ae_of_all _ (fun _ => zero_le_one)) - (by simpa using hg.norm) (by simpa using h1) - simpa only [mul_one,Real.rpow_two,one_pow,integral_const,smul_eq_mul,one_mul, - Measure.real,Measure.restrict_apply_univ,Real.volume_Ioc,sub_zero, - ENNReal.toReal_ofReal hh.le,Real.sqrt_eq_rpow,mul_comm] using hb - -theorem complex_interval_energy_oscillation {h : ℝ} (hh : 0 - ordConnected_Ioc.uIcc_subset hx hy (uIoc_subset_uIcc ht) - have hi : IntegrableOn (fun t => ‖deriv f t‖) (Ioc 0 h) := - (hd.integrable (by norm_num)).norm - have hi' : IntegrableOn (deriv f) (Ioc 0 h) := hd.integrable (by norm_num) - have hid : IntervalIntegrable (deriv f) volume x y := - intervalIntegrable_iff.mpr (hi'.mono_set hsub) - have hb : ‖f y-f x‖≤Real.sqrt h*Real.sqrt (∫ t in Ioc 0 h, ‖deriv f t‖^2) := by - calc - ‖f y-f x‖ = ‖∫ t in x..y, deriv f t‖ := - congrArg norm (intervalIntegral.integral_deriv_eq_sub hf hid).symm - _ ≤ ∫ t in uIoc x y, ‖deriv f t‖ := intervalIntegral.norm_integral_le_integral_norm_uIoc - _ ≤ ∫ t in Ioc 0 h, ‖deriv f t‖ := - setIntegral_mono_set hi (ae_of_all _ (fun _ => norm_nonneg _)) hsub.eventuallySubset - _ ≤ _ := integral_norm_le_interval_energy hh hd - have hE : 0≤∫ t in Ioc 0 h, ‖deriv f t‖^2 := integral_nonneg (fun _ => sq_nonneg _) - have hs := mul_nonneg (Real.sqrt_nonneg h) (Real.sqrt_nonneg (∫ t in Ioc 0 h, ‖deriv f t‖^2)) - have hsquare := mul_le_mul hb hb (norm_nonneg _) hs - simpa only [←pow_two,mul_pow,Real.sq_sqrt hh.le,Real.sq_sqrt hE] using hsquare - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.ComplexAngularEnergy + +namespace OAI + +/-! A vertical segment's squared oscillation is bounded by its height +times the one-dimensional derivative energy. -/ +noncomputable section +open Set Filter Function Complex Metric MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma integral_norm_le_interval_energy {h : ℝ} (hh : 0 (1:ℝ)) 2 (volume.restrict (Ioc 0 h)) := memLp_const 1 + have hb := integral_mul_le_Lp_mul_Lq_of_nonneg Real.HolderConjugate.two_two + (ae_of_all _ (fun y => norm_nonneg (g y))) (ae_of_all _ (fun _ => zero_le_one)) + (by simpa using hg.norm) (by simpa using h1) + simpa only [mul_one,Real.rpow_two,one_pow,integral_const,smul_eq_mul,one_mul, + Measure.real,Measure.restrict_apply_univ,Real.volume_Ioc,sub_zero, + ENNReal.toReal_ofReal hh.le,Real.sqrt_eq_rpow,mul_comm] using hb + +theorem complex_interval_energy_oscillation {h : ℝ} (hh : 0 + ordConnected_Ioc.uIcc_subset hx hy (uIoc_subset_uIcc ht) + have hi : IntegrableOn (fun t => ‖deriv f t‖) (Ioc 0 h) := + (hd.integrable (by norm_num)).norm + have hi' : IntegrableOn (deriv f) (Ioc 0 h) := hd.integrable (by norm_num) + have hid : IntervalIntegrable (deriv f) volume x y := + intervalIntegrable_iff.mpr (hi'.mono_set hsub) + have hb : ‖f y-f x‖≤Real.sqrt h*Real.sqrt (∫ t in Ioc 0 h, ‖deriv f t‖^2) := by + calc + ‖f y-f x‖ = ‖∫ t in x..y, deriv f t‖ := + congrArg norm (intervalIntegral.integral_deriv_eq_sub hf hid).symm + _ ≤ ∫ t in uIoc x y, ‖deriv f t‖ := intervalIntegral.norm_integral_le_integral_norm_uIoc + _ ≤ ∫ t in Ioc 0 h, ‖deriv f t‖ := + setIntegral_mono_set hi (ae_of_all _ (fun _ => norm_nonneg _)) hsub.eventuallySubset + _ ≤ _ := integral_norm_le_interval_energy hh hd + have hE : 0≤∫ t in Ioc 0 h, ‖deriv f t‖^2 := integral_nonneg (fun _ => sq_nonneg _) + have hs := mul_nonneg (Real.sqrt_nonneg h) (Real.sqrt_nonneg (∫ t in Ioc 0 h, ‖deriv f t‖^2)) + have hsquare := mul_le_mul hb hb (norm_nonneg _) hs + simpa only [←pow_two,mul_pow,Real.sq_sqrt hh.le,Real.sq_sqrt hE] using hsquare + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Loewner/VerticalMapDerivative.lean b/lean/OAI/Probability/SLE/Loewner/VerticalMapDerivative.lean index 153de25fa82b4d342960d2d50196d2e3767e3c04..ba726cce21d28a06b4f3e1d1ded92dd8097786fc 100644 --- a/lean/OAI/Probability/SLE/Loewner/VerticalMapDerivative.lean +++ b/lean/OAI/Probability/SLE/Loewner/VerticalMapDerivative.lean @@ -1,48 +1,48 @@ -import OAI.Probability.SLE.Loewner.LoewnerVerticalLimit - -namespace OAI - -/-! The derivative of a holomorphic inverse limit is determined by its -explicit values on the positive imaginary axis. -/ -noncomputable section -open Set Filter Complex -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma vertical_axis_derivative (f : ℂ → ℂ) (T : ℝ≥0) - (hf : DifferentiableOn ℂ f upperHalfPlane) - (haxis : ∀ y : ℝ, 0 < y → f ((y:ℂ)*I)=verticalReversePath y T) - {y : ℝ} (hy : 0 < y) : - deriv f ((y:ℂ)*I) = ((y/Real.sqrt (y^2+4*T) : ℝ) : ℂ) := by - have hp : 0 < y^2+4*(T:ℝ) := by positivity - have hd := (((hasDerivAt_id y).pow 2).add_const (4*(T:ℝ))).sqrt hp.ne' - have hreal : HasDerivAt (fun s : ℝ => verticalReversePath s T) - (((y/Real.sqrt (y^2+4*T) : ℝ) : ℂ)*I) y := by - convert! hd.ofReal_comp.mul_const I using 1 - congr 2 - simp only [Pi.pow_apply,id_eq,mul_one] - ring - have hhol := hf.differentiableAt (isOpen_upperHalfPlane.mem_nhds - (show (y:ℂ)*I ∈ upperHalfPlane by simpa [upperHalfPlane] using hy)) - have harg : HasDerivAt (fun s : ℝ => (s:ℂ)*I) I y := by - simpa using (hasDerivAt_id y).ofReal_comp.mul_const I - have hc := (hhol.restrictScalars ℝ).hasFDerivAt.comp_hasDerivAt y harg - have hval : (fderiv ℝ f ((y:ℂ)*I)) I = I*deriv f ((y:ℂ)*I) := by - rw [hhol.fderiv_restrictScalars (𝕜 := ℝ)] - change (fderiv ℂ f ((y:ℂ)*I)) I = _ - rw [fderiv_eq_smul_deriv,smul_eq_mul] - rw [hval] at hc - have he : (fun s : ℝ => f ((s:ℂ)*I)) =ᶠ[𝓝 y] - (fun s : ℝ => verticalReversePath s T) := by - filter_upwards [Ioi_mem_nhds hy] with s hs - exact haxis s hs - have heq := (hc.congr_of_eventuallyEq he.symm).unique hreal - rw [mul_comm I] at heq - exact mul_right_cancel₀ I_ne_zero heq - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.LoewnerVerticalLimit + +namespace OAI + +/-! The derivative of a holomorphic inverse limit is determined by its +explicit values on the positive imaginary axis. -/ +noncomputable section +open Set Filter Complex +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma vertical_axis_derivative (f : ℂ → ℂ) (T : ℝ≥0) + (hf : DifferentiableOn ℂ f upperHalfPlane) + (haxis : ∀ y : ℝ, 0 < y → f ((y:ℂ)*I)=verticalReversePath y T) + {y : ℝ} (hy : 0 < y) : + deriv f ((y:ℂ)*I) = ((y/Real.sqrt (y^2+4*T) : ℝ) : ℂ) := by + have hp : 0 < y^2+4*(T:ℝ) := by positivity + have hd := (((hasDerivAt_id y).pow 2).add_const (4*(T:ℝ))).sqrt hp.ne' + have hreal : HasDerivAt (fun s : ℝ => verticalReversePath s T) + (((y/Real.sqrt (y^2+4*T) : ℝ) : ℂ)*I) y := by + convert! hd.ofReal_comp.mul_const I using 1 + congr 2 + simp only [Pi.pow_apply,id_eq,mul_one] + ring + have hhol := hf.differentiableAt (isOpen_upperHalfPlane.mem_nhds + (show (y:ℂ)*I ∈ upperHalfPlane by simpa [upperHalfPlane] using hy)) + have harg : HasDerivAt (fun s : ℝ => (s:ℂ)*I) I y := by + simpa using (hasDerivAt_id y).ofReal_comp.mul_const I + have hc := (hhol.restrictScalars ℝ).hasFDerivAt.comp_hasDerivAt y harg + have hval : (fderiv ℝ f ((y:ℂ)*I)) I = I*deriv f ((y:ℂ)*I) := by + rw [hhol.fderiv_restrictScalars (𝕜 := ℝ)] + change (fderiv ℂ f ((y:ℂ)*I)) I = _ + rw [fderiv_eq_smul_deriv,smul_eq_mul] + rw [hval] at hc + have he : (fun s : ℝ => f ((s:ℂ)*I)) =ᶠ[𝓝 y] + (fun s : ℝ => verticalReversePath s T) := by + filter_upwards [Ioi_mem_nhds hy] with s hs + exact haxis s hs + have heq := (hc.congr_of_eventuallyEq he.symm).unique hreal + rw [mul_comm I] at heq + exact mul_right_cancel₀ I_ne_zero heq + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/LoewnerCharts.lean b/lean/OAI/Probability/SLE/LoewnerCharts.lean index e9acdfb4bc10e18a383d27a9584193446911edb0..13f5387060759328673e2e430c0a45c64c671597 100644 --- a/lean/OAI/Probability/SLE/LoewnerCharts.lean +++ b/lean/OAI/Probability/SLE/LoewnerCharts.lean @@ -1,617 +1,617 @@ -import Mathlib -import OAI.Probability.SLE.ClippedCoordinates - -namespace OAI - -/-! Loewner Charts for chordal SLE. -/ - -section -section - -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal -namespace SLEExactGauge - - -def upperHalfPlane : Set ℂ := {z | 0 < z.im} - - - -def survivingDomain (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : Set ℂ := - {z | z ∈ upperHalfPlane \ (γ '' Icc 0 t) ∧ - ¬ Bornology.IsBounded (connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z)} - - - -def IsCapacityTwoWitness (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) - (G F : ℝ≥0 → ℂ → ℂ) : Prop := - Continuous γ ∧ γ 0 = 0 ∧ (∀ t, 0 ≤ (γ t).im) ∧ - (∀ t, DifferentiableOn ℂ (G t) (survivingDomain γ t)) ∧ - (∀ t, DifferentiableOn ℂ (F t) upperHalfPlane) ∧ - (∀ t, MapsTo (G t) (survivingDomain γ t) upperHalfPlane) ∧ - (∀ t, MapsTo (F t) upperHalfPlane (survivingDomain γ t)) ∧ - (∀ t z, z ∈ survivingDomain γ t → F t (G t z) = z) ∧ - (∀ t z, z ∈ upperHalfPlane → G t (F t z) = z) ∧ - (∀ z ∈ upperHalfPlane, G 0 z = z) ∧ - (∀ t z, z ∈ survivingDomain γ t → - HasDerivWithinAt (fun u : ℝ => G (Real.toNNReal u) z) - (2 / (G t z - (U t : ℂ))) (Ici 0) (t : ℝ)) ∧ - (∀ t, Tendsto (fun z : ℂ => z * (G t z - z)) - (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) (𝓝 (2 * (t : ℂ)))) ∧ - (∀ t, Tendsto (fun y : ℝ => F t ((U t : ℂ) + (y : ℂ) * Complex.I)) - (𝓝[>] 0) (𝓝 (γ t))) - -lemma survivingDomain_antitone (γ : ℝ≥0 → ℂ) : Antitone (survivingDomain γ) := by - intro s t hst z hz - have hpast : γ '' Icc 0 s ⊆ γ '' Icc 0 t := by - rintro _ ⟨u, hu, rfl⟩ - exact ⟨u, ⟨hu.1, hu.2.trans hst⟩, rfl⟩ - have hb : upperHalfPlane \ (γ '' Icc 0 t) ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by - intro w hw - exact ⟨hw.1, fun hh => hw.2 (hpast hh)⟩ - exact ⟨hb hz.1, fun hbounded => hz.2 (hbounded.subset (connectedComponentIn_mono z hb))⟩ - -lemma isOpen_upperHalfPlane : IsOpen upperHalfPlane := isOpen_lt continuous_const Complex.continuous_im - -lemma isOpen_survivingDomain {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) (t : ℝ≥0) : - IsOpen (survivingDomain γ t) := by - have hb : IsOpen (upperHalfPlane \ (γ '' Icc 0 t)) := - isOpen_upperHalfPlane.sdiff (isCompact_Icc.image hγ).isClosed - apply isOpen_iff_forall_mem_open.mpr - intro z hz - refine ⟨connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z, ?_, - hb.connectedComponentIn, mem_connectedComponentIn hz.1⟩ - intro w hw - refine ⟨connectedComponentIn_subset _ _ hw, ?_⟩ - rw [← connectedComponentIn_eq hw] - exact hz.2 - -lemma IsCapacityTwoWitness.time_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun v : ℝ => G v.toNNReal z) - (2 / (G u.toNNReal z - (U u.toNNReal : ℂ))) (Icc 0 (T : ℝ)) u := by - rcases h with ⟨hγ, hγ0, hγim, hG, hF, hGmap, hFmap, hinv, hinv', h0, hode, hnorm, htip⟩ - have hlu : u.toNNReal ≤ T := by - exact_mod_cast (show (u.toNNReal : ℝ) ≤ (T : ℝ) by rw [Real.coe_toNNReal _ hu.1]; exact hu.2) - have hh := (hode u.toNNReal z (survivingDomain_antitone γ hlu hz)).mono - (show Icc (0 : ℝ) T ⊆ Ici 0 from Icc_subset_Ici_self) - simpa only [Real.coe_toNNReal _ hu.1] using hh - -lemma IsCapacityTwoWitness.time_continuousOn - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) : - ContinuousOn (fun v : ℝ => G v.toNNReal z) (Icc 0 (T : ℝ)) := - fun u hu => (h.time_derivative T z hz u hu).continuousWithinAt - -lemma IsCapacityTwoWitness.imag_positive - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ t) : 0 < (G t z).im := h.2.2.2.2.2.1 t hz - -lemma IsCapacityTwoWitness.centered_ne_zero - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ t) : G t z - (U t : ℂ) ≠ 0 := by - intro hh - have hi := congrArg Complex.im hh - simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hi - exact (h.imag_positive t z hz).ne' hi - -lemma IsCapacityTwoWitness.imag_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun v : ℝ => (G v.toNNReal z).im) - (-2 * (G u.toNNReal z).im / Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ))) - (Icc 0 (T : ℝ)) u := by - have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u (h.time_derivative T z hz u hu) - simpa [Function.comp_def, Complex.div_im, neg_div] using hh - -lemma IsCapacityTwoWitness.imag_antitoneOn - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) : - AntitoneOn (fun v : ℝ => (G v.toNNReal z).im) (Icc 0 (T : ℝ)) := by - apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _) - (Complex.continuous_im.comp_continuousOn (h.time_continuousOn T z hz)) - (fun u hu => (h.imag_derivative T z hz u (interior_subset hu)).mono interior_subset) - intro u hu - have hu' : u ∈ Icc 0 (T : ℝ) := interior_subset hu - have hle : u.toNNReal ≤ T := (Real.toNNReal_le_iff_le_coe).mpr hu'.2 - exact div_nonpos_of_nonpos_of_nonneg - (mul_nonpos_of_nonpos_of_nonneg (by norm_num) - (h.imag_positive u.toNNReal z (survivingDomain_antitone γ hle hz)).le) - (Complex.normSq_nonneg _) - - - -lemma complex_linear_ode_exp (T : ℝ) (hT : 0 ≤ T) (f c : ℝ → ℂ) - (hc : ContinuousOn c (Icc 0 T)) - (hf : ∀ u ∈ Icc 0 T, HasDerivWithinAt f (c u * f u) (Icc 0 T) u) : - f T = f 0 * Complex.exp (∫ u in (0 : ℝ)..T, c u) := by - by_cases hT0 : T = 0 - · simp [hT0] - have hTp : 0 < T := lt_of_le_of_ne hT (Ne.symm hT0) - let C (u : ℝ) : ℂ := c (projIcc 0 T hT u) - have hC : Continuous C := hc.comp_continuous - (continuous_subtype_val.comp continuous_projIcc) (fun u => (projIcc 0 T hT u).property) - have hCc (u : ℝ) (hu : u ∈ Icc 0 T) : C u = c u := by simp only [C, projIcc_of_mem hT hu] - let I (u : ℝ) : ℂ := ∫ v in (0 : ℝ)..u, C v - have hI (u : ℝ) : HasDerivAt I (C u) u := - intervalIntegral.integral_hasDerivAt_right (hC.intervalIntegrable _ _) - hC.aestronglyMeasurable.stronglyMeasurableAtFilter hC.continuousAt - have hd (u : ℝ) (hu : u ∈ Icc 0 T) : - HasDerivWithinAt (fun v => f v * Complex.exp (-I v)) 0 (Icc 0 T) u := by - convert (hf u hu).mul ((hI u).neg.cexp.hasDerivWithinAt) using 1 - try rfl - rw [hCc u hu] - ring - have hid : ∀ u ∈ Icc 0 T, DifferentiableWithinAt ℝ (fun v => f v * Complex.exp (-I v)) (Icc 0 T) u := - fun u hu => (hd u hu).differentiableWithinAt - have he := constant_of_derivWithin_zero hid (fun u hu => - (hd u ⟨hu.1, hu.2.le⟩).derivWithin (uniqueDiffOn_Icc hTp u ⟨hu.1, hu.2.le⟩)) T ⟨hT, le_rfl⟩ - have hfactor : f T * Complex.exp (-I T) = f 0 := by - simpa only [I, intervalIntegral.integral_same, neg_zero, Complex.exp_zero, mul_one] using he - have hInt : I T = ∫ u in (0 : ℝ)..T, c u := by - apply intervalIntegral.integral_congr - intro u hu - exact hCc u (by simpa only [uIcc_of_le hT] using hu) - calc - f T = (f T * Complex.exp (-I T)) * Complex.exp (I T) := by - rw [mul_assoc, ← Complex.exp_add, neg_add_cancel, Complex.exp_zero, mul_one] - _ = _ := by rw [hfactor, hInt] - -lemma IsCapacityTwoWitness.live_at_real_time - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (_h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ≤ (T : ℝ)) : - z ∈ survivingDomain γ u.toNNReal := - survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu) hz - -lemma IsCapacityTwoWitness.spatial_differentiableAt - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ t) : DifferentiableAt ℂ (G t) z := - (h.2.2.2.1 t z hz).differentiableAt ((isOpen_survivingDomain h.1 t).mem_nhds hz) - -lemma IsCapacityTwoWitness.initial_value - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (hz : z ∈ upperHalfPlane) : G 0 z = z := - h.2.2.2.2.2.2.2.2.2.1 z hz - -noncomputable def loewnerPairCoefficient - (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ) (w z : ℂ) (u : ℝ) : ℂ := - (-2 : ℂ) / ((G u.toNNReal w - (U u.toNNReal : ℂ)) * - (G u.toNNReal z - (U u.toNNReal : ℂ))) - -lemma IsCapacityTwoWitness.pair_coefficient_continuousOn - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) - (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) : - ContinuousOn (loewnerPairCoefficient U G w z) (Icc 0 (T : ℝ)) := by - have hUc : Continuous (fun u : ℝ => (U u.toNNReal : ℂ)) := - Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal) - exact continuousOn_const.div - (((h.time_continuousOn T w hw).sub hUc.continuousOn).mul - ((h.time_continuousOn T z hz).sub hUc.continuousOn)) - (fun u hu => mul_ne_zero (h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2)) - (h.centered_ne_zero _ z (h.live_at_real_time T z hz u hu.2))) - -lemma IsCapacityTwoWitness.difference_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) - (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) - (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun v : ℝ => G v.toNNReal w - G v.toNNReal z) - (loewnerPairCoefficient U G w z u * (G u.toNNReal w - G u.toNNReal z)) - (Icc 0 (T : ℝ)) u := by - have hw0 := h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2) - have hz0 := h.centered_ne_zero _ z (h.live_at_real_time T z hz u hu.2) - have he : loewnerPairCoefficient U G w z u * (G u.toNNReal w - G u.toNNReal z) = - 2 / (G u.toNNReal w - (U u.toNNReal : ℂ)) - - 2 / (G u.toNNReal z - (U u.toNNReal : ℂ)) := by - unfold loewnerPairCoefficient - field_simp - ring - rw [he] - exact (h.time_derivative T w hw u hu).sub (h.time_derivative T z hz u hu) - -lemma IsCapacityTwoWitness.difference_formula - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) - (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) : - G T w - G T z = (w - z) * Complex.exp - (∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G w z u) := by - have hh := complex_linear_ode_exp (T : ℝ) T.coe_nonneg - (fun v : ℝ => G v.toNNReal w - G v.toNNReal z) - (loewnerPairCoefficient U G w z) - (h.pair_coefficient_continuousOn hU T w z hw hz) - (h.difference_derivative T w z hw hz) - simpa only [Real.toNNReal_coe, Real.toNNReal_zero, - h.initial_value w hw.1.1, h.initial_value z hz.1.1] using hh - -lemma IsCapacityTwoWitness.imag_ge_terminal - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - (G T z).im ≤ (G u.toNNReal z).im := by - simpa only [Real.toNNReal_coe] using h.imag_antitoneOn T z hz hu ⟨T.coe_nonneg, le_rfl⟩ hu.2 - -lemma IsCapacityTwoWitness.centered_norm_ge_terminal - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - (G T z).im ≤ ‖G u.toNNReal z - (U u.toNNReal : ℂ)‖ := by - apply (h.imag_ge_terminal T z hz u hu).trans - simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using - Complex.im_le_norm (G u.toNNReal z - (U u.toNNReal : ℂ)) - -lemma IsCapacityTwoWitness.pair_coefficient_bound - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) - (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) - (c : ℝ) (hc : 0 < c) (hcw : c ≤ (G T w).im) (hcz : c ≤ (G T z).im) - (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - ‖loewnerPairCoefficient U G w z u‖ ≤ 2 / (c * c) := by - have hw' := hcw.trans (h.centered_norm_ge_terminal T w hw u hu) - have hz' := hcz.trans (h.centered_norm_ge_terminal T z hz u hu) - unfold loewnerPairCoefficient - rw [norm_div, norm_mul, norm_neg, Complex.norm_two] - exact div_le_div_of_nonneg_left (by norm_num) (mul_pos hc hc) - (mul_le_mul hw' hz' hc.le (norm_nonneg _)) - -lemma IsCapacityTwoWitness.pair_coefficient_intervalIntegrable - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) - (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) : - IntervalIntegrable (loewnerPairCoefficient U G w z) volume 0 (T : ℝ) := by - apply ContinuousOn.intervalIntegrable - simpa only [uIcc_of_le T.coe_nonneg] using h.pair_coefficient_continuousOn hU T w z hw hz - -lemma IsCapacityTwoWitness.pair_coefficient_tendsto - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ≤ (T : ℝ)) : - Tendsto (fun w => loewnerPairCoefficient U G w z u) (𝓝 z) - (𝓝 (loewnerPairCoefficient U G z z u)) := by - have hzc := h.centered_ne_zero _ z (h.live_at_real_time T z hz u hu) - have hG := (h.spatial_differentiableAt _ z (h.live_at_real_time T z hz u hu)).continuousAt - exact tendsto_const_nhds.div ((hG.sub_const _).mul_const _) (mul_ne_zero hzc hzc) - -lemma IsCapacityTwoWitness.pair_integral_tendsto - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) : - Tendsto (fun w => ∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G w z u) (𝓝 z) - (𝓝 (∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G z z u)) := by - let c : ℝ := (G T z).im / 2 - have hp := h.imag_positive T z hz - have hc : 0 < c := by dsimp [c]; positivity - have hcz : c ≤ (G T z).im := by dsimp [c]; linarith - have hlt : c < (G T z).im := by dsimp [c]; linarith - have hlim : Tendsto (fun w => (G T w).im) (𝓝 z) (𝓝 (G T z).im) := - Complex.continuous_im.continuousAt.comp (h.spatial_differentiableAt T z hz).continuousAt - have hmem : ∀ᶠ w in 𝓝 z, w ∈ survivingDomain γ T := - (isOpen_survivingDomain h.1 T).mem_nhds hz - have hnear : ∀ᶠ w in 𝓝 z, w ∈ survivingDomain γ T ∧ c < (G T w).im := - hmem.and (hlim.eventually (Ioi_mem_nhds hlt)) - simp_rw [intervalIntegral.integral_of_le T.coe_nonneg] - apply tendsto_integral_filter_of_dominated_convergence (fun _ => 2 / (c * c)) - · filter_upwards [hnear] with w hw - exact ((intervalIntegrable_iff_integrableOn_Ioc_of_le T.coe_nonneg).mp - (h.pair_coefficient_intervalIntegrable hU T w z hw.1 hz)).aestronglyMeasurable - · filter_upwards [hnear] with w hw - filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u hu - exact h.pair_coefficient_bound T w z hw.1 hz c hc hw.2.le hcz u ⟨hu.1.le, hu.2⟩ - · exact integrable_const _ - · filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u hu - exact h.pair_coefficient_tendsto T z hz u hu.2 - - - -theorem loewner_spatial_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) : - deriv (G T) z = Complex.exp - (∫ u in (0 : ℝ)..(T : ℝ), (-2 : ℂ) / (G u.toNNReal z - (U u.toNNReal : ℂ)) ^ 2) := by - let L (w : ℂ) := Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G w z u) - have hL : Tendsto L (𝓝 z) (𝓝 (L z)) := - Complex.continuous_exp.continuousAt.tendsto.comp (h.pair_integral_tendsto hU T z hz) - have he : slope (G T) z =ᶠ[𝓝[≠] z] L := by - have hnear : ∀ᶠ w in 𝓝 z, w ∈ survivingDomain γ T := (isOpen_survivingDomain h.1 T).mem_nhds hz - filter_upwards [hnear.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with w hw hwz - have hwz' : w - z ≠ 0 := sub_ne_zero.mpr hwz - rw [slope_def_field, h.difference_formula hU T w z hw hz] - change (w - z) * L w / (w - z) = L w - field_simp - have hs : Tendsto (slope (G T) z) (𝓝[≠] z) (𝓝 (L z)) := - (hL.mono_left nhdsWithin_le_nhds).congr' he.symm - have heq := tendsto_nhds_unique (h.spatial_differentiableAt T z hz).hasDerivAt.tendsto_slope hs - simpa only [L, loewnerPairCoefficient, pow_two] using heq - - -noncomputable def fullRadius (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) : ℝ := - 2 * (G t z).im / ‖deriv (G t) z‖ - -lemma IsCapacityTwoWitness.derivative_norm_positive - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) : - 0 < ‖deriv (G T) z‖ := by - rw [loewner_spatial_derivative hU h T z hz, Complex.norm_exp] - exact Real.exp_pos _ - -lemma IsCapacityTwoWitness.fullRadius_positive - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) : - 0 < fullRadius G T z := by - exact div_pos (mul_pos (by norm_num) (h.imag_positive T z hz)) - (h.derivative_norm_positive hU T z hz) - -lemma IsCapacityTwoWitness.derivative_norm_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) - (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun v : ℝ => ‖deriv (G v.toNNReal) z‖) - (‖deriv (G u.toNNReal) z‖ * (loewnerPairCoefficient U G z z u).re) - (Icc 0 (T : ℝ)) u := by - let c := loewnerPairCoefficient U G z z - let C (v : ℝ) : ℂ := c (projIcc 0 (T : ℝ) T.coe_nonneg v) - have hc : ContinuousOn c (Icc 0 (T : ℝ)) := h.pair_coefficient_continuousOn hU T z z hz hz - have hC : Continuous C := hc.comp_continuous - (continuous_subtype_val.comp continuous_projIcc) (fun v => (projIcc 0 (T : ℝ) T.coe_nonneg v).property) - have hCc (v : ℝ) (hv : v ∈ Icc 0 (T : ℝ)) : C v = c v := by simp only [C, projIcc_of_mem T.coe_nonneg hv] - let I (v : ℝ) : ℂ := ∫ t in (0 : ℝ)..v, C t - have hI (v : ℝ) : HasDerivAt I (C v) v := - intervalIntegral.integral_hasDerivAt_right (hC.intervalIntegrable _ _) - hC.aestronglyMeasurable.stronglyMeasurableAtFilter hC.continuousAt - have heq (v : ℝ) (hv : v ∈ Icc 0 (T : ℝ)) : - ‖deriv (G v.toNNReal) z‖ = Real.exp (I v).re := by - rw [loewner_spatial_derivative hU h v.toNNReal z (h.live_at_real_time T z hz v hv.2), Complex.norm_exp] - congr 2 - rw [Real.coe_toNNReal _ hv.1] - apply intervalIntegral.integral_congr - intro t ht - rw [uIcc_of_le hv.1] at ht - rw [hCc t ⟨ht.1, ht.2.trans hv.2⟩] - simp only [c, loewnerPairCoefficient, pow_two] - have hd := (Complex.reCLM.hasFDerivAt.comp_hasDerivAt u (hI u)).exp.hasDerivWithinAt (s := Icc 0 (T : ℝ)) - have hh := hd.congr heq (heq u hu) - simpa only [Function.comp_def, Complex.reCLM_apply, hCc u hu, ← heq u hu, c] using hh - -lemma loewnerPairCoefficient_re (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ) - (z : ℂ) (u : ℝ) : - (loewnerPairCoefficient U G z z u).re = - -2 * ((G u.toNNReal z - (U u.toNNReal : ℂ)).re ^ 2 - (G u.toNNReal z).im ^ 2) / - Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) ^ 2 := by - simp only [loewnerPairCoefficient, Complex.div_re, Complex.neg_re, Complex.re_ofNat, - Complex.neg_im, Complex.im_ofNat, neg_zero, zero_mul, zero_div, add_zero, - Complex.mul_re, Complex.normSq_mul, Complex.sub_im, Complex.ofReal_im, sub_zero, - pow_two] - -lemma IsCapacityTwoWitness.radius_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) - (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun v : ℝ => fullRadius G v.toNNReal z) - (-4 * (G u.toNNReal z).im ^ 2 / - Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) ^ 2 * fullRadius G u.toNNReal z) - (Icc 0 (T : ℝ)) u := by - have hlu := h.live_at_real_time T z hz u hu.2 - have hJ := h.derivative_norm_positive hU u.toNNReal z hlu - have hN : Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) ≠ 0 := - (Complex.normSq_pos.mpr (h.centered_ne_zero _ z hlu)).ne' - have hd := ((h.imag_derivative T z hz u hu).const_mul 2).div - (h.derivative_norm_derivative hU T z hz u hu) hJ.ne' - convert hd using 1 - try rfl - rw [loewnerPairCoefficient_re] - let X := (G u.toNNReal z - (U u.toNNReal : ℂ)).re - let Y := (G u.toNNReal z).im - let J := ‖deriv (G u.toNNReal) z‖ - let N := Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) - have hNeq : N = X ^ 2 + Y ^ 2 := by - simp only [N, X, Y, Complex.normSq_apply, Complex.sub_im, Complex.ofReal_im, sub_zero, pow_two] - change -4 * Y ^ 2 / N ^ 2 * (2 * Y / J) = - ((2 * (-2 * Y / N)) * J - 2 * Y * (J * (-2 * (X ^ 2 - Y ^ 2) / N ^ 2))) / J ^ 2 - have hn : N ≠ 0 := hN - have hj : J ≠ 0 := hJ.ne' - field_simp - rw [hNeq] - ring - - -theorem loewner_radius_clock_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) - (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt - (fun v : ℝ => Real.log ((2 * z.im) / fullRadius G v.toNNReal z)) - (4 * (G u.toNNReal z).im ^ 2 / - (Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ))) ^ 2) - (Icc 0 (T : ℝ)) u := by - have hp := h.fullRadius_positive hU u.toNNReal z (h.live_at_real_time T z hz u hu.2) - have hc : 2 * z.im ≠ 0 := ne_of_gt (mul_pos (by norm_num) hz.1.1) - have hr := h.radius_derivative hU T z hz u hu - have hd := ((hasDerivWithinAt_const u (Icc 0 (T : ℝ)) (2 * z.im)).div hr hp.ne').log - (div_ne_zero hc hp.ne') - convert hd using 1 - try rfl - simp only [Pi.div_apply] - field_simp [hp.ne', hc, (show z.im ≠ 0 from hz.1.1.ne')] - ring - -noncomputable def logLoewnerCoordinates (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ) : ℂ × ℝ := - (((G t.toNNReal z).re : ℂ) + (Real.log (G t.toNNReal z).im : ℂ) * Complex.I, - Real.log ‖deriv (G t.toNNReal) z‖) - -lemma IsCapacityTwoWitness.log_height_bounds - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - Real.log (G T z).im ≤ Real.log (G u.toNNReal z).im ∧ - Real.log (G u.toNNReal z).im ≤ Real.log z.im := by - have hanti := h.imag_antitoneOn T z hz - have hlow := hanti hu (show (T : ℝ) ∈ Icc 0 (T : ℝ) from ⟨T.coe_nonneg, le_rfl⟩) hu.2 - have hupp := hanti (show (0 : ℝ) ∈ Icc 0 (T : ℝ) from ⟨le_rfl, T.coe_nonneg⟩) hu hu.1 - simp only [Real.toNNReal_coe] at hlow - simp only [Real.toNNReal_zero, h.initial_value z hz.1.1] at hupp - exact ⟨Real.log_le_log (h.imag_positive T z hz) hlow, - Real.log_le_log (h.imag_positive _ z (h.live_at_real_time T z hz u hu.2)) hupp⟩ - -lemma clippedCenter_logCoordinates - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) - (hz : z ∈ survivingDomain γ T) (lo hi : ℝ) (hle : lo ≤ hi) - (hlo : lo ≤ Real.log (G T z).im) (hhi : Real.log z.im ≤ hi) - (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - clippedCenter lo hi hle (U u.toNNReal) (logLoewnerCoordinates G z u) = - G u.toNNReal z - (U u.toNNReal : ℂ) := by - obtain ⟨ha, hb⟩ := h.log_height_bounds T z hz u hu - have hmem : Real.log (G u.toNNReal z).im ∈ Icc lo hi := ⟨hlo.trans ha, hb.trans hhi⟩ - have him := h.imag_positive _ z (h.live_at_real_time T z hz u hu.2) - simp only [clippedCenter, clippedHeight, logLoewnerCoordinates, Complex.add_re, - Complex.ofReal_re, Complex.mul_re, Complex.I_re, mul_zero, Complex.ofReal_im, - Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, - mul_one, zero_add, projIcc_of_mem hle hmem, Real.exp_log him] - apply Complex.ext <;> simp - -lemma logLoewnerCoordinates_derivative - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) - (lo hi : ℝ) (hle : lo ≤ hi) - (hlo : lo ≤ Real.log (G T z).im) (hhi : Real.log z.im ≤ hi) - (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (logLoewnerCoordinates G z) - (clippedLogLoewnerField lo hi hle (U u.toNNReal) (logLoewnerCoordinates G z u)) - (Icc 0 (T : ℝ)) u := by - have hzlive := h.live_at_real_time T z hz u hu.2 - have hy := h.imag_positive _ z hzlive - have hj := h.derivative_norm_positive hU _ z hzlive - have hdre := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt u (h.time_derivative T z hz u hu) - have hdY := (h.imag_derivative T z hz u hu).log hy.ne' - have hdJ := (h.derivative_norm_derivative hU T z hz u hu).log hj.ne' - have hd := ((Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt u hdre).add - ((Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt u hdY).mul_const Complex.I)).prodMk hdJ - convert hd using 1 <;> try rfl - unfold clippedLogLoewnerField - rw [clippedCenter_logCoordinates h T z hz lo hi hle hlo hhi u hu] - dsimp only [Function.comp_def, Complex.reCLM_apply, Complex.ofRealCLM_apply] - apply Prod.ext - · apply Complex.ext - · simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, mul_zero, - Complex.ofReal_im, Complex.I_im, zero_mul, sub_self, add_zero, - div_eq_mul_inv, Complex.mul_re, Complex.re_ofNat, Complex.im_ofNat, - zero_mul, sub_zero] - · simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.I_im, - Complex.ofReal_re, mul_one, Complex.I_re, mul_zero, add_zero, zero_add, - Complex.normSq_inv] - field_simp [hy.ne'] - · change (-2 * (G u.toNNReal z - (U u.toNNReal : ℂ))⁻¹ ^ 2).re = - (_ * (loewnerPairCoefficient U G z z u).re) / _ - rw [mul_div_cancel_left₀ _ hj.ne'] - congr 1 - simp only [loewnerPairCoefficient, div_eq_mul_inv, mul_inv_rev, pow_two] - -lemma hasDerivWithinAt_Ici_of_Icc - {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] - {f : ℝ → E} {f' : E} {a b u : ℝ} (hu : u ∈ Ico a b) - (hf : HasDerivWithinAt f f' (Icc a b) u) : HasDerivWithinAt f f' (Ici u) u := by - apply hf.mono_of_mem_nhdsWithin - filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (Iic_mem_nhds hu.2)] with x hx hxT - exact ⟨hu.1.trans hx, hxT⟩ - -lemma logLoewnerCoordinates_initial - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (z : ℂ) (hz : z ∈ survivingDomain γ 0) : - logLoewnerCoordinates G z 0 = ((z.re : ℂ) + (Real.log z.im : ℂ) * Complex.I, 0) := by - simp only [logLoewnerCoordinates, Real.toNNReal_zero, h.initial_value z hz.1.1, - loewner_spatial_derivative hU h 0 z hz, NNReal.coe_zero, - intervalIntegral.integral_same, Complex.exp_zero, norm_one, Real.log_one] - - - -theorem clipped_log_loewner_agrees - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) - (lo hi : ℝ) (hle : lo ≤ hi) - (hlo : lo ≤ Real.log (G T z).im) (hhi : Real.log z.im ≤ hi) - (Q : ℝ → ℂ × ℝ) - (hQ0 : Q 0 = ((z.re : ℂ) + (Real.log z.im : ℂ) * Complex.I, 0)) - (hQ : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q - (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t) : - EqOn Q (logLoewnerCoordinates G z) (Icc 0 (T : ℝ)) := by - obtain ⟨L, K, hb, hl⟩ := clippedLogLoewnerField_bounds lo hi hle - have hd := logLoewnerCoordinates_derivative hU h T z hz lo hi hle hlo hhi - apply ODE_solution_unique (fun t => hl (U t.toNNReal)) - (fun t ht => (hQ t ht).continuousWithinAt) - (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hQ t ⟨ht.1, ht.2.le⟩)) - (fun t ht => (hd t ht).continuousWithinAt) - (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hd t ⟨ht.1, ht.2.le⟩)) - exact hQ0.trans (logLoewnerCoordinates_initial hU h z (survivingDomain_antitone γ bot_le hz)).symm - - - -theorem loewner_survival_of_positive_solution - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) - (T : ℝ≥0) (W : ℝ → ℂ) - (hWp : ∀ u ∈ Icc 0 (T : ℝ), 0 < (W u).im) - (hWd : ∀ u ∈ Icc 0 (T : ℝ), HasDerivWithinAt W - (2 / (W u - (U u.toNNReal : ℂ))) (Icc 0 (T : ℝ)) u) : - W 0 ∈ survivingDomain γ T ∧ G T (W 0) = W T := by - let w := F T (W T) - have hWT : W T ∈ upperHalfPlane := hWp T ⟨T.coe_nonneg, le_rfl⟩ - have hw : w ∈ survivingDomain γ T := h.2.2.2.2.2.2.1 T hWT - have hGw : G T w = W T := h.2.2.2.2.2.2.2.2.1 T (W T) hWT - have hWc : ContinuousOn W (Icc 0 (T : ℝ)) := fun u hu => (hWd u hu).continuousWithinAt - have hWnz (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : W u - (U u.toNNReal : ℂ) ≠ 0 := by - intro he - have hh := congrArg Complex.im he - simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hh - exact (hWp u hu).ne' hh - let c (u : ℝ) : ℂ := (-2 : ℂ) / - ((G u.toNNReal w - (U u.toNNReal : ℂ)) * (W u - (U u.toNNReal : ℂ))) - have hUc : Continuous (fun u : ℝ => (U u.toNNReal : ℂ)) := - Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal) - have hc : ContinuousOn c (Icc 0 (T : ℝ)) := - continuousOn_const.div - (((h.time_continuousOn T w hw).sub hUc.continuousOn).mul (hWc.sub hUc.continuousOn)) - (fun u hu => mul_ne_zero (h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2)) (hWnz u hu)) - have hd (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun s : ℝ => G s.toNNReal w - W s) - (c u * (G u.toNNReal w - W u)) (Icc 0 (T : ℝ)) u := by - have hg := h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2) - have hn := hWnz u hu - have he : c u * (G u.toNNReal w - W u) = - 2 / (G u.toNNReal w - (U u.toNNReal : ℂ)) - 2 / (W u - (U u.toNNReal : ℂ)) := by - dsimp only [c] - field_simp - ring - rw [he] - exact (h.time_derivative T w hw u hu).sub (hWd u hu) - have he := complex_linear_ode_exp (T : ℝ) T.coe_nonneg - (fun s : ℝ => G s.toNNReal w - W s) c hc hd - simp only [Real.toNNReal_coe, hGw, sub_self, Real.toNNReal_zero, - h.initial_value w hw.1.1] at he - have hw0 : w = W 0 := sub_eq_zero.mp ((mul_eq_zero.mp he.symm).resolve_right (Complex.exp_ne_zero _)) - exact ⟨hw0 ▸ hw, hw0 ▸ hGw⟩ - -end SLEExactGauge -end - - -end - -end OAI +import Mathlib +import OAI.Probability.SLE.ClippedCoordinates + +namespace OAI + +/-! Loewner Charts for chordal SLE. -/ + +section +section + +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal +namespace SLEExactGauge + + +def upperHalfPlane : Set ℂ := {z | 0 < z.im} + + + +def survivingDomain (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : Set ℂ := + {z | z ∈ upperHalfPlane \ (γ '' Icc 0 t) ∧ + ¬ Bornology.IsBounded (connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z)} + + + +def IsCapacityTwoWitness (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) + (G F : ℝ≥0 → ℂ → ℂ) : Prop := + Continuous γ ∧ γ 0 = 0 ∧ (∀ t, 0 ≤ (γ t).im) ∧ + (∀ t, DifferentiableOn ℂ (G t) (survivingDomain γ t)) ∧ + (∀ t, DifferentiableOn ℂ (F t) upperHalfPlane) ∧ + (∀ t, MapsTo (G t) (survivingDomain γ t) upperHalfPlane) ∧ + (∀ t, MapsTo (F t) upperHalfPlane (survivingDomain γ t)) ∧ + (∀ t z, z ∈ survivingDomain γ t → F t (G t z) = z) ∧ + (∀ t z, z ∈ upperHalfPlane → G t (F t z) = z) ∧ + (∀ z ∈ upperHalfPlane, G 0 z = z) ∧ + (∀ t z, z ∈ survivingDomain γ t → + HasDerivWithinAt (fun u : ℝ => G (Real.toNNReal u) z) + (2 / (G t z - (U t : ℂ))) (Ici 0) (t : ℝ)) ∧ + (∀ t, Tendsto (fun z : ℂ => z * (G t z - z)) + (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) (𝓝 (2 * (t : ℂ)))) ∧ + (∀ t, Tendsto (fun y : ℝ => F t ((U t : ℂ) + (y : ℂ) * Complex.I)) + (𝓝[>] 0) (𝓝 (γ t))) + +lemma survivingDomain_antitone (γ : ℝ≥0 → ℂ) : Antitone (survivingDomain γ) := by + intro s t hst z hz + have hpast : γ '' Icc 0 s ⊆ γ '' Icc 0 t := by + rintro _ ⟨u, hu, rfl⟩ + exact ⟨u, ⟨hu.1, hu.2.trans hst⟩, rfl⟩ + have hb : upperHalfPlane \ (γ '' Icc 0 t) ⊆ upperHalfPlane \ (γ '' Icc 0 s) := by + intro w hw + exact ⟨hw.1, fun hh => hw.2 (hpast hh)⟩ + exact ⟨hb hz.1, fun hbounded => hz.2 (hbounded.subset (connectedComponentIn_mono z hb))⟩ + +lemma isOpen_upperHalfPlane : IsOpen upperHalfPlane := isOpen_lt continuous_const Complex.continuous_im + +lemma isOpen_survivingDomain {γ : ℝ≥0 → ℂ} (hγ : Continuous γ) (t : ℝ≥0) : + IsOpen (survivingDomain γ t) := by + have hb : IsOpen (upperHalfPlane \ (γ '' Icc 0 t)) := + isOpen_upperHalfPlane.sdiff (isCompact_Icc.image hγ).isClosed + apply isOpen_iff_forall_mem_open.mpr + intro z hz + refine ⟨connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z, ?_, + hb.connectedComponentIn, mem_connectedComponentIn hz.1⟩ + intro w hw + refine ⟨connectedComponentIn_subset _ _ hw, ?_⟩ + rw [← connectedComponentIn_eq hw] + exact hz.2 + +lemma IsCapacityTwoWitness.time_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun v : ℝ => G v.toNNReal z) + (2 / (G u.toNNReal z - (U u.toNNReal : ℂ))) (Icc 0 (T : ℝ)) u := by + rcases h with ⟨hγ, hγ0, hγim, hG, hF, hGmap, hFmap, hinv, hinv', h0, hode, hnorm, htip⟩ + have hlu : u.toNNReal ≤ T := by + exact_mod_cast (show (u.toNNReal : ℝ) ≤ (T : ℝ) by rw [Real.coe_toNNReal _ hu.1]; exact hu.2) + have hh := (hode u.toNNReal z (survivingDomain_antitone γ hlu hz)).mono + (show Icc (0 : ℝ) T ⊆ Ici 0 from Icc_subset_Ici_self) + simpa only [Real.coe_toNNReal _ hu.1] using hh + +lemma IsCapacityTwoWitness.time_continuousOn + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) : + ContinuousOn (fun v : ℝ => G v.toNNReal z) (Icc 0 (T : ℝ)) := + fun u hu => (h.time_derivative T z hz u hu).continuousWithinAt + +lemma IsCapacityTwoWitness.imag_positive + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ t) : 0 < (G t z).im := h.2.2.2.2.2.1 t hz + +lemma IsCapacityTwoWitness.centered_ne_zero + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ t) : G t z - (U t : ℂ) ≠ 0 := by + intro hh + have hi := congrArg Complex.im hh + simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hi + exact (h.imag_positive t z hz).ne' hi + +lemma IsCapacityTwoWitness.imag_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun v : ℝ => (G v.toNNReal z).im) + (-2 * (G u.toNNReal z).im / Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ))) + (Icc 0 (T : ℝ)) u := by + have hh := Complex.imCLM.hasFDerivAt.comp_hasDerivWithinAt u (h.time_derivative T z hz u hu) + simpa [Function.comp_def, Complex.div_im, neg_div] using hh + +lemma IsCapacityTwoWitness.imag_antitoneOn + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) : + AntitoneOn (fun v : ℝ => (G v.toNNReal z).im) (Icc 0 (T : ℝ)) := by + apply antitoneOn_of_hasDerivWithinAt_nonpos (convex_Icc _ _) + (Complex.continuous_im.comp_continuousOn (h.time_continuousOn T z hz)) + (fun u hu => (h.imag_derivative T z hz u (interior_subset hu)).mono interior_subset) + intro u hu + have hu' : u ∈ Icc 0 (T : ℝ) := interior_subset hu + have hle : u.toNNReal ≤ T := (Real.toNNReal_le_iff_le_coe).mpr hu'.2 + exact div_nonpos_of_nonpos_of_nonneg + (mul_nonpos_of_nonpos_of_nonneg (by norm_num) + (h.imag_positive u.toNNReal z (survivingDomain_antitone γ hle hz)).le) + (Complex.normSq_nonneg _) + + + +lemma complex_linear_ode_exp (T : ℝ) (hT : 0 ≤ T) (f c : ℝ → ℂ) + (hc : ContinuousOn c (Icc 0 T)) + (hf : ∀ u ∈ Icc 0 T, HasDerivWithinAt f (c u * f u) (Icc 0 T) u) : + f T = f 0 * Complex.exp (∫ u in (0 : ℝ)..T, c u) := by + by_cases hT0 : T = 0 + · simp [hT0] + have hTp : 0 < T := lt_of_le_of_ne hT (Ne.symm hT0) + let C (u : ℝ) : ℂ := c (projIcc 0 T hT u) + have hC : Continuous C := hc.comp_continuous + (continuous_subtype_val.comp continuous_projIcc) (fun u => (projIcc 0 T hT u).property) + have hCc (u : ℝ) (hu : u ∈ Icc 0 T) : C u = c u := by simp only [C, projIcc_of_mem hT hu] + let I (u : ℝ) : ℂ := ∫ v in (0 : ℝ)..u, C v + have hI (u : ℝ) : HasDerivAt I (C u) u := + intervalIntegral.integral_hasDerivAt_right (hC.intervalIntegrable _ _) + hC.aestronglyMeasurable.stronglyMeasurableAtFilter hC.continuousAt + have hd (u : ℝ) (hu : u ∈ Icc 0 T) : + HasDerivWithinAt (fun v => f v * Complex.exp (-I v)) 0 (Icc 0 T) u := by + convert (hf u hu).mul ((hI u).neg.cexp.hasDerivWithinAt) using 1 + try rfl + rw [hCc u hu] + ring + have hid : ∀ u ∈ Icc 0 T, DifferentiableWithinAt ℝ (fun v => f v * Complex.exp (-I v)) (Icc 0 T) u := + fun u hu => (hd u hu).differentiableWithinAt + have he := constant_of_derivWithin_zero hid (fun u hu => + (hd u ⟨hu.1, hu.2.le⟩).derivWithin (uniqueDiffOn_Icc hTp u ⟨hu.1, hu.2.le⟩)) T ⟨hT, le_rfl⟩ + have hfactor : f T * Complex.exp (-I T) = f 0 := by + simpa only [I, intervalIntegral.integral_same, neg_zero, Complex.exp_zero, mul_one] using he + have hInt : I T = ∫ u in (0 : ℝ)..T, c u := by + apply intervalIntegral.integral_congr + intro u hu + exact hCc u (by simpa only [uIcc_of_le hT] using hu) + calc + f T = (f T * Complex.exp (-I T)) * Complex.exp (I T) := by + rw [mul_assoc, ← Complex.exp_add, neg_add_cancel, Complex.exp_zero, mul_one] + _ = _ := by rw [hfactor, hInt] + +lemma IsCapacityTwoWitness.live_at_real_time + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (_h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ≤ (T : ℝ)) : + z ∈ survivingDomain γ u.toNNReal := + survivingDomain_antitone γ (Real.toNNReal_le_iff_le_coe.mpr hu) hz + +lemma IsCapacityTwoWitness.spatial_differentiableAt + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (t : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ t) : DifferentiableAt ℂ (G t) z := + (h.2.2.2.1 t z hz).differentiableAt ((isOpen_survivingDomain h.1 t).mem_nhds hz) + +lemma IsCapacityTwoWitness.initial_value + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (z : ℂ) (hz : z ∈ upperHalfPlane) : G 0 z = z := + h.2.2.2.2.2.2.2.2.2.1 z hz + +noncomputable def loewnerPairCoefficient + (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ) (w z : ℂ) (u : ℝ) : ℂ := + (-2 : ℂ) / ((G u.toNNReal w - (U u.toNNReal : ℂ)) * + (G u.toNNReal z - (U u.toNNReal : ℂ))) + +lemma IsCapacityTwoWitness.pair_coefficient_continuousOn + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) + (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) : + ContinuousOn (loewnerPairCoefficient U G w z) (Icc 0 (T : ℝ)) := by + have hUc : Continuous (fun u : ℝ => (U u.toNNReal : ℂ)) := + Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal) + exact continuousOn_const.div + (((h.time_continuousOn T w hw).sub hUc.continuousOn).mul + ((h.time_continuousOn T z hz).sub hUc.continuousOn)) + (fun u hu => mul_ne_zero (h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2)) + (h.centered_ne_zero _ z (h.live_at_real_time T z hz u hu.2))) + +lemma IsCapacityTwoWitness.difference_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) + (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) + (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun v : ℝ => G v.toNNReal w - G v.toNNReal z) + (loewnerPairCoefficient U G w z u * (G u.toNNReal w - G u.toNNReal z)) + (Icc 0 (T : ℝ)) u := by + have hw0 := h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2) + have hz0 := h.centered_ne_zero _ z (h.live_at_real_time T z hz u hu.2) + have he : loewnerPairCoefficient U G w z u * (G u.toNNReal w - G u.toNNReal z) = + 2 / (G u.toNNReal w - (U u.toNNReal : ℂ)) - + 2 / (G u.toNNReal z - (U u.toNNReal : ℂ)) := by + unfold loewnerPairCoefficient + field_simp + ring + rw [he] + exact (h.time_derivative T w hw u hu).sub (h.time_derivative T z hz u hu) + +lemma IsCapacityTwoWitness.difference_formula + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) + (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) : + G T w - G T z = (w - z) * Complex.exp + (∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G w z u) := by + have hh := complex_linear_ode_exp (T : ℝ) T.coe_nonneg + (fun v : ℝ => G v.toNNReal w - G v.toNNReal z) + (loewnerPairCoefficient U G w z) + (h.pair_coefficient_continuousOn hU T w z hw hz) + (h.difference_derivative T w z hw hz) + simpa only [Real.toNNReal_coe, Real.toNNReal_zero, + h.initial_value w hw.1.1, h.initial_value z hz.1.1] using hh + +lemma IsCapacityTwoWitness.imag_ge_terminal + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + (G T z).im ≤ (G u.toNNReal z).im := by + simpa only [Real.toNNReal_coe] using h.imag_antitoneOn T z hz hu ⟨T.coe_nonneg, le_rfl⟩ hu.2 + +lemma IsCapacityTwoWitness.centered_norm_ge_terminal + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + (G T z).im ≤ ‖G u.toNNReal z - (U u.toNNReal : ℂ)‖ := by + apply (h.imag_ge_terminal T z hz u hu).trans + simpa only [Complex.sub_im, Complex.ofReal_im, sub_zero] using + Complex.im_le_norm (G u.toNNReal z - (U u.toNNReal : ℂ)) + +lemma IsCapacityTwoWitness.pair_coefficient_bound + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) + (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) + (c : ℝ) (hc : 0 < c) (hcw : c ≤ (G T w).im) (hcz : c ≤ (G T z).im) + (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + ‖loewnerPairCoefficient U G w z u‖ ≤ 2 / (c * c) := by + have hw' := hcw.trans (h.centered_norm_ge_terminal T w hw u hu) + have hz' := hcz.trans (h.centered_norm_ge_terminal T z hz u hu) + unfold loewnerPairCoefficient + rw [norm_div, norm_mul, norm_neg, Complex.norm_two] + exact div_le_div_of_nonneg_left (by norm_num) (mul_pos hc hc) + (mul_le_mul hw' hz' hc.le (norm_nonneg _)) + +lemma IsCapacityTwoWitness.pair_coefficient_intervalIntegrable + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w z : ℂ) + (hw : w ∈ survivingDomain γ T) (hz : z ∈ survivingDomain γ T) : + IntervalIntegrable (loewnerPairCoefficient U G w z) volume 0 (T : ℝ) := by + apply ContinuousOn.intervalIntegrable + simpa only [uIcc_of_le T.coe_nonneg] using h.pair_coefficient_continuousOn hU T w z hw hz + +lemma IsCapacityTwoWitness.pair_coefficient_tendsto + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ≤ (T : ℝ)) : + Tendsto (fun w => loewnerPairCoefficient U G w z u) (𝓝 z) + (𝓝 (loewnerPairCoefficient U G z z u)) := by + have hzc := h.centered_ne_zero _ z (h.live_at_real_time T z hz u hu) + have hG := (h.spatial_differentiableAt _ z (h.live_at_real_time T z hz u hu)).continuousAt + exact tendsto_const_nhds.div ((hG.sub_const _).mul_const _) (mul_ne_zero hzc hzc) + +lemma IsCapacityTwoWitness.pair_integral_tendsto + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) : + Tendsto (fun w => ∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G w z u) (𝓝 z) + (𝓝 (∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G z z u)) := by + let c : ℝ := (G T z).im / 2 + have hp := h.imag_positive T z hz + have hc : 0 < c := by dsimp [c]; positivity + have hcz : c ≤ (G T z).im := by dsimp [c]; linarith + have hlt : c < (G T z).im := by dsimp [c]; linarith + have hlim : Tendsto (fun w => (G T w).im) (𝓝 z) (𝓝 (G T z).im) := + Complex.continuous_im.continuousAt.comp (h.spatial_differentiableAt T z hz).continuousAt + have hmem : ∀ᶠ w in 𝓝 z, w ∈ survivingDomain γ T := + (isOpen_survivingDomain h.1 T).mem_nhds hz + have hnear : ∀ᶠ w in 𝓝 z, w ∈ survivingDomain γ T ∧ c < (G T w).im := + hmem.and (hlim.eventually (Ioi_mem_nhds hlt)) + simp_rw [intervalIntegral.integral_of_le T.coe_nonneg] + apply tendsto_integral_filter_of_dominated_convergence (fun _ => 2 / (c * c)) + · filter_upwards [hnear] with w hw + exact ((intervalIntegrable_iff_integrableOn_Ioc_of_le T.coe_nonneg).mp + (h.pair_coefficient_intervalIntegrable hU T w z hw.1 hz)).aestronglyMeasurable + · filter_upwards [hnear] with w hw + filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u hu + exact h.pair_coefficient_bound T w z hw.1 hz c hc hw.2.le hcz u ⟨hu.1.le, hu.2⟩ + · exact integrable_const _ + · filter_upwards [ae_restrict_mem (μ := volume) measurableSet_Ioc] with u hu + exact h.pair_coefficient_tendsto T z hz u hu.2 + + + +theorem loewner_spatial_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) : + deriv (G T) z = Complex.exp + (∫ u in (0 : ℝ)..(T : ℝ), (-2 : ℂ) / (G u.toNNReal z - (U u.toNNReal : ℂ)) ^ 2) := by + let L (w : ℂ) := Complex.exp (∫ u in (0 : ℝ)..(T : ℝ), loewnerPairCoefficient U G w z u) + have hL : Tendsto L (𝓝 z) (𝓝 (L z)) := + Complex.continuous_exp.continuousAt.tendsto.comp (h.pair_integral_tendsto hU T z hz) + have he : slope (G T) z =ᶠ[𝓝[≠] z] L := by + have hnear : ∀ᶠ w in 𝓝 z, w ∈ survivingDomain γ T := (isOpen_survivingDomain h.1 T).mem_nhds hz + filter_upwards [hnear.filter_mono nhdsWithin_le_nhds, self_mem_nhdsWithin] with w hw hwz + have hwz' : w - z ≠ 0 := sub_ne_zero.mpr hwz + rw [slope_def_field, h.difference_formula hU T w z hw hz] + change (w - z) * L w / (w - z) = L w + field_simp + have hs : Tendsto (slope (G T) z) (𝓝[≠] z) (𝓝 (L z)) := + (hL.mono_left nhdsWithin_le_nhds).congr' he.symm + have heq := tendsto_nhds_unique (h.spatial_differentiableAt T z hz).hasDerivAt.tendsto_slope hs + simpa only [L, loewnerPairCoefficient, pow_two] using heq + + +noncomputable def fullRadius (G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) : ℝ := + 2 * (G t z).im / ‖deriv (G t) z‖ + +lemma IsCapacityTwoWitness.derivative_norm_positive + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) : + 0 < ‖deriv (G T) z‖ := by + rw [loewner_spatial_derivative hU h T z hz, Complex.norm_exp] + exact Real.exp_pos _ + +lemma IsCapacityTwoWitness.fullRadius_positive + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) : + 0 < fullRadius G T z := by + exact div_pos (mul_pos (by norm_num) (h.imag_positive T z hz)) + (h.derivative_norm_positive hU T z hz) + +lemma IsCapacityTwoWitness.derivative_norm_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) + (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun v : ℝ => ‖deriv (G v.toNNReal) z‖) + (‖deriv (G u.toNNReal) z‖ * (loewnerPairCoefficient U G z z u).re) + (Icc 0 (T : ℝ)) u := by + let c := loewnerPairCoefficient U G z z + let C (v : ℝ) : ℂ := c (projIcc 0 (T : ℝ) T.coe_nonneg v) + have hc : ContinuousOn c (Icc 0 (T : ℝ)) := h.pair_coefficient_continuousOn hU T z z hz hz + have hC : Continuous C := hc.comp_continuous + (continuous_subtype_val.comp continuous_projIcc) (fun v => (projIcc 0 (T : ℝ) T.coe_nonneg v).property) + have hCc (v : ℝ) (hv : v ∈ Icc 0 (T : ℝ)) : C v = c v := by simp only [C, projIcc_of_mem T.coe_nonneg hv] + let I (v : ℝ) : ℂ := ∫ t in (0 : ℝ)..v, C t + have hI (v : ℝ) : HasDerivAt I (C v) v := + intervalIntegral.integral_hasDerivAt_right (hC.intervalIntegrable _ _) + hC.aestronglyMeasurable.stronglyMeasurableAtFilter hC.continuousAt + have heq (v : ℝ) (hv : v ∈ Icc 0 (T : ℝ)) : + ‖deriv (G v.toNNReal) z‖ = Real.exp (I v).re := by + rw [loewner_spatial_derivative hU h v.toNNReal z (h.live_at_real_time T z hz v hv.2), Complex.norm_exp] + congr 2 + rw [Real.coe_toNNReal _ hv.1] + apply intervalIntegral.integral_congr + intro t ht + rw [uIcc_of_le hv.1] at ht + rw [hCc t ⟨ht.1, ht.2.trans hv.2⟩] + simp only [c, loewnerPairCoefficient, pow_two] + have hd := (Complex.reCLM.hasFDerivAt.comp_hasDerivAt u (hI u)).exp.hasDerivWithinAt (s := Icc 0 (T : ℝ)) + have hh := hd.congr heq (heq u hu) + simpa only [Function.comp_def, Complex.reCLM_apply, hCc u hu, ← heq u hu, c] using hh + +lemma loewnerPairCoefficient_re (U : ℝ≥0 → ℝ) (G : ℝ≥0 → ℂ → ℂ) + (z : ℂ) (u : ℝ) : + (loewnerPairCoefficient U G z z u).re = + -2 * ((G u.toNNReal z - (U u.toNNReal : ℂ)).re ^ 2 - (G u.toNNReal z).im ^ 2) / + Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) ^ 2 := by + simp only [loewnerPairCoefficient, Complex.div_re, Complex.neg_re, Complex.re_ofNat, + Complex.neg_im, Complex.im_ofNat, neg_zero, zero_mul, zero_div, add_zero, + Complex.mul_re, Complex.normSq_mul, Complex.sub_im, Complex.ofReal_im, sub_zero, + pow_two] + +lemma IsCapacityTwoWitness.radius_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) + (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun v : ℝ => fullRadius G v.toNNReal z) + (-4 * (G u.toNNReal z).im ^ 2 / + Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) ^ 2 * fullRadius G u.toNNReal z) + (Icc 0 (T : ℝ)) u := by + have hlu := h.live_at_real_time T z hz u hu.2 + have hJ := h.derivative_norm_positive hU u.toNNReal z hlu + have hN : Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) ≠ 0 := + (Complex.normSq_pos.mpr (h.centered_ne_zero _ z hlu)).ne' + have hd := ((h.imag_derivative T z hz u hu).const_mul 2).div + (h.derivative_norm_derivative hU T z hz u hu) hJ.ne' + convert hd using 1 + try rfl + rw [loewnerPairCoefficient_re] + let X := (G u.toNNReal z - (U u.toNNReal : ℂ)).re + let Y := (G u.toNNReal z).im + let J := ‖deriv (G u.toNNReal) z‖ + let N := Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ)) + have hNeq : N = X ^ 2 + Y ^ 2 := by + simp only [N, X, Y, Complex.normSq_apply, Complex.sub_im, Complex.ofReal_im, sub_zero, pow_two] + change -4 * Y ^ 2 / N ^ 2 * (2 * Y / J) = + ((2 * (-2 * Y / N)) * J - 2 * Y * (J * (-2 * (X ^ 2 - Y ^ 2) / N ^ 2))) / J ^ 2 + have hn : N ≠ 0 := hN + have hj : J ≠ 0 := hJ.ne' + field_simp + rw [hNeq] + ring + + +theorem loewner_radius_clock_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) + (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt + (fun v : ℝ => Real.log ((2 * z.im) / fullRadius G v.toNNReal z)) + (4 * (G u.toNNReal z).im ^ 2 / + (Complex.normSq (G u.toNNReal z - (U u.toNNReal : ℂ))) ^ 2) + (Icc 0 (T : ℝ)) u := by + have hp := h.fullRadius_positive hU u.toNNReal z (h.live_at_real_time T z hz u hu.2) + have hc : 2 * z.im ≠ 0 := ne_of_gt (mul_pos (by norm_num) hz.1.1) + have hr := h.radius_derivative hU T z hz u hu + have hd := ((hasDerivWithinAt_const u (Icc 0 (T : ℝ)) (2 * z.im)).div hr hp.ne').log + (div_ne_zero hc hp.ne') + convert hd using 1 + try rfl + simp only [Pi.div_apply] + field_simp [hp.ne', hc, (show z.im ≠ 0 from hz.1.1.ne')] + ring + +noncomputable def logLoewnerCoordinates (G : ℝ≥0 → ℂ → ℂ) (z : ℂ) (t : ℝ) : ℂ × ℝ := + (((G t.toNNReal z).re : ℂ) + (Real.log (G t.toNNReal z).im : ℂ) * Complex.I, + Real.log ‖deriv (G t.toNNReal) z‖) + +lemma IsCapacityTwoWitness.log_height_bounds + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + Real.log (G T z).im ≤ Real.log (G u.toNNReal z).im ∧ + Real.log (G u.toNNReal z).im ≤ Real.log z.im := by + have hanti := h.imag_antitoneOn T z hz + have hlow := hanti hu (show (T : ℝ) ∈ Icc 0 (T : ℝ) from ⟨T.coe_nonneg, le_rfl⟩) hu.2 + have hupp := hanti (show (0 : ℝ) ∈ Icc 0 (T : ℝ) from ⟨le_rfl, T.coe_nonneg⟩) hu hu.1 + simp only [Real.toNNReal_coe] at hlow + simp only [Real.toNNReal_zero, h.initial_value z hz.1.1] at hupp + exact ⟨Real.log_le_log (h.imag_positive T z hz) hlow, + Real.log_le_log (h.imag_positive _ z (h.live_at_real_time T z hz u hu.2)) hupp⟩ + +lemma clippedCenter_logCoordinates + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (z : ℂ) + (hz : z ∈ survivingDomain γ T) (lo hi : ℝ) (hle : lo ≤ hi) + (hlo : lo ≤ Real.log (G T z).im) (hhi : Real.log z.im ≤ hi) + (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + clippedCenter lo hi hle (U u.toNNReal) (logLoewnerCoordinates G z u) = + G u.toNNReal z - (U u.toNNReal : ℂ) := by + obtain ⟨ha, hb⟩ := h.log_height_bounds T z hz u hu + have hmem : Real.log (G u.toNNReal z).im ∈ Icc lo hi := ⟨hlo.trans ha, hb.trans hhi⟩ + have him := h.imag_positive _ z (h.live_at_real_time T z hz u hu.2) + simp only [clippedCenter, clippedHeight, logLoewnerCoordinates, Complex.add_re, + Complex.ofReal_re, Complex.mul_re, Complex.I_re, mul_zero, Complex.ofReal_im, + Complex.I_im, sub_self, add_zero, Complex.add_im, Complex.mul_im, + mul_one, zero_add, projIcc_of_mem hle hmem, Real.exp_log him] + apply Complex.ext <;> simp + +lemma logLoewnerCoordinates_derivative + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) + (lo hi : ℝ) (hle : lo ≤ hi) + (hlo : lo ≤ Real.log (G T z).im) (hhi : Real.log z.im ≤ hi) + (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (logLoewnerCoordinates G z) + (clippedLogLoewnerField lo hi hle (U u.toNNReal) (logLoewnerCoordinates G z u)) + (Icc 0 (T : ℝ)) u := by + have hzlive := h.live_at_real_time T z hz u hu.2 + have hy := h.imag_positive _ z hzlive + have hj := h.derivative_norm_positive hU _ z hzlive + have hdre := Complex.reCLM.hasFDerivAt.comp_hasDerivWithinAt u (h.time_derivative T z hz u hu) + have hdY := (h.imag_derivative T z hz u hu).log hy.ne' + have hdJ := (h.derivative_norm_derivative hU T z hz u hu).log hj.ne' + have hd := ((Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt u hdre).add + ((Complex.ofRealCLM.hasFDerivAt.comp_hasDerivWithinAt u hdY).mul_const Complex.I)).prodMk hdJ + convert hd using 1 <;> try rfl + unfold clippedLogLoewnerField + rw [clippedCenter_logCoordinates h T z hz lo hi hle hlo hhi u hu] + dsimp only [Function.comp_def, Complex.reCLM_apply, Complex.ofRealCLM_apply] + apply Prod.ext + · apply Complex.ext + · simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, mul_zero, + Complex.ofReal_im, Complex.I_im, zero_mul, sub_self, add_zero, + div_eq_mul_inv, Complex.mul_re, Complex.re_ofNat, Complex.im_ofNat, + zero_mul, sub_zero] + · simp only [Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.I_im, + Complex.ofReal_re, mul_one, Complex.I_re, mul_zero, add_zero, zero_add, + Complex.normSq_inv] + field_simp [hy.ne'] + · change (-2 * (G u.toNNReal z - (U u.toNNReal : ℂ))⁻¹ ^ 2).re = + (_ * (loewnerPairCoefficient U G z z u).re) / _ + rw [mul_div_cancel_left₀ _ hj.ne'] + congr 1 + simp only [loewnerPairCoefficient, div_eq_mul_inv, mul_inv_rev, pow_two] + +lemma hasDerivWithinAt_Ici_of_Icc + {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] + {f : ℝ → E} {f' : E} {a b u : ℝ} (hu : u ∈ Ico a b) + (hf : HasDerivWithinAt f f' (Icc a b) u) : HasDerivWithinAt f f' (Ici u) u := by + apply hf.mono_of_mem_nhdsWithin + filter_upwards [self_mem_nhdsWithin, mem_nhdsWithin_of_mem_nhds (Iic_mem_nhds hu.2)] with x hx hxT + exact ⟨hu.1.trans hx, hxT⟩ + +lemma logLoewnerCoordinates_initial + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (z : ℂ) (hz : z ∈ survivingDomain γ 0) : + logLoewnerCoordinates G z 0 = ((z.re : ℂ) + (Real.log z.im : ℂ) * Complex.I, 0) := by + simp only [logLoewnerCoordinates, Real.toNNReal_zero, h.initial_value z hz.1.1, + loewner_spatial_derivative hU h 0 z hz, NNReal.coe_zero, + intervalIntegral.integral_same, Complex.exp_zero, norm_one, Real.log_one] + + + +theorem clipped_log_loewner_agrees + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ T) + (lo hi : ℝ) (hle : lo ≤ hi) + (hlo : lo ≤ Real.log (G T z).im) (hhi : Real.log z.im ≤ hi) + (Q : ℝ → ℂ × ℝ) + (hQ0 : Q 0 = ((z.re : ℂ) + (Real.log z.im : ℂ) * Complex.I, 0)) + (hQ : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q + (clippedLogLoewnerField lo hi hle (U t.toNNReal) (Q t)) (Icc 0 (T : ℝ)) t) : + EqOn Q (logLoewnerCoordinates G z) (Icc 0 (T : ℝ)) := by + obtain ⟨L, K, hb, hl⟩ := clippedLogLoewnerField_bounds lo hi hle + have hd := logLoewnerCoordinates_derivative hU h T z hz lo hi hle hlo hhi + apply ODE_solution_unique (fun t => hl (U t.toNNReal)) + (fun t ht => (hQ t ht).continuousWithinAt) + (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hQ t ⟨ht.1, ht.2.le⟩)) + (fun t ht => (hd t ht).continuousWithinAt) + (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hd t ⟨ht.1, ht.2.le⟩)) + exact hQ0.trans (logLoewnerCoordinates_initial hU h z (survivingDomain_antitone γ bot_le hz)).symm + + + +theorem loewner_survival_of_positive_solution + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (h : IsCapacityTwoWitness U γ G F) + (T : ℝ≥0) (W : ℝ → ℂ) + (hWp : ∀ u ∈ Icc 0 (T : ℝ), 0 < (W u).im) + (hWd : ∀ u ∈ Icc 0 (T : ℝ), HasDerivWithinAt W + (2 / (W u - (U u.toNNReal : ℂ))) (Icc 0 (T : ℝ)) u) : + W 0 ∈ survivingDomain γ T ∧ G T (W 0) = W T := by + let w := F T (W T) + have hWT : W T ∈ upperHalfPlane := hWp T ⟨T.coe_nonneg, le_rfl⟩ + have hw : w ∈ survivingDomain γ T := h.2.2.2.2.2.2.1 T hWT + have hGw : G T w = W T := h.2.2.2.2.2.2.2.2.1 T (W T) hWT + have hWc : ContinuousOn W (Icc 0 (T : ℝ)) := fun u hu => (hWd u hu).continuousWithinAt + have hWnz (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : W u - (U u.toNNReal : ℂ) ≠ 0 := by + intro he + have hh := congrArg Complex.im he + simp only [Complex.sub_im, Complex.ofReal_im, sub_zero, Complex.zero_im] at hh + exact (hWp u hu).ne' hh + let c (u : ℝ) : ℂ := (-2 : ℂ) / + ((G u.toNNReal w - (U u.toNNReal : ℂ)) * (W u - (U u.toNNReal : ℂ))) + have hUc : Continuous (fun u : ℝ => (U u.toNNReal : ℂ)) := + Complex.continuous_ofReal.comp (hU.comp continuous_real_toNNReal) + have hc : ContinuousOn c (Icc 0 (T : ℝ)) := + continuousOn_const.div + (((h.time_continuousOn T w hw).sub hUc.continuousOn).mul (hWc.sub hUc.continuousOn)) + (fun u hu => mul_ne_zero (h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2)) (hWnz u hu)) + have hd (u : ℝ) (hu : u ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun s : ℝ => G s.toNNReal w - W s) + (c u * (G u.toNNReal w - W u)) (Icc 0 (T : ℝ)) u := by + have hg := h.centered_ne_zero _ w (h.live_at_real_time T w hw u hu.2) + have hn := hWnz u hu + have he : c u * (G u.toNNReal w - W u) = + 2 / (G u.toNNReal w - (U u.toNNReal : ℂ)) - 2 / (W u - (U u.toNNReal : ℂ)) := by + dsimp only [c] + field_simp + ring + rw [he] + exact (h.time_derivative T w hw u hu).sub (hWd u hu) + have he := complex_linear_ode_exp (T : ℝ) T.coe_nonneg + (fun s : ℝ => G s.toNNReal w - W s) c hc hd + simp only [Real.toNNReal_coe, hGw, sub_self, Real.toNNReal_zero, + h.initial_value w hw.1.1] at he + have hw0 : w = W 0 := sub_eq_zero.mp ((mul_eq_zero.mp he.symm).resolve_right (Complex.exp_ne_zero _)) + exact ⟨hw0 ▸ hw, hw0 ▸ hGw⟩ + +end SLEExactGauge +end + + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/LoewnerReflection.lean b/lean/OAI/Probability/SLE/LoewnerReflection.lean index cc44cef37452d15e06728c711c2c195900fecf49..07b955253b7b445593f429f6c66b5e60e72a3e6f 100644 --- a/lean/OAI/Probability/SLE/LoewnerReflection.lean +++ b/lean/OAI/Probability/SLE/LoewnerReflection.lean @@ -1,783 +1,783 @@ -import Mathlib -import OAI.Probability.SLE.LoewnerCharts - -namespace OAI - -/-! Loewner Reflection for chordal SLE. -/ - -section -section - -open Set Filter Complex Metric ComplexConjugate -open scoped Topology NNReal -namespace SLEExactGauge - - - -noncomputable def loewnerMirror : ℂ ≃ₜ ℂ where - toFun z := -conj z - invFun z := -conj z - left_inv z := by simp - right_inv z := by simp - continuous_toFun := Complex.continuous_conj.neg - continuous_invFun := Complex.continuous_conj.neg - -@[simp] lemma loewnerMirror_apply (z : ℂ) : loewnerMirror z = -conj z := rfl -@[simp] lemma loewnerMirror_twice (z : ℂ) : loewnerMirror (loewnerMirror z) = z := by simp -@[simp] lemma loewnerMirror_im (z : ℂ) : (loewnerMirror z).im = z.im := by simp -@[simp] lemma loewnerMirror_re (z : ℂ) : (loewnerMirror z).re = -z.re := by simp -@[simp] lemma loewnerMirror_dist (z w : ℂ) : dist (loewnerMirror z) (loewnerMirror w) = dist z w := by - simp only [loewnerMirror_apply,dist_neg_neg,Complex.dist_conj_conj] -@[simp] lemma loewnerMirror_norm (z : ℂ) : ‖loewnerMirror z‖ = ‖z‖ := by simp - -lemma loewnerMirror_bounded (s : Set ℂ) : - Bornology.IsBounded (loewnerMirror '' s) ↔ Bornology.IsBounded s := by - rw [Metric.isBounded_image_iff,Metric.isBounded_iff] - simp only [loewnerMirror_dist] - -lemma loewnerMirror_free (γ : ℝ≥0 → ℂ) (t : ℝ≥0) (z : ℂ) : - loewnerMirror z ∈ upperHalfPlane \ ((loewnerMirror ∘ γ) '' Icc 0 t) ↔ - z ∈ upperHalfPlane \ (γ '' Icc 0 t) := by - constructor - · rintro ⟨hup,hn⟩ - refine ⟨by simpa only [upperHalfPlane,mem_ofPred_eq,loewnerMirror_im] using hup,?_⟩ - rintro ⟨u,hu,rfl⟩ - exact hn ⟨u,hu,rfl⟩ - · rintro ⟨hup,hn⟩ - refine ⟨by simpa only [upperHalfPlane,mem_ofPred_eq,loewnerMirror_im] using hup,?_⟩ - rintro ⟨u,hu,he⟩ - exact hn ⟨u,hu,loewnerMirror.injective he⟩ - -lemma loewnerMirror_free_image (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : - loewnerMirror '' (upperHalfPlane \ (γ '' Icc 0 t)) = - upperHalfPlane \ ((loewnerMirror ∘ γ) '' Icc 0 t) := by - ext z - constructor - · rintro ⟨w,hw,rfl⟩ - exact (loewnerMirror_free γ t w).mpr hw - · intro hz - refine ⟨loewnerMirror z,?_,loewnerMirror_twice z⟩ - apply (loewnerMirror_free γ t (loewnerMirror z)).mp - simpa only [loewnerMirror_twice] using hz - -lemma loewnerMirror_surviving (γ : ℝ≥0 → ℂ) (t : ℝ≥0) (z : ℂ) : - loewnerMirror z ∈ survivingDomain (loewnerMirror ∘ γ) t ↔ - z ∈ survivingDomain γ t := by - have hcc (hz : z ∈ upperHalfPlane \ (γ '' Icc 0 t)) : - loewnerMirror '' connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z = - connectedComponentIn (upperHalfPlane \ ((loewnerMirror ∘ γ) '' Icc 0 t)) (loewnerMirror z) := by - rw [loewnerMirror.image_connectedComponentIn hz,loewnerMirror_free_image] - constructor - · rintro ⟨hz,hun⟩ - have hz' := (loewnerMirror_free γ t z).mp hz - refine ⟨hz',?_⟩ - intro hb - apply hun - rw [← hcc hz'] - exact (loewnerMirror_bounded _).mpr hb - · rintro ⟨hz,hun⟩ - refine ⟨(loewnerMirror_free γ t z).mpr hz,?_⟩ - intro hb - apply hun - apply (loewnerMirror_bounded _).mp - rwa [hcc hz] - -lemma mirror_conjugate_differentiable {f : ℂ → ℂ} {z : ℂ} - (hf : DifferentiableAt ℂ f (loewnerMirror z)) : - DifferentiableAt ℂ (fun w => loewnerMirror (f (loewnerMirror w))) z := by - have hg : DifferentiableAt ℂ (fun w => f (-w)) (conj z) := - hf.comp (conj z) (differentiableAt_id.neg) - have hh := hg.conj_conj.neg - convert! hh using 1 - simp only [Complex.conj_conj] - -end SLEExactGauge - -namespace SLEExactGauge - - - -lemma IsCapacityTwoWitness.mirror - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (h : IsCapacityTwoWitness U γ G F) : - IsCapacityTwoWitness (fun t => -U t) (loewnerMirror ∘ γ) - (fun t z => loewnerMirror (G t (loewnerMirror z))) - (fun t z => loewnerMirror (F t (loewnerMirror z))) := by - have hm (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain (loewnerMirror ∘ γ) t) : - loewnerMirror z ∈ survivingDomain γ t := by - apply (loewnerMirror_surviving γ t (loewnerMirror z)).mp - simpa only [loewnerMirror_twice] using hz - have hup (z : ℂ) : loewnerMirror z ∈ upperHalfPlane ↔ z ∈ upperHalfPlane := by - simp only [upperHalfPlane,mem_ofPred_eq,loewnerMirror_im] - rcases h with ⟨hc,h0,him,hG,hF,hGm,hFm,hFG,hGF,hG0,hode,hnorm,htip⟩ - refine ⟨loewnerMirror.continuous.comp hc,by simp [h0],?_,?_,?_,?_,?_,?_,?_,?_,?_,?_,?_⟩ - · intro t; simpa only [Function.comp_apply,loewnerMirror_im] using him t - · intro t z hz - exact (mirror_conjugate_differentiable ((hG t (loewnerMirror z) (hm t z hz)).differentiableAt - ((isOpen_survivingDomain hc t).mem_nhds (hm t z hz)))).differentiableWithinAt - · intro t z hz - exact (mirror_conjugate_differentiable ((hF t (loewnerMirror z) ((hup z).mpr hz)).differentiableAt - ((isOpen_lt continuous_const Complex.continuous_im).mem_nhds ((hup z).mpr hz)))).differentiableWithinAt - · intro t z hz - exact (hup _).mpr (hGm t (hm t z hz)) - · intro t z hz - exact (loewnerMirror_surviving γ t _).mpr (hFm t ((hup z).mpr hz)) - · intro t z hz - dsimp only - rw [loewnerMirror_twice,hFG t _ (hm t z hz),loewnerMirror_twice] - · intro t z hz - dsimp only - rw [loewnerMirror_twice,hGF t _ ((hup z).mpr hz),loewnerMirror_twice] - · intro z hz - dsimp only - rw [hG0 _ ((hup z).mpr hz),loewnerMirror_twice] - · intro t z hz - have hd := (hode t (loewnerMirror z) (hm t z hz)).star.neg - convert! hd using 1 - dsimp only - simp only [loewnerMirror_apply,Complex.ofReal_neg,neg_sub_neg,Complex.star_def] - rw [← neg_sub,div_neg] - simp only [map_div₀,map_sub,map_ofNat,Complex.conj_ofReal] - · intro t - have hmir : Tendsto loewnerMirror - (cocompact ℂ ⊓ 𝓟 (survivingDomain (loewnerMirror ∘ γ) t)) - (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) := by - apply tendsto_inf.mpr - constructor - · exact (show Tendsto loewnerMirror (cocompact ℂ) (cocompact ℂ) from - le_of_eq loewnerMirror.map_cocompact).mono_left inf_le_left - · apply tendsto_principal.mpr - exact (eventually_inf_principal.mpr (Filter.Eventually.of_forall fun z hz => hm t z hz)) - have hh := Complex.continuous_conj.continuousAt.tendsto.comp ((hnorm t).comp hmir) - convert! hh using 1 - · ext z - simp only [Function.comp_def,loewnerMirror_apply,map_mul,map_sub,map_neg, - Complex.conj_conj] - ring - · simp only [map_mul,map_ofNat,Complex.conj_ofReal] - · intro t - have hh := loewnerMirror.continuous.continuousAt.tendsto.comp (htip t) - convert! hh using 1 - ext y - apply Complex.ext <;> simp [add_comm] - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory Complex -open scoped Topology NNReal -namespace SLEExactGauge - -noncomputable def rightCenter (a u : ℝ) (z : ℂ) : ℂ := - ((max a (z.re - u) : ℝ) : ℂ) + ((z.im : ℝ) : ℂ) * I - -noncomputable def rightField (a u : ℝ) (z : ℂ) : ℂ := 2 * (rightCenter a u z)⁻¹ - -lemma rightCenter_re (a u : ℝ) (z : ℂ) : (rightCenter a u z).re = max a (z.re-u) := by - simp [rightCenter] - -lemma rightCenter_norm (a u : ℝ) (z : ℂ) : a ≤ ‖rightCenter a u z‖ := - (le_max_left a (z.re-u)).trans ((rightCenter_re a u z).symm.le.trans (Complex.re_le_norm _)) - -lemma rightCenter_ne_zero {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : rightCenter a u z ≠ 0 := - norm_pos_iff.mp (ha.trans_le (rightCenter_norm a u z)) - -lemma rightCenter_eq {a u : ℝ} {z : ℂ} (hz : a ≤ z.re-u) : rightCenter a u z = z - (u : ℂ) := by - apply Complex.ext <;> simp [rightCenter, max_eq_right hz] - -lemma rightCenter_dist (a u : ℝ) (z w : ℂ) : - dist (rightCenter a u z) (rightCenter a u w) ≤ 2 * dist z w := by - have hr : |max a (z.re-u) - max a (w.re-u)| ≤ ‖z-w‖ := by - have h := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul - (z.re-u) (w.re-u) - have hh : |max a (z.re-u) - max a (w.re-u)| ≤ |z.re-w.re| := by - simpa only [Real.dist_eq,NNReal.coe_one,one_mul,id_eq,sub_sub_sub_cancel_right] using h - exact hh.trans (Complex.abs_re_le_norm (z-w)) - have hi : |z.im-w.im| ≤ ‖z-w‖ := Complex.abs_im_le_norm (z-w) - have he : rightCenter a u z - rightCenter a u w = - ((max a (z.re-u) - max a (w.re-u) : ℝ) : ℂ) + ((z.im-w.im : ℝ) : ℂ)*I := by - simp only [rightCenter,Complex.ofReal_sub]; ring - rw [dist_eq_norm,he] - have hb := norm_add_le ((max a (z.re-u)-max a (w.re-u) : ℝ) : ℂ) (((z.im-w.im : ℝ) : ℂ)*I) - simp only [norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one] at hb - rw [dist_eq_norm] - linarith - -lemma rightField_bound {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : - ‖rightField a u z‖ ≤ 2 * a⁻¹ := by - have hi : ‖(rightCenter a u z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv] - exact inv_anti₀ ha (rightCenter_norm a u z) - simpa only [rightField, norm_mul, norm_ofNat] using mul_le_mul_of_nonneg_left hi (by norm_num : (0 : ℝ) ≤ 2) - -lemma rightField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : - LipschitzWith ⟨4 * a⁻¹ ^ 2, by positivity⟩ (rightField a u) := by - apply LipschitzWith.of_dist_le_mul - intro z w - have hz : ‖(rightCenter a u z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv]; exact inv_anti₀ ha (rightCenter_norm a u z) - have hw : ‖(rightCenter a u w)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv]; exact inv_anti₀ ha (rightCenter_norm a u w) - rw [dist_eq_norm, rightField, rightField, ← mul_sub, norm_mul, norm_ofNat, - inv_sub_inv' (rightCenter_ne_zero ha _ _) (rightCenter_ne_zero ha _ _), norm_mul, norm_mul] - calc - _ ≤ 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) := by - gcongr - simpa only [← dist_eq_norm, dist_comm] using rightCenter_dist a u z w - _ = _ := by change 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) = (4 * a⁻¹ ^ 2) * dist z w; ring - -lemma continuous_rightField {a : ℝ} (ha : 0 < a) : - Continuous (fun p : ℝ × ℂ => rightField a p.1 p.2) := by - apply continuous_const.mul - apply Continuous.inv₀ - · exact ((Complex.continuous_ofReal.comp (continuous_const.max - ((Complex.continuous_re.comp continuous_snd).sub continuous_fst))).add - ((Complex.continuous_ofReal.comp (Complex.continuous_im.comp continuous_snd)).mul continuous_const)) - · intro p; exact rightCenter_ne_zero ha _ _ - -lemma rightField_eq {a u : ℝ} {z : ℂ} (hz : a ≤ z.re-u) : - rightField a u z = 2/(z-(u:ℂ)) := by - simp only [rightField,rightCenter_eq hz,div_eq_mul_inv] - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory Complex -open scoped Topology NNReal -namespace SLEExactGauge - -noncomputable def reverseCenter (a u : ℝ) (z : ℂ) : ℂ := - ((z.re - u : ℝ) : ℂ) + ((max a z.im : ℝ) : ℂ) * I - -noncomputable def reverseField (a u : ℝ) (z : ℂ) : ℂ := -2 * (reverseCenter a u z)⁻¹ - -lemma reverseCenter_im (a u : ℝ) (z : ℂ) : (reverseCenter a u z).im = max a z.im := by - simp [reverseCenter] - -lemma reverseCenter_norm (a u : ℝ) (z : ℂ) : a ≤ ‖reverseCenter a u z‖ := - (le_max_left a z.im).trans ((reverseCenter_im a u z).symm.le.trans (Complex.im_le_norm _)) - -lemma reverseCenter_ne_zero {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : reverseCenter a u z ≠ 0 := - norm_pos_iff.mp (ha.trans_le (reverseCenter_norm a u z)) - -lemma reverseCenter_eq {a u : ℝ} {z : ℂ} (hz : a ≤ z.im) : reverseCenter a u z = z - (u : ℂ) := by - apply Complex.ext <;> simp [reverseCenter, max_eq_right hz] - -lemma reverseCenter_dist (a u : ℝ) (z w : ℂ) : - dist (reverseCenter a u z) (reverseCenter a u w) ≤ 2 * dist z w := by - have hr : |z.re - w.re| ≤ ‖z - w‖ := Complex.abs_re_le_norm (z - w) - have hi : |max a z.im - max a w.im| ≤ ‖z - w‖ := by - have h := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul z.im w.im - have hh : |max a z.im - max a w.im| ≤ |z.im - w.im| := by - simpa only [Real.dist_eq, NNReal.coe_one, one_mul, id_eq] using h - exact hh.trans (Complex.abs_im_le_norm (z - w)) - have he : reverseCenter a u z - reverseCenter a u w = - ((z.re - w.re : ℝ) : ℂ) + ((max a z.im - max a w.im : ℝ) : ℂ) * I := by - simp only [reverseCenter, Complex.ofReal_sub] - ring - rw [dist_eq_norm, he] - have hb := norm_add_le ((z.re - w.re : ℝ) : ℂ) (((max a z.im - max a w.im : ℝ) : ℂ) * I) - simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, mul_one] at hb - rw [dist_eq_norm] - linarith - -lemma reverseField_bound {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : - ‖reverseField a u z‖ ≤ 2 * a⁻¹ := by - have hi : ‖(reverseCenter a u z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv] - exact inv_anti₀ ha (reverseCenter_norm a u z) - simpa only [reverseField, norm_mul, norm_neg, norm_ofNat] using mul_le_mul_of_nonneg_left hi (by norm_num : (0 : ℝ) ≤ 2) - -lemma reverseField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : - LipschitzWith ⟨4 * a⁻¹ ^ 2, by positivity⟩ (reverseField a u) := by - apply LipschitzWith.of_dist_le_mul - intro z w - have hz : ‖(reverseCenter a u z)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv]; exact inv_anti₀ ha (reverseCenter_norm a u z) - have hw : ‖(reverseCenter a u w)⁻¹‖ ≤ a⁻¹ := by - rw [norm_inv]; exact inv_anti₀ ha (reverseCenter_norm a u w) - rw [dist_eq_norm, reverseField, reverseField, ← mul_sub, norm_mul, norm_neg, norm_ofNat, - inv_sub_inv' (reverseCenter_ne_zero ha _ _) (reverseCenter_ne_zero ha _ _), norm_mul, norm_mul] - calc - _ ≤ 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) := by - gcongr - simpa only [← dist_eq_norm, dist_comm] using reverseCenter_dist a u z w - _ = _ := by change 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) = (4 * a⁻¹ ^ 2) * dist z w; ring - -lemma continuous_reverseField {a : ℝ} (ha : 0 < a) : - Continuous (fun p : ℝ × ℂ => reverseField a p.1 p.2) := by - apply continuous_const.mul - apply Continuous.inv₀ - · exact ((Complex.continuous_ofReal.comp ((Complex.continuous_re.comp continuous_snd).sub continuous_fst)).add - ((Complex.continuous_ofReal.comp (continuous_const.max (Complex.continuous_im.comp continuous_snd))).mul continuous_const)) - · intro p; exact reverseCenter_ne_zero ha _ _ - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory Complex -open scoped Topology NNReal -namespace SLEExactGauge - - - -lemma exists_random_initial_bounded_flow - {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E] - [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] - (V : ℝ → Ω → E → E) (q₀ : Ω → E) (hq₀ : Measurable q₀) (L K : ℝ≥0) - (hV : Measurable (fun p : (ℝ × Ω) × E => V p.1.1 p.1.2 p.2)) - (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x)) - (hl : ∀ ω t, LipschitzWith K (V t ω)) - (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) : - ∃ X : ℝ → Ω → E, - Measurable (fun p : ℝ × Ω => X p.1 p.2) ∧ - (∀ ω, X 0 ω = q₀ ω) ∧ - (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => X s ω) (V t ω (X t ω)) (Ici 0) t) := by - let W (t : ℝ) (ω : Ω) (x : E) := V t ω (q₀ ω + x) - have hW : Measurable (fun p : (ℝ × Ω) × E => W p.1.1 p.1.2 p.2) := - hV.comp (measurable_fst.prodMk ((hq₀.comp (measurable_snd.comp measurable_fst)).add measurable_snd)) - have hWl (ω : Ω) (t : ℝ) : LipschitzWith K (W t ω) := by - apply LipschitzWith.of_dist_le_mul - intro x y - simpa only [W, dist_add_left] using (hl ω t).dist_le_mul (q₀ ω + x) (q₀ ω + y) - obtain ⟨Y, hYm, _, hY0, hYd, _⟩ := exists_measurable_global_bounded_ode_picard - W 0 L K hW (fun ω x => hc ω (q₀ ω + x)) hWl (fun ω t x => hb ω t (q₀ ω + x)) - refine ⟨fun t ω => q₀ ω + Y t ω, (hq₀.comp measurable_snd).add hYm, ?_, ?_⟩ - · intro ω; change q₀ ω + Y 0 ω = q₀ ω; rw [hY0, add_zero] - · intro ω t ht - exact (hYd ω t ht).const_add (q₀ ω) - -lemma exists_measurable_reverse_flow - {Ω : Type*} [MeasurableSpace Ω] - (U : ℝ → Ω → ℝ) (hUm : Measurable (fun p : ℝ × Ω => U p.1 p.2)) - (hUc : ∀ ω, Continuous (fun t => U t ω)) - (w : Ω → ℂ) (hwm : Measurable w) {a : ℝ} (ha : 0 < a) : - ∃ Q : ℝ → Ω → ℂ, - Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧ - (∀ ω, Q 0 ω = w ω) ∧ - (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) - (reverseField a (U t ω) (Q t ω)) (Ici 0) t) := by - have hm : Measurable (fun p : (ℝ × Ω) × ℂ => reverseField a (U p.1.1 p.1.2) p.2) := by - have hj : Measurable (fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) := - (hUm.comp measurable_fst).prodMk measurable_snd - have hf : Measurable (fun q : ℝ × ℂ => reverseField a q.1 q.2) := - (continuous_reverseField ha).measurable - exact Measurable.comp (g := fun q : ℝ × ℂ => reverseField a q.1 q.2) - (f := fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) hf hj - have hc (ω : Ω) (z : ℂ) : Continuous (fun t : ℝ => reverseField a (U t ω) z) := - Continuous.comp (g := fun q : ℝ × ℂ => reverseField a q.1 q.2) - (f := fun t : ℝ => (U t ω, z)) (continuous_reverseField ha) - ((hUc ω).prodMk continuous_const) - exact exists_random_initial_bounded_flow (fun t ω z => reverseField a (U t ω) z) - w hwm ⟨2 * a⁻¹, by positivity⟩ ⟨4 * a⁻¹ ^ 2, by positivity⟩ hm hc - (fun ω t => reverseField_lipschitz ha _) (fun ω t z => reverseField_bound ha _ _) - -variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - -lemma IsCapacityTwoWitness.reverse_path_derivative - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w : ℂ) {a : ℝ} - (ha : 0 < a) (haw : a ≤ w.im) (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : - HasDerivWithinAt (fun t : ℝ => G ((T : ℝ) - t).toNNReal (F T w)) - (reverseField a (U ((T : ℝ) - s).toNNReal) (G ((T : ℝ) - s).toNNReal (F T w))) - (Icc 0 (T : ℝ)) s := by - have hw : w ∈ upperHalfPlane := ha.trans_le haw - have hz : F T w ∈ survivingDomain γ T := h.2.2.2.2.2.2.1 T hw - have hi : G T (F T w) = w := h.2.2.2.2.2.2.2.2.1 T w hw - have hmaps : MapsTo (fun t : ℝ => (T : ℝ) - t) (Icc 0 (T : ℝ)) (Icc 0 (T : ℝ)) := by - intro t ht; constructor <;> linarith [ht.1, ht.2] - have hd := (h.time_derivative T (F T w) hz ((T : ℝ) - s) (hmaps hs)).scomp s - ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt hmaps - have hh : a ≤ (G ((T : ℝ) - s).toNNReal (F T w)).im := by - have hh := h.imag_antitoneOn T (F T w) hz (hmaps hs) ⟨T.coe_nonneg, le_rfl⟩ (hmaps hs).2 - simp only [Real.toNNReal_coe, hi] at hh - exact haw.trans hh - simp only [Function.comp_def, neg_smul, one_smul] at hd - simpa only [reverseField, reverseCenter_eq hh, div_eq_mul_inv, neg_mul] using hd - - - -theorem IsCapacityTwoWitness.reverse_flow_identifies - (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w : ℂ) {a : ℝ} - (ha : 0 < a) (haw : a ≤ w.im) (Q : ℝ → ℂ) (hQ0 : Q 0 = w) - (hQd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q - (reverseField a (U ((T : ℝ) - t).toNNReal) (Q t)) (Icc 0 (T : ℝ)) t) : - Q T = F T w := by - have hw : w ∈ upperHalfPlane := ha.trans_le haw - have hp := h.reverse_path_derivative T w ha haw - have he : EqOn Q (fun t : ℝ => G ((T : ℝ) - t).toNNReal (F T w)) (Icc 0 (T : ℝ)) := by - apply ODE_solution_unique (fun t => reverseField_lipschitz ha (U ((T : ℝ) - t).toNNReal)) - (fun t ht => (hQd t ht).continuousWithinAt) - (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hQd t ⟨ht.1, ht.2.le⟩)) - (fun t ht => (hp t ht).continuousWithinAt) - (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hp t ⟨ht.1, ht.2.le⟩)) - simpa only [sub_zero, Real.toNNReal_coe, h.2.2.2.2.2.2.2.2.1 T w hw] using hQ0 - have hh := he ⟨T.coe_nonneg, le_rfl⟩ - simpa only [sub_self, Real.toNNReal_zero, h.initial_value (F T w) - (h.2.2.2.2.2.2.1 T hw).1.1] using hh - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory Complex -open scoped Topology NNReal -namespace SLEExactGauge - -lemma exists_measurable_right_flow - {Ω : Type*} [MeasurableSpace Ω] - (U : ℝ → Ω → ℝ) (hUm : Measurable (fun p : ℝ × Ω => U p.1 p.2)) - (hUc : ∀ ω, Continuous (fun t => U t ω)) - (w : Ω → ℂ) (hwm : Measurable w) {a : ℝ} (ha : 0 < a) : - ∃ Q : ℝ → Ω → ℂ, - Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧ - (∀ ω, Q 0 ω = w ω) ∧ - (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) - (rightField a (U t ω) (Q t ω)) (Ici 0) t) := by - have hm : Measurable (fun p : (ℝ × Ω) × ℂ => rightField a (U p.1.1 p.1.2) p.2) := by - have hj : Measurable (fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) := - (hUm.comp measurable_fst).prodMk measurable_snd - have hf : Measurable (fun q : ℝ × ℂ => rightField a q.1 q.2) := - (continuous_rightField ha).measurable - exact Measurable.comp (g := fun q : ℝ × ℂ => rightField a q.1 q.2) - (f := fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) hf hj - have hc (ω : Ω) (z : ℂ) : Continuous (fun t : ℝ => rightField a (U t ω) z) := - Continuous.comp (g := fun q : ℝ × ℂ => rightField a q.1 q.2) - (f := fun t : ℝ => (U t ω, z)) (continuous_rightField ha) - ((hUc ω).prodMk continuous_const) - exact exists_random_initial_bounded_flow (fun t ω z => rightField a (U t ω) z) - w hwm ⟨2 * a⁻¹, by positivity⟩ ⟨4 * a⁻¹ ^ 2, by positivity⟩ hm hc - (fun ω t => rightField_lipschitz ha _) (fun ω t z => rightField_bound ha _ _) - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory -open scoped Topology NNReal -namespace SLEExactGauge - - - -noncomputable def positiveBoundaryField (a u x : ℝ) : ℝ := - 2 * (max a (x-u))⁻¹ - -lemma positiveBoundaryField_eq {a u x : ℝ} (h : a ≤ x-u) : - positiveBoundaryField a u x = 2/(x-u) := by - simp only [positiveBoundaryField,max_eq_right h,div_eq_mul_inv] - -lemma positiveBoundaryField_bound {a : ℝ} (ha : 0 < a) (u x : ℝ) : - ‖positiveBoundaryField a u x‖ ≤ 2*a⁻¹ := by - have hp : 0 < max a (x-u) := ha.trans_le (le_max_left _ _) - simp only [positiveBoundaryField,Real.norm_eq_abs,abs_mul,abs_of_pos (by norm_num : (0:ℝ) < 2),abs_inv, - abs_of_pos hp] - exact mul_le_mul_of_nonneg_left (inv_anti₀ ha (le_max_left _ _)) (by norm_num) - -lemma positiveBoundaryField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : - LipschitzWith ⟨2*a⁻¹^2,by positivity⟩ (positiveBoundaryField a u) := by - apply LipschitzWith.of_dist_le_mul - intro x y - have hx : 0 < max a (x-u) := ha.trans_le (le_max_left _ _) - have hy : 0 < max a (y-u) := ha.trans_le (le_max_left _ _) - have hi (z : ℝ) : ‖(max a (z-u))⁻¹‖ ≤ a⁻¹ := by - rw [Real.norm_eq_abs,abs_inv,abs_of_pos (ha.trans_le (le_max_left _ _))] - exact inv_anti₀ ha (le_max_left _ _) - have hm : ‖max a (y-u)-max a (x-u)‖ ≤ dist x y := by - have hh := ((LipschitzWith.id : LipschitzWith 1 (fun z : ℝ => z)).const_max a).dist_le_mul - (y-u) (x-u) - simpa only [NNReal.coe_one,one_mul,id_eq,dist_sub_right,dist_eq_norm,norm_sub_rev,sub_sub_sub_cancel_right] using hh - rw [dist_eq_norm,positiveBoundaryField,positiveBoundaryField,← mul_sub,norm_mul, - Real.norm_of_nonneg (by norm_num : (0:ℝ) ≤ 2), - inv_sub_inv' hx.ne' hy.ne',norm_mul,norm_mul] - calc - _ ≤ 2*(a⁻¹ * dist x y * a⁻¹) := by gcongr <;> exact hi _ - _ = _ := by change 2*(a⁻¹*dist x y*a⁻¹)=(2*a⁻¹^2)*dist x y; ring - -lemma continuous_positiveBoundaryField {a : ℝ} (ha : 0 < a) : - Continuous (fun p : ℝ × ℝ => positiveBoundaryField a p.1 p.2) := by - apply continuous_const.mul - apply Continuous.inv₀ - · exact continuous_const.max (continuous_snd.sub continuous_fst) - · intro p; exact (ha.trans_le (le_max_left a (p.2-p.1))).ne' - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal -namespace SLEExactGauge - - - -theorem positive_boundary_global_adapted_solution - {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ) - (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2)) - (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω)) - (a : ℝ) (ha : 0 < a) (q₀ : ℝ) : - ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℝ), - Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧ - (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧ - (∀ ω, Q 0 ω = q₀) ∧ - (∀ ω t, 0 ≤ t → - HasDerivWithinAt (fun s => Q s ω) - (positiveBoundaryField a (U t.toNNReal ω) (Q t ω)) - (Ici 0) t) ∧ - Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable)) - (fun t : ℝ≥0 => Q t) := by - let L : ℝ≥0 := ⟨2*a⁻¹,by positivity⟩ - let K : ℝ≥0 := ⟨2*a⁻¹^2,by positivity⟩ - have hbound := positiveBoundaryField_bound ha - have hlip := positiveBoundaryField_lipschitz ha - let V (t : ℝ) (ω : Ω) (q : ℝ) := positiveBoundaryField a (U t.toNNReal ω) q - have hF := continuous_positiveBoundaryField ha - have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) := - hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd) - have hm : Measurable (fun q : (ℝ × Ω) × (ℝ) => V q.1.1 q.1.2 q.2) := by - change Measurable ((fun p : ℝ × (ℝ) => positiveBoundaryField a p.1 p.2) ∘ - (fun q : (ℝ × Ω) × (ℝ) => (U q.1.1.toNNReal q.1.2, q.2))) - exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd) - have hcont (ω : Ω) (q : ℝ) : Continuous (fun t : ℝ => V t ω q) := by - change Continuous ((fun p : ℝ × (ℝ) => positiveBoundaryField a p.1 p.2) ∘ - (fun t : ℝ => (U t.toNNReal ω, q))) - exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const) - obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ := - exists_measurable_global_bounded_ode_picard V q₀ L K hm hcont - (fun ω t => hlip (U t.toNNReal ω)) (fun ω t q => hbound (U t.toNNReal ω) q) - refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩ - intro t - have hUfrozen := measurable_driver_frozen U hU hc t - let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t - let : MeasurableSpace Ω := m - let W (s : ℝ) (ω : Ω) (q : ℝ) := - positiveBoundaryField a (U (min s.toNNReal t) ω) q - have hW : Measurable (fun q : (ℝ × Ω) × (ℝ) => W q.1.1 q.1.2 q.2) := by - change Measurable ((fun p : ℝ × (ℝ) => positiveBoundaryField a p.1 p.2) ∘ - (fun q : (ℝ × Ω) × (ℝ) => (U (min q.1.1.toNNReal t) q.1.2, q.2))) - exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd) - have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n (t : ℝ) ω = picardApprox V q₀ n (t : ℝ) ω := by - apply picardApprox_congr_on q₀ (t : ℝ) _ n (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩ ω - intro s hs ω q - have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hs.2 - simp only [W, V, min_eq_left hh] - apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n (t : ℝ) ω) - · intro n - exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left - · apply tendsto_pi_nhds.mpr - intro ω - simpa only [heq, max_eq_right t.coe_nonneg] using hQconv t ω - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory Complex -open scoped Topology NNReal -namespace SLEExactGauge - -noncomputable def boundaryClockField (a u : ℝ) (q : ℂ) : ℂ := - (positiveBoundaryField a u q.re : ℂ) + ((max a (q.re-u))⁻¹^2 : ℝ) * I - -lemma clipped_real_inv_bound {a : ℝ} (ha : 0 < a) (x : ℝ) : - ‖(max a x)⁻¹‖ ≤ a⁻¹ := by - rw [Real.norm_eq_abs,abs_inv,abs_of_pos (ha.trans_le (le_max_left _ _))] - exact inv_anti₀ ha (le_max_left _ _) - -lemma clipped_real_inv_lipschitz {a : ℝ} (ha : 0 < a) : - LipschitzWith ⟨a⁻¹^2,by positivity⟩ (fun x : ℝ => (max a x)⁻¹) := by - apply LipschitzWith.of_dist_le_mul - intro x y - have hx : max a x ≠ 0 := (ha.trans_le (le_max_left _ _)).ne' - have hy : max a y ≠ 0 := (ha.trans_le (le_max_left _ _)).ne' - have hmax : ‖max a y-max a x‖ ≤ dist x y := by - have hh := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul y x - simpa only [Real.dist_eq,NNReal.coe_one,one_mul,id_eq,Real.norm_eq_abs,abs_sub_comm] using hh - rw [dist_eq_norm,inv_sub_inv' hx hy,norm_mul,norm_mul] - calc - _ ≤ a⁻¹*dist x y*a⁻¹ := by gcongr <;> exact clipped_real_inv_bound ha _ - _ = _ := by change a⁻¹*dist x y*a⁻¹ = a⁻¹^2*dist x y; ring - -lemma boundaryClockField_bound {a : ℝ} (ha : 0 < a) (u : ℝ) (q : ℂ) : - ‖boundaryClockField a u q‖ ≤ 2*a⁻¹+a⁻¹^2 := by - have h0 := positiveBoundaryField_bound ha u q.re - have h1 := clipped_real_inv_bound ha (q.re-u) - have h2 : ‖(max a (q.re-u))⁻¹^2‖ ≤ a⁻¹^2 := by - rw [norm_pow] - gcongr - have hh := norm_add_le (positiveBoundaryField a u q.re : ℂ) (((max a (q.re-u))⁻¹^2 : ℝ)*I) - simp only [norm_mul,Complex.norm_real,Complex.norm_I,mul_one] at hh - exact hh.trans (add_le_add h0 h2) - -lemma boundaryClockField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : - LipschitzWith ⟨2*a⁻¹^2+2*a⁻¹^3,by positivity⟩ (boundaryClockField a u) := by - apply LipschitzWith.of_dist_le_mul - intro q w - have hd : dist (q.re-u) (w.re-u) ≤ dist q w := by - simpa only [dist_sub_right,dist_eq_norm,Complex.sub_re,Real.norm_eq_abs,sub_sub_sub_cancel_right] using Complex.abs_re_le_norm (q-w) - have hi : ‖(max a (q.re-u))⁻¹-(max a (w.re-u))⁻¹‖ ≤ a⁻¹^2*dist q w := by - exact ((clipped_real_inv_lipschitz ha).dist_le_mul _ _).trans (mul_le_mul_of_nonneg_left hd (sq_nonneg _)) - have hsum : ‖(max a (q.re-u))⁻¹+(max a (w.re-u))⁻¹‖ ≤ 2*a⁻¹ := by - exact (norm_add_le _ _).trans (by linarith [clipped_real_inv_bound ha (q.re-u),clipped_real_inv_bound ha (w.re-u)]) - have hsq : ‖(max a (q.re-u))⁻¹^2-(max a (w.re-u))⁻¹^2‖ ≤ 2*a⁻¹^3*dist q w := by - rw [sq_sub_sq,norm_mul,mul_comm] - calc - _ ≤ (a⁻¹^2*dist q w)*(2*a⁻¹) := mul_le_mul hi hsum (norm_nonneg _) (by positivity) - _ = _ := by ring - have hr := (positiveBoundaryField_lipschitz ha u).dist_le_mul q.re w.re - have hr' : ‖positiveBoundaryField a u q.re-positiveBoundaryField a u w.re‖ ≤ 2*a⁻¹^2*dist q w := by - rw [← dist_eq_norm] - apply hr.trans - change (2*a⁻¹^2)*dist q.re w.re ≤ (2*a⁻¹^2)*dist q w - exact mul_le_mul_of_nonneg_left (by simpa only [dist_sub_right] using hd) (by positivity) - rw [dist_eq_norm] - have he : boundaryClockField a u q-boundaryClockField a u w = - ((positiveBoundaryField a u q.re-positiveBoundaryField a u w.re:ℝ):ℂ)+ - (((max a (q.re-u))⁻¹^2-(max a (w.re-u))⁻¹^2:ℝ):ℂ)*I := by - simp only [boundaryClockField,Complex.ofReal_sub]; ring - rw [he] - have hn := norm_add_le ((positiveBoundaryField a u q.re-positiveBoundaryField a u w.re:ℝ):ℂ) - ((((max a (q.re-u))⁻¹^2-(max a (w.re-u))⁻¹^2:ℝ):ℂ)*I) - simp only [norm_mul,Complex.norm_real,Complex.norm_I,mul_one] at hn - calc - _ ≤ 2*a⁻¹^2*dist q w+2*a⁻¹^3*dist q w := hn.trans (add_le_add hr' hsq) - _ = _ := by change _ = (2*a⁻¹^2+2*a⁻¹^3)*dist q w; ring - -lemma continuous_boundaryClockField {a : ℝ} (ha : 0 < a) : - Continuous (fun p : ℝ × ℂ => boundaryClockField a p.1 p.2) := by - apply Continuous.add - · exact Complex.continuous_ofReal.comp ((continuous_positiveBoundaryField ha).comp - (continuous_fst.prodMk (Complex.continuous_re.comp continuous_snd))) - · apply Continuous.mul _ continuous_const - apply Complex.continuous_ofReal.comp - apply Continuous.pow - apply Continuous.inv₀ - · exact continuous_const.max ((Complex.continuous_re.comp continuous_snd).sub continuous_fst) - · intro p; exact (ha.trans_le (le_max_left _ _)).ne' - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal -namespace SLEExactGauge - - - -theorem boundary_clock_global_adapted_solution - {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ) - (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2)) - (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω)) - (κ a : ℝ) (ha : 0 < a) (q₀ : ℂ) : - ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ), - Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧ - (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧ - (∀ ω, Q 0 ω = q₀) ∧ - (∀ ω t, 0 ≤ t → - HasDerivWithinAt (fun s => Q s ω) - (boundaryClockField a (Real.sqrt κ*U t.toNNReal ω) (Q t ω)) - (Ici 0) t) ∧ - Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable)) - (fun t : ℝ≥0 => Q t) := by - let L : ℝ≥0 := ⟨2*a⁻¹+a⁻¹^2,by positivity⟩ - let K : ℝ≥0 := ⟨2*a⁻¹^2+2*a⁻¹^3,by positivity⟩ - have hbound := boundaryClockField_bound ha - have hlip := boundaryClockField_lipschitz ha - let V (t : ℝ) (ω : Ω) (q : ℂ) := boundaryClockField a (Real.sqrt κ*U t.toNNReal ω) q - have hF : Continuous (fun p : ℝ × ℂ => boundaryClockField a (Real.sqrt κ*p.1) p.2) := - (continuous_boundaryClockField ha).comp ((continuous_const.mul continuous_fst).prodMk continuous_snd) - have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) := - hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd) - have hm : Measurable (fun q : (ℝ × Ω) × (ℂ) => V q.1.1 q.1.2 q.2) := by - change Measurable ((fun p : ℝ × (ℂ) => boundaryClockField a (Real.sqrt κ*p.1) p.2) ∘ - (fun q : (ℝ × Ω) × (ℂ) => (U q.1.1.toNNReal q.1.2, q.2))) - exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd) - have hcont (ω : Ω) (q : ℂ) : Continuous (fun t : ℝ => V t ω q) := by - change Continuous ((fun p : ℝ × (ℂ) => boundaryClockField a (Real.sqrt κ*p.1) p.2) ∘ - (fun t : ℝ => (U t.toNNReal ω, q))) - exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const) - obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ := - exists_measurable_global_bounded_ode_picard V q₀ L K hm hcont - (fun ω t => hlip (Real.sqrt κ*U t.toNNReal ω)) (fun ω t q => hbound (Real.sqrt κ*U t.toNNReal ω) q) - refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩ - intro t - have hUfrozen := measurable_driver_frozen U hU hc t - let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t - let : MeasurableSpace Ω := m - let W (s : ℝ) (ω : Ω) (q : ℂ) := - boundaryClockField a (Real.sqrt κ*U (min s.toNNReal t) ω) q - have hW : Measurable (fun q : (ℝ × Ω) × (ℂ) => W q.1.1 q.1.2 q.2) := by - change Measurable ((fun p : ℝ × (ℂ) => boundaryClockField a (Real.sqrt κ*p.1) p.2) ∘ - (fun q : (ℝ × Ω) × (ℂ) => (U (min q.1.1.toNNReal t) q.1.2, q.2))) - exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd) - have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n (t : ℝ) ω = picardApprox V q₀ n (t : ℝ) ω := by - apply picardApprox_congr_on q₀ (t : ℝ) _ n (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩ ω - intro s hs ω q - have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hs.2 - simp only [W, V, min_eq_left hh] - apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n (t : ℝ) ω) - · intro n - exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left - · apply tendsto_pi_nhds.mpr - intro ω - simpa only [heq, max_eq_right t.coe_nonneg] using hQconv t ω - -end SLEExactGauge -end - - -section - -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -namespace SLEExactGauge - - - -structure BoundaryClockSolution {Ω : Type*} [MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (κ a : ℝ) (q₀ : ℂ) where - L : ℝ≥0 - Q : ℝ → Ω → ℂ - measurable : Measurable (fun p : ℝ × Ω => Q p.1 p.2) - lipschitz : ∀ ω, LipschitzWith L (fun t => Q t ω) - initial : ∀ ω, Q 0 ω = q₀ - derivative : ∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) - (boundaryClockField a (Real.sqrt κ * B t.toNNReal ω) (Q t ω)) (Ici 0) t - adapted : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) - (fun t : ℝ≥0 => Q t) - -lemma exists_boundaryClockSolution - {Ω : Type*} [MeasurableSpace Ω] - (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) - (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ a : ℝ) (ha : 0 < a) (q₀ : ℂ) : - Nonempty (BoundaryClockSolution B hm κ a q₀) := by - obtain ⟨L,Q,hQm,hQl,hQ0,hQd,hQa⟩ := boundary_clock_global_adapted_solution B hBm hBc κ a ha q₀ - exact ⟨⟨L,Q,hQm,hQl,hQ0,hQd,hQa⟩⟩ - -end SLEExactGauge -end - - -end - -end OAI +import Mathlib +import OAI.Probability.SLE.LoewnerCharts + +namespace OAI + +/-! Loewner Reflection for chordal SLE. -/ + +section +section + +open Set Filter Complex Metric ComplexConjugate +open scoped Topology NNReal +namespace SLEExactGauge + + + +noncomputable def loewnerMirror : ℂ ≃ₜ ℂ where + toFun z := -conj z + invFun z := -conj z + left_inv z := by simp + right_inv z := by simp + continuous_toFun := Complex.continuous_conj.neg + continuous_invFun := Complex.continuous_conj.neg + +@[simp] lemma loewnerMirror_apply (z : ℂ) : loewnerMirror z = -conj z := rfl +@[simp] lemma loewnerMirror_twice (z : ℂ) : loewnerMirror (loewnerMirror z) = z := by simp +@[simp] lemma loewnerMirror_im (z : ℂ) : (loewnerMirror z).im = z.im := by simp +@[simp] lemma loewnerMirror_re (z : ℂ) : (loewnerMirror z).re = -z.re := by simp +@[simp] lemma loewnerMirror_dist (z w : ℂ) : dist (loewnerMirror z) (loewnerMirror w) = dist z w := by + simp only [loewnerMirror_apply,dist_neg_neg,Complex.dist_conj_conj] +@[simp] lemma loewnerMirror_norm (z : ℂ) : ‖loewnerMirror z‖ = ‖z‖ := by simp + +lemma loewnerMirror_bounded (s : Set ℂ) : + Bornology.IsBounded (loewnerMirror '' s) ↔ Bornology.IsBounded s := by + rw [Metric.isBounded_image_iff,Metric.isBounded_iff] + simp only [loewnerMirror_dist] + +lemma loewnerMirror_free (γ : ℝ≥0 → ℂ) (t : ℝ≥0) (z : ℂ) : + loewnerMirror z ∈ upperHalfPlane \ ((loewnerMirror ∘ γ) '' Icc 0 t) ↔ + z ∈ upperHalfPlane \ (γ '' Icc 0 t) := by + constructor + · rintro ⟨hup,hn⟩ + refine ⟨by simpa only [upperHalfPlane,mem_ofPred_eq,loewnerMirror_im] using hup,?_⟩ + rintro ⟨u,hu,rfl⟩ + exact hn ⟨u,hu,rfl⟩ + · rintro ⟨hup,hn⟩ + refine ⟨by simpa only [upperHalfPlane,mem_ofPred_eq,loewnerMirror_im] using hup,?_⟩ + rintro ⟨u,hu,he⟩ + exact hn ⟨u,hu,loewnerMirror.injective he⟩ + +lemma loewnerMirror_free_image (γ : ℝ≥0 → ℂ) (t : ℝ≥0) : + loewnerMirror '' (upperHalfPlane \ (γ '' Icc 0 t)) = + upperHalfPlane \ ((loewnerMirror ∘ γ) '' Icc 0 t) := by + ext z + constructor + · rintro ⟨w,hw,rfl⟩ + exact (loewnerMirror_free γ t w).mpr hw + · intro hz + refine ⟨loewnerMirror z,?_,loewnerMirror_twice z⟩ + apply (loewnerMirror_free γ t (loewnerMirror z)).mp + simpa only [loewnerMirror_twice] using hz + +lemma loewnerMirror_surviving (γ : ℝ≥0 → ℂ) (t : ℝ≥0) (z : ℂ) : + loewnerMirror z ∈ survivingDomain (loewnerMirror ∘ γ) t ↔ + z ∈ survivingDomain γ t := by + have hcc (hz : z ∈ upperHalfPlane \ (γ '' Icc 0 t)) : + loewnerMirror '' connectedComponentIn (upperHalfPlane \ (γ '' Icc 0 t)) z = + connectedComponentIn (upperHalfPlane \ ((loewnerMirror ∘ γ) '' Icc 0 t)) (loewnerMirror z) := by + rw [loewnerMirror.image_connectedComponentIn hz,loewnerMirror_free_image] + constructor + · rintro ⟨hz,hun⟩ + have hz' := (loewnerMirror_free γ t z).mp hz + refine ⟨hz',?_⟩ + intro hb + apply hun + rw [← hcc hz'] + exact (loewnerMirror_bounded _).mpr hb + · rintro ⟨hz,hun⟩ + refine ⟨(loewnerMirror_free γ t z).mpr hz,?_⟩ + intro hb + apply hun + apply (loewnerMirror_bounded _).mp + rwa [hcc hz] + +lemma mirror_conjugate_differentiable {f : ℂ → ℂ} {z : ℂ} + (hf : DifferentiableAt ℂ f (loewnerMirror z)) : + DifferentiableAt ℂ (fun w => loewnerMirror (f (loewnerMirror w))) z := by + have hg : DifferentiableAt ℂ (fun w => f (-w)) (conj z) := + hf.comp (conj z) (differentiableAt_id.neg) + have hh := hg.conj_conj.neg + convert! hh using 1 + simp only [Complex.conj_conj] + +end SLEExactGauge + +namespace SLEExactGauge + + + +lemma IsCapacityTwoWitness.mirror + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (h : IsCapacityTwoWitness U γ G F) : + IsCapacityTwoWitness (fun t => -U t) (loewnerMirror ∘ γ) + (fun t z => loewnerMirror (G t (loewnerMirror z))) + (fun t z => loewnerMirror (F t (loewnerMirror z))) := by + have hm (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain (loewnerMirror ∘ γ) t) : + loewnerMirror z ∈ survivingDomain γ t := by + apply (loewnerMirror_surviving γ t (loewnerMirror z)).mp + simpa only [loewnerMirror_twice] using hz + have hup (z : ℂ) : loewnerMirror z ∈ upperHalfPlane ↔ z ∈ upperHalfPlane := by + simp only [upperHalfPlane,mem_ofPred_eq,loewnerMirror_im] + rcases h with ⟨hc,h0,him,hG,hF,hGm,hFm,hFG,hGF,hG0,hode,hnorm,htip⟩ + refine ⟨loewnerMirror.continuous.comp hc,by simp [h0],?_,?_,?_,?_,?_,?_,?_,?_,?_,?_,?_⟩ + · intro t; simpa only [Function.comp_apply,loewnerMirror_im] using him t + · intro t z hz + exact (mirror_conjugate_differentiable ((hG t (loewnerMirror z) (hm t z hz)).differentiableAt + ((isOpen_survivingDomain hc t).mem_nhds (hm t z hz)))).differentiableWithinAt + · intro t z hz + exact (mirror_conjugate_differentiable ((hF t (loewnerMirror z) ((hup z).mpr hz)).differentiableAt + ((isOpen_lt continuous_const Complex.continuous_im).mem_nhds ((hup z).mpr hz)))).differentiableWithinAt + · intro t z hz + exact (hup _).mpr (hGm t (hm t z hz)) + · intro t z hz + exact (loewnerMirror_surviving γ t _).mpr (hFm t ((hup z).mpr hz)) + · intro t z hz + dsimp only + rw [loewnerMirror_twice,hFG t _ (hm t z hz),loewnerMirror_twice] + · intro t z hz + dsimp only + rw [loewnerMirror_twice,hGF t _ ((hup z).mpr hz),loewnerMirror_twice] + · intro z hz + dsimp only + rw [hG0 _ ((hup z).mpr hz),loewnerMirror_twice] + · intro t z hz + have hd := (hode t (loewnerMirror z) (hm t z hz)).star.neg + convert! hd using 1 + dsimp only + simp only [loewnerMirror_apply,Complex.ofReal_neg,neg_sub_neg,Complex.star_def] + rw [← neg_sub,div_neg] + simp only [map_div₀,map_sub,map_ofNat,Complex.conj_ofReal] + · intro t + have hmir : Tendsto loewnerMirror + (cocompact ℂ ⊓ 𝓟 (survivingDomain (loewnerMirror ∘ γ) t)) + (cocompact ℂ ⊓ 𝓟 (survivingDomain γ t)) := by + apply tendsto_inf.mpr + constructor + · exact (show Tendsto loewnerMirror (cocompact ℂ) (cocompact ℂ) from + le_of_eq loewnerMirror.map_cocompact).mono_left inf_le_left + · apply tendsto_principal.mpr + exact (eventually_inf_principal.mpr (Filter.Eventually.of_forall fun z hz => hm t z hz)) + have hh := Complex.continuous_conj.continuousAt.tendsto.comp ((hnorm t).comp hmir) + convert! hh using 1 + · ext z + simp only [Function.comp_def,loewnerMirror_apply,map_mul,map_sub,map_neg, + Complex.conj_conj] + ring + · simp only [map_mul,map_ofNat,Complex.conj_ofReal] + · intro t + have hh := loewnerMirror.continuous.continuousAt.tendsto.comp (htip t) + convert! hh using 1 + ext y + apply Complex.ext <;> simp [add_comm] + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory Complex +open scoped Topology NNReal +namespace SLEExactGauge + +noncomputable def rightCenter (a u : ℝ) (z : ℂ) : ℂ := + ((max a (z.re - u) : ℝ) : ℂ) + ((z.im : ℝ) : ℂ) * I + +noncomputable def rightField (a u : ℝ) (z : ℂ) : ℂ := 2 * (rightCenter a u z)⁻¹ + +lemma rightCenter_re (a u : ℝ) (z : ℂ) : (rightCenter a u z).re = max a (z.re-u) := by + simp [rightCenter] + +lemma rightCenter_norm (a u : ℝ) (z : ℂ) : a ≤ ‖rightCenter a u z‖ := + (le_max_left a (z.re-u)).trans ((rightCenter_re a u z).symm.le.trans (Complex.re_le_norm _)) + +lemma rightCenter_ne_zero {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : rightCenter a u z ≠ 0 := + norm_pos_iff.mp (ha.trans_le (rightCenter_norm a u z)) + +lemma rightCenter_eq {a u : ℝ} {z : ℂ} (hz : a ≤ z.re-u) : rightCenter a u z = z - (u : ℂ) := by + apply Complex.ext <;> simp [rightCenter, max_eq_right hz] + +lemma rightCenter_dist (a u : ℝ) (z w : ℂ) : + dist (rightCenter a u z) (rightCenter a u w) ≤ 2 * dist z w := by + have hr : |max a (z.re-u) - max a (w.re-u)| ≤ ‖z-w‖ := by + have h := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul + (z.re-u) (w.re-u) + have hh : |max a (z.re-u) - max a (w.re-u)| ≤ |z.re-w.re| := by + simpa only [Real.dist_eq,NNReal.coe_one,one_mul,id_eq,sub_sub_sub_cancel_right] using h + exact hh.trans (Complex.abs_re_le_norm (z-w)) + have hi : |z.im-w.im| ≤ ‖z-w‖ := Complex.abs_im_le_norm (z-w) + have he : rightCenter a u z - rightCenter a u w = + ((max a (z.re-u) - max a (w.re-u) : ℝ) : ℂ) + ((z.im-w.im : ℝ) : ℂ)*I := by + simp only [rightCenter,Complex.ofReal_sub]; ring + rw [dist_eq_norm,he] + have hb := norm_add_le ((max a (z.re-u)-max a (w.re-u) : ℝ) : ℂ) (((z.im-w.im : ℝ) : ℂ)*I) + simp only [norm_mul,Complex.norm_real,Real.norm_eq_abs,Complex.norm_I,mul_one] at hb + rw [dist_eq_norm] + linarith + +lemma rightField_bound {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : + ‖rightField a u z‖ ≤ 2 * a⁻¹ := by + have hi : ‖(rightCenter a u z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv] + exact inv_anti₀ ha (rightCenter_norm a u z) + simpa only [rightField, norm_mul, norm_ofNat] using mul_le_mul_of_nonneg_left hi (by norm_num : (0 : ℝ) ≤ 2) + +lemma rightField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : + LipschitzWith ⟨4 * a⁻¹ ^ 2, by positivity⟩ (rightField a u) := by + apply LipschitzWith.of_dist_le_mul + intro z w + have hz : ‖(rightCenter a u z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv]; exact inv_anti₀ ha (rightCenter_norm a u z) + have hw : ‖(rightCenter a u w)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv]; exact inv_anti₀ ha (rightCenter_norm a u w) + rw [dist_eq_norm, rightField, rightField, ← mul_sub, norm_mul, norm_ofNat, + inv_sub_inv' (rightCenter_ne_zero ha _ _) (rightCenter_ne_zero ha _ _), norm_mul, norm_mul] + calc + _ ≤ 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) := by + gcongr + simpa only [← dist_eq_norm, dist_comm] using rightCenter_dist a u z w + _ = _ := by change 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) = (4 * a⁻¹ ^ 2) * dist z w; ring + +lemma continuous_rightField {a : ℝ} (ha : 0 < a) : + Continuous (fun p : ℝ × ℂ => rightField a p.1 p.2) := by + apply continuous_const.mul + apply Continuous.inv₀ + · exact ((Complex.continuous_ofReal.comp (continuous_const.max + ((Complex.continuous_re.comp continuous_snd).sub continuous_fst))).add + ((Complex.continuous_ofReal.comp (Complex.continuous_im.comp continuous_snd)).mul continuous_const)) + · intro p; exact rightCenter_ne_zero ha _ _ + +lemma rightField_eq {a u : ℝ} {z : ℂ} (hz : a ≤ z.re-u) : + rightField a u z = 2/(z-(u:ℂ)) := by + simp only [rightField,rightCenter_eq hz,div_eq_mul_inv] + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory Complex +open scoped Topology NNReal +namespace SLEExactGauge + +noncomputable def reverseCenter (a u : ℝ) (z : ℂ) : ℂ := + ((z.re - u : ℝ) : ℂ) + ((max a z.im : ℝ) : ℂ) * I + +noncomputable def reverseField (a u : ℝ) (z : ℂ) : ℂ := -2 * (reverseCenter a u z)⁻¹ + +lemma reverseCenter_im (a u : ℝ) (z : ℂ) : (reverseCenter a u z).im = max a z.im := by + simp [reverseCenter] + +lemma reverseCenter_norm (a u : ℝ) (z : ℂ) : a ≤ ‖reverseCenter a u z‖ := + (le_max_left a z.im).trans ((reverseCenter_im a u z).symm.le.trans (Complex.im_le_norm _)) + +lemma reverseCenter_ne_zero {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : reverseCenter a u z ≠ 0 := + norm_pos_iff.mp (ha.trans_le (reverseCenter_norm a u z)) + +lemma reverseCenter_eq {a u : ℝ} {z : ℂ} (hz : a ≤ z.im) : reverseCenter a u z = z - (u : ℂ) := by + apply Complex.ext <;> simp [reverseCenter, max_eq_right hz] + +lemma reverseCenter_dist (a u : ℝ) (z w : ℂ) : + dist (reverseCenter a u z) (reverseCenter a u w) ≤ 2 * dist z w := by + have hr : |z.re - w.re| ≤ ‖z - w‖ := Complex.abs_re_le_norm (z - w) + have hi : |max a z.im - max a w.im| ≤ ‖z - w‖ := by + have h := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul z.im w.im + have hh : |max a z.im - max a w.im| ≤ |z.im - w.im| := by + simpa only [Real.dist_eq, NNReal.coe_one, one_mul, id_eq] using h + exact hh.trans (Complex.abs_im_le_norm (z - w)) + have he : reverseCenter a u z - reverseCenter a u w = + ((z.re - w.re : ℝ) : ℂ) + ((max a z.im - max a w.im : ℝ) : ℂ) * I := by + simp only [reverseCenter, Complex.ofReal_sub] + ring + rw [dist_eq_norm, he] + have hb := norm_add_le ((z.re - w.re : ℝ) : ℂ) (((max a z.im - max a w.im : ℝ) : ℂ) * I) + simp only [norm_mul, Complex.norm_real, Real.norm_eq_abs, Complex.norm_I, mul_one] at hb + rw [dist_eq_norm] + linarith + +lemma reverseField_bound {a : ℝ} (ha : 0 < a) (u : ℝ) (z : ℂ) : + ‖reverseField a u z‖ ≤ 2 * a⁻¹ := by + have hi : ‖(reverseCenter a u z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv] + exact inv_anti₀ ha (reverseCenter_norm a u z) + simpa only [reverseField, norm_mul, norm_neg, norm_ofNat] using mul_le_mul_of_nonneg_left hi (by norm_num : (0 : ℝ) ≤ 2) + +lemma reverseField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : + LipschitzWith ⟨4 * a⁻¹ ^ 2, by positivity⟩ (reverseField a u) := by + apply LipschitzWith.of_dist_le_mul + intro z w + have hz : ‖(reverseCenter a u z)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv]; exact inv_anti₀ ha (reverseCenter_norm a u z) + have hw : ‖(reverseCenter a u w)⁻¹‖ ≤ a⁻¹ := by + rw [norm_inv]; exact inv_anti₀ ha (reverseCenter_norm a u w) + rw [dist_eq_norm, reverseField, reverseField, ← mul_sub, norm_mul, norm_neg, norm_ofNat, + inv_sub_inv' (reverseCenter_ne_zero ha _ _) (reverseCenter_ne_zero ha _ _), norm_mul, norm_mul] + calc + _ ≤ 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) := by + gcongr + simpa only [← dist_eq_norm, dist_comm] using reverseCenter_dist a u z w + _ = _ := by change 2 * (a⁻¹ * (2 * dist z w) * a⁻¹) = (4 * a⁻¹ ^ 2) * dist z w; ring + +lemma continuous_reverseField {a : ℝ} (ha : 0 < a) : + Continuous (fun p : ℝ × ℂ => reverseField a p.1 p.2) := by + apply continuous_const.mul + apply Continuous.inv₀ + · exact ((Complex.continuous_ofReal.comp ((Complex.continuous_re.comp continuous_snd).sub continuous_fst)).add + ((Complex.continuous_ofReal.comp (continuous_const.max (Complex.continuous_im.comp continuous_snd))).mul continuous_const)) + · intro p; exact reverseCenter_ne_zero ha _ _ + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory Complex +open scoped Topology NNReal +namespace SLEExactGauge + + + +lemma exists_random_initial_bounded_flow + {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E] + [CompleteSpace E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] + (V : ℝ → Ω → E → E) (q₀ : Ω → E) (hq₀ : Measurable q₀) (L K : ℝ≥0) + (hV : Measurable (fun p : (ℝ × Ω) × E => V p.1.1 p.1.2 p.2)) + (hc : ∀ ω x, Continuous (fun t : ℝ => V t ω x)) + (hl : ∀ ω t, LipschitzWith K (V t ω)) + (hb : ∀ ω t x, ‖V t ω x‖ ≤ L) : + ∃ X : ℝ → Ω → E, + Measurable (fun p : ℝ × Ω => X p.1 p.2) ∧ + (∀ ω, X 0 ω = q₀ ω) ∧ + (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => X s ω) (V t ω (X t ω)) (Ici 0) t) := by + let W (t : ℝ) (ω : Ω) (x : E) := V t ω (q₀ ω + x) + have hW : Measurable (fun p : (ℝ × Ω) × E => W p.1.1 p.1.2 p.2) := + hV.comp (measurable_fst.prodMk ((hq₀.comp (measurable_snd.comp measurable_fst)).add measurable_snd)) + have hWl (ω : Ω) (t : ℝ) : LipschitzWith K (W t ω) := by + apply LipschitzWith.of_dist_le_mul + intro x y + simpa only [W, dist_add_left] using (hl ω t).dist_le_mul (q₀ ω + x) (q₀ ω + y) + obtain ⟨Y, hYm, _, hY0, hYd, _⟩ := exists_measurable_global_bounded_ode_picard + W 0 L K hW (fun ω x => hc ω (q₀ ω + x)) hWl (fun ω t x => hb ω t (q₀ ω + x)) + refine ⟨fun t ω => q₀ ω + Y t ω, (hq₀.comp measurable_snd).add hYm, ?_, ?_⟩ + · intro ω; change q₀ ω + Y 0 ω = q₀ ω; rw [hY0, add_zero] + · intro ω t ht + exact (hYd ω t ht).const_add (q₀ ω) + +lemma exists_measurable_reverse_flow + {Ω : Type*} [MeasurableSpace Ω] + (U : ℝ → Ω → ℝ) (hUm : Measurable (fun p : ℝ × Ω => U p.1 p.2)) + (hUc : ∀ ω, Continuous (fun t => U t ω)) + (w : Ω → ℂ) (hwm : Measurable w) {a : ℝ} (ha : 0 < a) : + ∃ Q : ℝ → Ω → ℂ, + Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧ + (∀ ω, Q 0 ω = w ω) ∧ + (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) + (reverseField a (U t ω) (Q t ω)) (Ici 0) t) := by + have hm : Measurable (fun p : (ℝ × Ω) × ℂ => reverseField a (U p.1.1 p.1.2) p.2) := by + have hj : Measurable (fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) := + (hUm.comp measurable_fst).prodMk measurable_snd + have hf : Measurable (fun q : ℝ × ℂ => reverseField a q.1 q.2) := + (continuous_reverseField ha).measurable + exact Measurable.comp (g := fun q : ℝ × ℂ => reverseField a q.1 q.2) + (f := fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) hf hj + have hc (ω : Ω) (z : ℂ) : Continuous (fun t : ℝ => reverseField a (U t ω) z) := + Continuous.comp (g := fun q : ℝ × ℂ => reverseField a q.1 q.2) + (f := fun t : ℝ => (U t ω, z)) (continuous_reverseField ha) + ((hUc ω).prodMk continuous_const) + exact exists_random_initial_bounded_flow (fun t ω z => reverseField a (U t ω) z) + w hwm ⟨2 * a⁻¹, by positivity⟩ ⟨4 * a⁻¹ ^ 2, by positivity⟩ hm hc + (fun ω t => reverseField_lipschitz ha _) (fun ω t z => reverseField_bound ha _ _) + +variable {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + +lemma IsCapacityTwoWitness.reverse_path_derivative + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w : ℂ) {a : ℝ} + (ha : 0 < a) (haw : a ≤ w.im) (s : ℝ) (hs : s ∈ Icc 0 (T : ℝ)) : + HasDerivWithinAt (fun t : ℝ => G ((T : ℝ) - t).toNNReal (F T w)) + (reverseField a (U ((T : ℝ) - s).toNNReal) (G ((T : ℝ) - s).toNNReal (F T w))) + (Icc 0 (T : ℝ)) s := by + have hw : w ∈ upperHalfPlane := ha.trans_le haw + have hz : F T w ∈ survivingDomain γ T := h.2.2.2.2.2.2.1 T hw + have hi : G T (F T w) = w := h.2.2.2.2.2.2.2.2.1 T w hw + have hmaps : MapsTo (fun t : ℝ => (T : ℝ) - t) (Icc 0 (T : ℝ)) (Icc 0 (T : ℝ)) := by + intro t ht; constructor <;> linarith [ht.1, ht.2] + have hd := (h.time_derivative T (F T w) hz ((T : ℝ) - s) (hmaps hs)).scomp s + ((hasDerivAt_id s).const_sub (T : ℝ)).hasDerivWithinAt hmaps + have hh : a ≤ (G ((T : ℝ) - s).toNNReal (F T w)).im := by + have hh := h.imag_antitoneOn T (F T w) hz (hmaps hs) ⟨T.coe_nonneg, le_rfl⟩ (hmaps hs).2 + simp only [Real.toNNReal_coe, hi] at hh + exact haw.trans hh + simp only [Function.comp_def, neg_smul, one_smul] at hd + simpa only [reverseField, reverseCenter_eq hh, div_eq_mul_inv, neg_mul] using hd + + + +theorem IsCapacityTwoWitness.reverse_flow_identifies + (h : IsCapacityTwoWitness U γ G F) (T : ℝ≥0) (w : ℂ) {a : ℝ} + (ha : 0 < a) (haw : a ≤ w.im) (Q : ℝ → ℂ) (hQ0 : Q 0 = w) + (hQd : ∀ t ∈ Icc 0 (T : ℝ), HasDerivWithinAt Q + (reverseField a (U ((T : ℝ) - t).toNNReal) (Q t)) (Icc 0 (T : ℝ)) t) : + Q T = F T w := by + have hw : w ∈ upperHalfPlane := ha.trans_le haw + have hp := h.reverse_path_derivative T w ha haw + have he : EqOn Q (fun t : ℝ => G ((T : ℝ) - t).toNNReal (F T w)) (Icc 0 (T : ℝ)) := by + apply ODE_solution_unique (fun t => reverseField_lipschitz ha (U ((T : ℝ) - t).toNNReal)) + (fun t ht => (hQd t ht).continuousWithinAt) + (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hQd t ⟨ht.1, ht.2.le⟩)) + (fun t ht => (hp t ht).continuousWithinAt) + (fun t ht => hasDerivWithinAt_Ici_of_Icc ht (hp t ⟨ht.1, ht.2.le⟩)) + simpa only [sub_zero, Real.toNNReal_coe, h.2.2.2.2.2.2.2.2.1 T w hw] using hQ0 + have hh := he ⟨T.coe_nonneg, le_rfl⟩ + simpa only [sub_self, Real.toNNReal_zero, h.initial_value (F T w) + (h.2.2.2.2.2.2.1 T hw).1.1] using hh + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory Complex +open scoped Topology NNReal +namespace SLEExactGauge + +lemma exists_measurable_right_flow + {Ω : Type*} [MeasurableSpace Ω] + (U : ℝ → Ω → ℝ) (hUm : Measurable (fun p : ℝ × Ω => U p.1 p.2)) + (hUc : ∀ ω, Continuous (fun t => U t ω)) + (w : Ω → ℂ) (hwm : Measurable w) {a : ℝ} (ha : 0 < a) : + ∃ Q : ℝ → Ω → ℂ, + Measurable (fun p : ℝ × Ω => Q p.1 p.2) ∧ + (∀ ω, Q 0 ω = w ω) ∧ + (∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) + (rightField a (U t ω) (Q t ω)) (Ici 0) t) := by + have hm : Measurable (fun p : (ℝ × Ω) × ℂ => rightField a (U p.1.1 p.1.2) p.2) := by + have hj : Measurable (fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) := + (hUm.comp measurable_fst).prodMk measurable_snd + have hf : Measurable (fun q : ℝ × ℂ => rightField a q.1 q.2) := + (continuous_rightField ha).measurable + exact Measurable.comp (g := fun q : ℝ × ℂ => rightField a q.1 q.2) + (f := fun p : (ℝ × Ω) × ℂ => (U p.1.1 p.1.2, p.2)) hf hj + have hc (ω : Ω) (z : ℂ) : Continuous (fun t : ℝ => rightField a (U t ω) z) := + Continuous.comp (g := fun q : ℝ × ℂ => rightField a q.1 q.2) + (f := fun t : ℝ => (U t ω, z)) (continuous_rightField ha) + ((hUc ω).prodMk continuous_const) + exact exists_random_initial_bounded_flow (fun t ω z => rightField a (U t ω) z) + w hwm ⟨2 * a⁻¹, by positivity⟩ ⟨4 * a⁻¹ ^ 2, by positivity⟩ hm hc + (fun ω t => rightField_lipschitz ha _) (fun ω t z => rightField_bound ha _ _) + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory +open scoped Topology NNReal +namespace SLEExactGauge + + + +noncomputable def positiveBoundaryField (a u x : ℝ) : ℝ := + 2 * (max a (x-u))⁻¹ + +lemma positiveBoundaryField_eq {a u x : ℝ} (h : a ≤ x-u) : + positiveBoundaryField a u x = 2/(x-u) := by + simp only [positiveBoundaryField,max_eq_right h,div_eq_mul_inv] + +lemma positiveBoundaryField_bound {a : ℝ} (ha : 0 < a) (u x : ℝ) : + ‖positiveBoundaryField a u x‖ ≤ 2*a⁻¹ := by + have hp : 0 < max a (x-u) := ha.trans_le (le_max_left _ _) + simp only [positiveBoundaryField,Real.norm_eq_abs,abs_mul,abs_of_pos (by norm_num : (0:ℝ) < 2),abs_inv, + abs_of_pos hp] + exact mul_le_mul_of_nonneg_left (inv_anti₀ ha (le_max_left _ _)) (by norm_num) + +lemma positiveBoundaryField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : + LipschitzWith ⟨2*a⁻¹^2,by positivity⟩ (positiveBoundaryField a u) := by + apply LipschitzWith.of_dist_le_mul + intro x y + have hx : 0 < max a (x-u) := ha.trans_le (le_max_left _ _) + have hy : 0 < max a (y-u) := ha.trans_le (le_max_left _ _) + have hi (z : ℝ) : ‖(max a (z-u))⁻¹‖ ≤ a⁻¹ := by + rw [Real.norm_eq_abs,abs_inv,abs_of_pos (ha.trans_le (le_max_left _ _))] + exact inv_anti₀ ha (le_max_left _ _) + have hm : ‖max a (y-u)-max a (x-u)‖ ≤ dist x y := by + have hh := ((LipschitzWith.id : LipschitzWith 1 (fun z : ℝ => z)).const_max a).dist_le_mul + (y-u) (x-u) + simpa only [NNReal.coe_one,one_mul,id_eq,dist_sub_right,dist_eq_norm,norm_sub_rev,sub_sub_sub_cancel_right] using hh + rw [dist_eq_norm,positiveBoundaryField,positiveBoundaryField,← mul_sub,norm_mul, + Real.norm_of_nonneg (by norm_num : (0:ℝ) ≤ 2), + inv_sub_inv' hx.ne' hy.ne',norm_mul,norm_mul] + calc + _ ≤ 2*(a⁻¹ * dist x y * a⁻¹) := by gcongr <;> exact hi _ + _ = _ := by change 2*(a⁻¹*dist x y*a⁻¹)=(2*a⁻¹^2)*dist x y; ring + +lemma continuous_positiveBoundaryField {a : ℝ} (ha : 0 < a) : + Continuous (fun p : ℝ × ℝ => positiveBoundaryField a p.1 p.2) := by + apply continuous_const.mul + apply Continuous.inv₀ + · exact continuous_const.max (continuous_snd.sub continuous_fst) + · intro p; exact (ha.trans_le (le_max_left a (p.2-p.1))).ne' + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal +namespace SLEExactGauge + + + +theorem positive_boundary_global_adapted_solution + {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ) + (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2)) + (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω)) + (a : ℝ) (ha : 0 < a) (q₀ : ℝ) : + ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℝ), + Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧ + (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧ + (∀ ω, Q 0 ω = q₀) ∧ + (∀ ω t, 0 ≤ t → + HasDerivWithinAt (fun s => Q s ω) + (positiveBoundaryField a (U t.toNNReal ω) (Q t ω)) + (Ici 0) t) ∧ + Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable)) + (fun t : ℝ≥0 => Q t) := by + let L : ℝ≥0 := ⟨2*a⁻¹,by positivity⟩ + let K : ℝ≥0 := ⟨2*a⁻¹^2,by positivity⟩ + have hbound := positiveBoundaryField_bound ha + have hlip := positiveBoundaryField_lipschitz ha + let V (t : ℝ) (ω : Ω) (q : ℝ) := positiveBoundaryField a (U t.toNNReal ω) q + have hF := continuous_positiveBoundaryField ha + have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) := + hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd) + have hm : Measurable (fun q : (ℝ × Ω) × (ℝ) => V q.1.1 q.1.2 q.2) := by + change Measurable ((fun p : ℝ × (ℝ) => positiveBoundaryField a p.1 p.2) ∘ + (fun q : (ℝ × Ω) × (ℝ) => (U q.1.1.toNNReal q.1.2, q.2))) + exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd) + have hcont (ω : Ω) (q : ℝ) : Continuous (fun t : ℝ => V t ω q) := by + change Continuous ((fun p : ℝ × (ℝ) => positiveBoundaryField a p.1 p.2) ∘ + (fun t : ℝ => (U t.toNNReal ω, q))) + exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const) + obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ := + exists_measurable_global_bounded_ode_picard V q₀ L K hm hcont + (fun ω t => hlip (U t.toNNReal ω)) (fun ω t q => hbound (U t.toNNReal ω) q) + refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩ + intro t + have hUfrozen := measurable_driver_frozen U hU hc t + let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t + let : MeasurableSpace Ω := m + let W (s : ℝ) (ω : Ω) (q : ℝ) := + positiveBoundaryField a (U (min s.toNNReal t) ω) q + have hW : Measurable (fun q : (ℝ × Ω) × (ℝ) => W q.1.1 q.1.2 q.2) := by + change Measurable ((fun p : ℝ × (ℝ) => positiveBoundaryField a p.1 p.2) ∘ + (fun q : (ℝ × Ω) × (ℝ) => (U (min q.1.1.toNNReal t) q.1.2, q.2))) + exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd) + have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n (t : ℝ) ω = picardApprox V q₀ n (t : ℝ) ω := by + apply picardApprox_congr_on q₀ (t : ℝ) _ n (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩ ω + intro s hs ω q + have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hs.2 + simp only [W, V, min_eq_left hh] + apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n (t : ℝ) ω) + · intro n + exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left + · apply tendsto_pi_nhds.mpr + intro ω + simpa only [heq, max_eq_right t.coe_nonneg] using hQconv t ω + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory Complex +open scoped Topology NNReal +namespace SLEExactGauge + +noncomputable def boundaryClockField (a u : ℝ) (q : ℂ) : ℂ := + (positiveBoundaryField a u q.re : ℂ) + ((max a (q.re-u))⁻¹^2 : ℝ) * I + +lemma clipped_real_inv_bound {a : ℝ} (ha : 0 < a) (x : ℝ) : + ‖(max a x)⁻¹‖ ≤ a⁻¹ := by + rw [Real.norm_eq_abs,abs_inv,abs_of_pos (ha.trans_le (le_max_left _ _))] + exact inv_anti₀ ha (le_max_left _ _) + +lemma clipped_real_inv_lipschitz {a : ℝ} (ha : 0 < a) : + LipschitzWith ⟨a⁻¹^2,by positivity⟩ (fun x : ℝ => (max a x)⁻¹) := by + apply LipschitzWith.of_dist_le_mul + intro x y + have hx : max a x ≠ 0 := (ha.trans_le (le_max_left _ _)).ne' + have hy : max a y ≠ 0 := (ha.trans_le (le_max_left _ _)).ne' + have hmax : ‖max a y-max a x‖ ≤ dist x y := by + have hh := ((LipschitzWith.id : LipschitzWith 1 (fun x : ℝ => x)).const_max a).dist_le_mul y x + simpa only [Real.dist_eq,NNReal.coe_one,one_mul,id_eq,Real.norm_eq_abs,abs_sub_comm] using hh + rw [dist_eq_norm,inv_sub_inv' hx hy,norm_mul,norm_mul] + calc + _ ≤ a⁻¹*dist x y*a⁻¹ := by gcongr <;> exact clipped_real_inv_bound ha _ + _ = _ := by change a⁻¹*dist x y*a⁻¹ = a⁻¹^2*dist x y; ring + +lemma boundaryClockField_bound {a : ℝ} (ha : 0 < a) (u : ℝ) (q : ℂ) : + ‖boundaryClockField a u q‖ ≤ 2*a⁻¹+a⁻¹^2 := by + have h0 := positiveBoundaryField_bound ha u q.re + have h1 := clipped_real_inv_bound ha (q.re-u) + have h2 : ‖(max a (q.re-u))⁻¹^2‖ ≤ a⁻¹^2 := by + rw [norm_pow] + gcongr + have hh := norm_add_le (positiveBoundaryField a u q.re : ℂ) (((max a (q.re-u))⁻¹^2 : ℝ)*I) + simp only [norm_mul,Complex.norm_real,Complex.norm_I,mul_one] at hh + exact hh.trans (add_le_add h0 h2) + +lemma boundaryClockField_lipschitz {a : ℝ} (ha : 0 < a) (u : ℝ) : + LipschitzWith ⟨2*a⁻¹^2+2*a⁻¹^3,by positivity⟩ (boundaryClockField a u) := by + apply LipschitzWith.of_dist_le_mul + intro q w + have hd : dist (q.re-u) (w.re-u) ≤ dist q w := by + simpa only [dist_sub_right,dist_eq_norm,Complex.sub_re,Real.norm_eq_abs,sub_sub_sub_cancel_right] using Complex.abs_re_le_norm (q-w) + have hi : ‖(max a (q.re-u))⁻¹-(max a (w.re-u))⁻¹‖ ≤ a⁻¹^2*dist q w := by + exact ((clipped_real_inv_lipschitz ha).dist_le_mul _ _).trans (mul_le_mul_of_nonneg_left hd (sq_nonneg _)) + have hsum : ‖(max a (q.re-u))⁻¹+(max a (w.re-u))⁻¹‖ ≤ 2*a⁻¹ := by + exact (norm_add_le _ _).trans (by linarith [clipped_real_inv_bound ha (q.re-u),clipped_real_inv_bound ha (w.re-u)]) + have hsq : ‖(max a (q.re-u))⁻¹^2-(max a (w.re-u))⁻¹^2‖ ≤ 2*a⁻¹^3*dist q w := by + rw [sq_sub_sq,norm_mul,mul_comm] + calc + _ ≤ (a⁻¹^2*dist q w)*(2*a⁻¹) := mul_le_mul hi hsum (norm_nonneg _) (by positivity) + _ = _ := by ring + have hr := (positiveBoundaryField_lipschitz ha u).dist_le_mul q.re w.re + have hr' : ‖positiveBoundaryField a u q.re-positiveBoundaryField a u w.re‖ ≤ 2*a⁻¹^2*dist q w := by + rw [← dist_eq_norm] + apply hr.trans + change (2*a⁻¹^2)*dist q.re w.re ≤ (2*a⁻¹^2)*dist q w + exact mul_le_mul_of_nonneg_left (by simpa only [dist_sub_right] using hd) (by positivity) + rw [dist_eq_norm] + have he : boundaryClockField a u q-boundaryClockField a u w = + ((positiveBoundaryField a u q.re-positiveBoundaryField a u w.re:ℝ):ℂ)+ + (((max a (q.re-u))⁻¹^2-(max a (w.re-u))⁻¹^2:ℝ):ℂ)*I := by + simp only [boundaryClockField,Complex.ofReal_sub]; ring + rw [he] + have hn := norm_add_le ((positiveBoundaryField a u q.re-positiveBoundaryField a u w.re:ℝ):ℂ) + ((((max a (q.re-u))⁻¹^2-(max a (w.re-u))⁻¹^2:ℝ):ℂ)*I) + simp only [norm_mul,Complex.norm_real,Complex.norm_I,mul_one] at hn + calc + _ ≤ 2*a⁻¹^2*dist q w+2*a⁻¹^3*dist q w := hn.trans (add_le_add hr' hsq) + _ = _ := by change _ = (2*a⁻¹^2+2*a⁻¹^3)*dist q w; ring + +lemma continuous_boundaryClockField {a : ℝ} (ha : 0 < a) : + Continuous (fun p : ℝ × ℂ => boundaryClockField a p.1 p.2) := by + apply Continuous.add + · exact Complex.continuous_ofReal.comp ((continuous_positiveBoundaryField ha).comp + (continuous_fst.prodMk (Complex.continuous_re.comp continuous_snd))) + · apply Continuous.mul _ continuous_const + apply Complex.continuous_ofReal.comp + apply Continuous.pow + apply Continuous.inv₀ + · exact continuous_const.max ((Complex.continuous_re.comp continuous_snd).sub continuous_fst) + · intro p; exact (ha.trans_le (le_max_left _ _)).ne' + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal +namespace SLEExactGauge + + + +theorem boundary_clock_global_adapted_solution + {Ω : Type*} [MeasurableSpace Ω] (U : ℝ≥0 → Ω → ℝ) + (hU : Measurable (fun q : ℝ≥0 × Ω => U q.1 q.2)) + (hc : ∀ ω, Continuous (fun t : ℝ≥0 => U t ω)) + (κ a : ℝ) (ha : 0 < a) (q₀ : ℂ) : + ∃ (L : ℝ≥0) (Q : ℝ → Ω → ℂ), + Measurable (fun q : ℝ × Ω => Q q.1 q.2) ∧ + (∀ ω, LipschitzWith L (fun t => Q t ω)) ∧ + (∀ ω, Q 0 ω = q₀) ∧ + (∀ ω t, 0 ≤ t → + HasDerivWithinAt (fun s => Q s ω) + (boundaryClockField a (Real.sqrt κ*U t.toNNReal ω) (Q t ω)) + (Ici 0) t) ∧ + Adapted (Filtration.natural U (fun t => (hU.comp (measurable_prodMk_left (x := t))).stronglyMeasurable)) + (fun t : ℝ≥0 => Q t) := by + let L : ℝ≥0 := ⟨2*a⁻¹+a⁻¹^2,by positivity⟩ + let K : ℝ≥0 := ⟨2*a⁻¹^2+2*a⁻¹^3,by positivity⟩ + have hbound := boundaryClockField_bound ha + have hlip := boundaryClockField_lipschitz ha + let V (t : ℝ) (ω : Ω) (q : ℂ) := boundaryClockField a (Real.sqrt κ*U t.toNNReal ω) q + have hF : Continuous (fun p : ℝ × ℂ => boundaryClockField a (Real.sqrt κ*p.1) p.2) := + (continuous_boundaryClockField ha).comp ((continuous_const.mul continuous_fst).prodMk continuous_snd) + have hUr : Measurable (fun q : ℝ × Ω => U q.1.toNNReal q.2) := + hU.comp ((continuous_real_toNNReal.measurable.comp measurable_fst).prodMk measurable_snd) + have hm : Measurable (fun q : (ℝ × Ω) × (ℂ) => V q.1.1 q.1.2 q.2) := by + change Measurable ((fun p : ℝ × (ℂ) => boundaryClockField a (Real.sqrt κ*p.1) p.2) ∘ + (fun q : (ℝ × Ω) × (ℂ) => (U q.1.1.toNNReal q.1.2, q.2))) + exact hF.measurable.comp ((hUr.comp measurable_fst).prodMk measurable_snd) + have hcont (ω : Ω) (q : ℂ) : Continuous (fun t : ℝ => V t ω q) := by + change Continuous ((fun p : ℝ × (ℂ) => boundaryClockField a (Real.sqrt κ*p.1) p.2) ∘ + (fun t : ℝ => (U t.toNNReal ω, q))) + exact hF.comp (((hc ω).comp continuous_real_toNNReal).prodMk continuous_const) + obtain ⟨Q, hQm, hQl, hQ0, hQd, hQconv⟩ := + exists_measurable_global_bounded_ode_picard V q₀ L K hm hcont + (fun ω t => hlip (Real.sqrt κ*U t.toNNReal ω)) (fun ω t q => hbound (Real.sqrt κ*U t.toNNReal ω) q) + refine ⟨L, Q, hQm, hQl, hQ0, hQd, ?_⟩ + intro t + have hUfrozen := measurable_driver_frozen U hU hc t + let m := Filtration.natural U (fun t => (hU.comp measurable_prodMk_left).stronglyMeasurable) t + let : MeasurableSpace Ω := m + let W (s : ℝ) (ω : Ω) (q : ℂ) := + boundaryClockField a (Real.sqrt κ*U (min s.toNNReal t) ω) q + have hW : Measurable (fun q : (ℝ × Ω) × (ℂ) => W q.1.1 q.1.2 q.2) := by + change Measurable ((fun p : ℝ × (ℂ) => boundaryClockField a (Real.sqrt κ*p.1) p.2) ∘ + (fun q : (ℝ × Ω) × (ℂ) => (U (min q.1.1.toNNReal t) q.1.2, q.2))) + exact hF.measurable.comp ((hUfrozen.comp measurable_fst).prodMk measurable_snd) + have heq (n : ℕ) (ω : Ω) : picardApprox W q₀ n (t : ℝ) ω = picardApprox V q₀ n (t : ℝ) ω := by + apply picardApprox_congr_on q₀ (t : ℝ) _ n (t : ℝ) ⟨t.coe_nonneg, le_rfl⟩ ω + intro s hs ω q + have hh : s.toNNReal ≤ t := Real.toNNReal_le_iff_le_coe.mpr hs.2 + simp only [W, V, min_eq_left hh] + apply measurable_of_tendsto_metrizable (f := fun n ω => picardApprox W q₀ n (t : ℝ) ω) + · intro n + exact (picardApprox_measurable W q₀ hW n).comp measurable_prodMk_left + · apply tendsto_pi_nhds.mpr + intro ω + simpa only [heq, max_eq_right t.coe_nonneg] using hQconv t ω + +end SLEExactGauge +end + + +section + +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +namespace SLEExactGauge + + + +structure BoundaryClockSolution {Ω : Type*} [MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (κ a : ℝ) (q₀ : ℂ) where + L : ℝ≥0 + Q : ℝ → Ω → ℂ + measurable : Measurable (fun p : ℝ × Ω => Q p.1 p.2) + lipschitz : ∀ ω, LipschitzWith L (fun t => Q t ω) + initial : ∀ ω, Q 0 ω = q₀ + derivative : ∀ ω t, 0 ≤ t → HasDerivWithinAt (fun s => Q s ω) + (boundaryClockField a (Real.sqrt κ * B t.toNNReal ω) (Q t ω)) (Ici 0) t + adapted : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) + (fun t : ℝ≥0 => Q t) + +lemma exists_boundaryClockSolution + {Ω : Type*} [MeasurableSpace Ω] + (B : ℝ≥0 → Ω → ℝ) (hm : ∀ t, Measurable (B t)) + (hBm : Measurable (fun p : ℝ≥0 × Ω => B p.1 p.2)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ a : ℝ) (ha : 0 < a) (q₀ : ℂ) : + Nonempty (BoundaryClockSolution B hm κ a q₀) := by + obtain ⟨L,Q,hQm,hQl,hQ0,hQd,hQa⟩ := boundary_clock_global_adapted_solution B hBm hBc κ a ha q₀ + exact ⟨⟨L,Q,hQm,hQl,hQ0,hQd,hQa⟩⟩ + +end SLEExactGauge +end + + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/LowerPositivity.lean b/lean/OAI/Probability/SLE/LowerPositivity.lean index a138894f5a6e7e4fbd87a170c10eb71e3a7eb83c..7dbd420f4539ff70d5bb14ea1eb71ec607b07e23 100644 --- a/lean/OAI/Probability/SLE/LowerPositivity.lean +++ b/lean/OAI/Probability/SLE/LowerPositivity.lean @@ -1,72 +1,72 @@ -import Mathlib -import OAI.Probability.SLE.LowerRestartConditionalAllKappa - -namespace OAI - -/-! Lower Positivity for chordal SLE. -/ - -section -section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -open Set Filter MeasureTheory -open scoped Topology NNReal -open Set Filter MeasureTheory Complex -open MeasureTheory ProbabilityTheory Set Filter -open Set Filter Metric Complex InnerProductSpace -open scoped Topology Laplacian -open Set Filter Metric Complex MeasureTheory -open Set Filter Metric Complex MeasureTheory InnerProductSpace -open scoped Topology Laplacian ENNReal -open scoped Topology ENNReal -open scoped Topology Laplacian RealInnerProductSpace -open Set Filter Metric Complex -open scoped Topology -open Set Filter Metric -open Set Filter MeasureTheory Metric -open Set Filter Complex -open scoped Topology ComplexConjugate NNReal -open scoped Topology ENNReal NNReal ComplexConjugate -open Set Filter Metric MeasureTheory Complex -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped Topology ENNReal NNReal -open Set Filter MeasureTheory ProbabilityTheory Metric Classical -open Set Filter -open scoped Topology NNReal ENNReal ComplexConjugate -open scoped ENNReal NNReal -noncomputable section -open Set MeasureTheory ProbabilityTheory Filter -open scoped ENNReal NNReal Topology -section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal - -namespace SLEExactGauge - -def segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t -universe v_lower - - -def SourceLowerMainTarget : Prop := - ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P → - ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) → - ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P → - ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) → - ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → - 0 < hausdorffGauge h (segment (γ ω) s t) -end SLEExactGauge - -namespace SLEExactGauge -theorem sourceLowerMain_proved : SourceLowerMainTarget := by - intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf - let : IsProbabilityMeasure P := hP - exact source_lower_universe hκ hκ8 hSLE h hh hf -end SLEExactGauge -end - - -end -end -end - -end OAI +import Mathlib +import OAI.Probability.SLE.LowerRestartConditionalAllKappa + +namespace OAI + +/-! Lower Positivity for chordal SLE. -/ + +section +section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +open Set Filter MeasureTheory +open scoped Topology NNReal +open Set Filter MeasureTheory Complex +open MeasureTheory ProbabilityTheory Set Filter +open Set Filter Metric Complex InnerProductSpace +open scoped Topology Laplacian +open Set Filter Metric Complex MeasureTheory +open Set Filter Metric Complex MeasureTheory InnerProductSpace +open scoped Topology Laplacian ENNReal +open scoped Topology ENNReal +open scoped Topology Laplacian RealInnerProductSpace +open Set Filter Metric Complex +open scoped Topology +open Set Filter Metric +open Set Filter MeasureTheory Metric +open Set Filter Complex +open scoped Topology ComplexConjugate NNReal +open scoped Topology ENNReal NNReal ComplexConjugate +open Set Filter Metric MeasureTheory Complex +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped Topology ENNReal NNReal +open Set Filter MeasureTheory ProbabilityTheory Metric Classical +open Set Filter +open scoped Topology NNReal ENNReal ComplexConjugate +open scoped ENNReal NNReal +noncomputable section +open Set MeasureTheory ProbabilityTheory Filter +open scoped ENNReal NNReal Topology +section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +def segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : Set ℂ := γ '' Icc s t +universe v_lower + + +def SourceLowerMainTarget : Prop := + ∀ (Ω : Type v_lower) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P → + ∀ κ : ℝ, 0 < κ → κ < ((8 : ℕ) : ℝ) → + ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P → + ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) → + ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → + 0 < hausdorffGauge h (segment (γ ω) s t) +end SLEExactGauge + +namespace SLEExactGauge +theorem sourceLowerMain_proved : SourceLowerMainTarget := by + intro Ω m P hP κ hκ hκ8 γ hSLE h hh hf + let : IsProbabilityMeasure P := hP + exact source_lower_universe hκ hκ8 hSLE h hh hf +end SLEExactGauge +end + + +end +end +end + +end OAI diff --git a/lean/OAI/Probability/SLE/LowerRestartConditionalAllKappa.lean b/lean/OAI/Probability/SLE/LowerRestartConditionalAllKappa.lean index 35bbd175954a79a4c7479192dd1c1014b3070d62..647a4d84443378c40efbf79af70fc703a4b7f38a 100644 --- a/lean/OAI/Probability/SLE/LowerRestartConditionalAllKappa.lean +++ b/lean/OAI/Probability/SLE/LowerRestartConditionalAllKappa.lean @@ -1,386 +1,386 @@ -import Mathlib -import OAI.Probability.SLE.SimplePastMesh - -namespace OAI - -/-! Lower Restart Conditional All Kappa for chordal SLE. -/ - -section -section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -open Set Filter MeasureTheory -open scoped Topology NNReal -open Set Filter MeasureTheory Complex -open MeasureTheory ProbabilityTheory Set Filter -open Set Filter Metric Complex InnerProductSpace -open scoped Topology Laplacian -open Set Filter Metric Complex MeasureTheory -open Set Filter Metric Complex MeasureTheory InnerProductSpace -open scoped Topology Laplacian ENNReal -open scoped Topology ENNReal -open scoped Topology Laplacian RealInnerProductSpace -open Set Filter Metric Complex -open scoped Topology -open Set Filter Metric -open Set Filter MeasureTheory Metric -open Set Filter Complex -open scoped Topology ComplexConjugate NNReal -open scoped Topology ENNReal NNReal ComplexConjugate -open Set Filter Metric MeasureTheory Complex -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped Topology ENNReal NNReal -open Set Filter MeasureTheory ProbabilityTheory Metric Classical -open Set Filter -open scoped Topology NNReal ENNReal ComplexConjugate -open scoped ENNReal NNReal -noncomputable section -open Set MeasureTheory ProbabilityTheory Filter -open scoped ENNReal NNReal Topology -section -open Set Filter MeasureTheory ProbabilityTheory Complex Metric -open scoped Topology NNReal ENNReal -namespace SLEExactGauge -noncomputable section -local instance lowerRestartAllKappaRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance lowerRestartAllKappaComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance lowerRestartAllKappaRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ -local instance lowerRestartAllKappaComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ -lemma source_translation_gauge_positive_allKappa (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) - (b : ℂ) {S : Set ℂ} (hS : IsCompact S) - (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => z+b) '' S) := by - by_contra hn - have hz : hausdorffGauge h ((fun z => z+b) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le - have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf - (D := univ) (f := fun z => z-b) isOpen_univ (hS.image (continuous_id.add_const b)) - (subset_univ _) (by fun_prop) hz - have hi : (fun z => z-b) '' ((fun z => z+b) '' S) = S := by - rw [←image_comp] - simp only [Function.comp_def,add_sub_cancel_right,image_id'] - change hausdorffGauge h ((fun z => z-b) '' ((fun z => z+b) '' S)) = 0 at he - rw [hi] at he - exact hp.ne' he - -lemma source_restart_segment_positive_allKappa (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) - (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) - (hw : IsCapacityTwoWitness U γ G F) (s δ : ℝ≥0) (η : C(ℝ≥0,ℂ)) - (hid : ∀ t : ℝ≥0, η t ∈ upperHalfPlane → γ (s+t) = F s (η t + (U s:ℂ))) - {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) - (hp : 0 < hausdorffGauge h (K ∩ (η '' Icc 0 δ))) : - 0 < hausdorffGauge h (γ '' Icc s (s+δ)) := by - let L := K ∩ (η '' Icc 0 δ) - have hL : IsCompact L := hK.inter_right (isCompact_Icc.image η.continuous).isClosed - have hLH : L ⊆ upperHalfPlane := inter_subset_left.trans hKH - have htrans := source_translation_gauge_positive_allKappa κ hκ hκ8 h hh hf (U s:ℂ) hL hp - have htrK : IsCompact ((fun z => z+(U s:ℂ)) '' L) := hL.image (continuous_id.add_const _) - have htrH : (fun z => z+(U s:ℂ)) '' L ⊆ upperHalfPlane := by - rintro _ ⟨z,hz,rfl⟩ - simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,add_zero] using hLH hz - have hpositive := source_inverse_chart_gauge_positive κ hκ hκ8 h hh hf U γ G F hw s htrK htrH htrans - apply hpositive.trans_le (measure_mono ?_) - rintro _ ⟨_,⟨z,hz,rfl⟩,rfl⟩ - obtain ⟨u,hu,rfl⟩ := hz.2 - refine ⟨s+u,⟨le_add_of_nonneg_right hu.1,add_le_add le_rfl hu.2⟩,?_⟩ - exact hid u (hKH hz.1) - -lemma RegularSLEPresentation.source_segment_positive_measurable_at_allKappa - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} - (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h) - (s t : ℝ≥0) : - MeasurableSet[a.filtration t] {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} := by - let f : a.samples → C(ℝ≥0,ℂ) := fun ω => - ⟨fun u => γ ω (min u t), (a.trace_continuous ω).comp (continuous_id.min continuous_const)⟩ - have hm : @Measurable a.samples C(ℝ≥0,ℂ) (a.filtration t) (borel _) f := by - let : MeasurableSpace a.samples := a.filtration t - apply source_path_measurable - intro u - exact (a.trace_adapted (min u t)).mono (a.filtration.mono (min_le_right u t)) le_rfl - have heq : {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} = - f ⁻¹' {g : C(ℝ≥0,ℂ) | 0 < hausdorffGauge h (univ ∩ (g '' Icc s t))} := by - ext ω - simp only [mem_preimage,mem_ofPred_eq,univ_inter] - have hi : γ ω '' Icc s t = f ω '' Icc s t := by - apply image_congr - intro u hu - simp only [f,ContinuousMap.coe_mk,min_eq_left hu.2] - rw [hi] - rw [heq] - exact (sourceTracePositiveMeasurable_proved h hh univ isClosed_univ s t).preimage hm - -lemma RegularSLEPresentation.source_segment_null_measurable_at_allKappa - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} - (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h) (s t : ℝ≥0) : - MeasurableSet[a.filtration t] {ω : a.samples | hausdorffGauge h (γ ω '' Icc s t) = 0} := by - simpa only [compl_ofPred,not_lt,nonpos_iff_eq_zero] using - (a.source_segment_positive_measurable_at_allKappa h hh s t).compl - -lemma RegularSLEPresentation.source_uniform_null_slot_bound_allKappa - {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h) - (hf : HasSmallRadiusFormula h (hFormula κ)) : - ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ → - ∀ A : Set a.samples, MeasurableSet[a.filtration s] A → - (P.comap Subtype.val) (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0}) ≤ - (1-p)*(P.comap Subtype.val) A := by - have hk : 0 < κ := by linarith - let Q := P.comap (Subtype.val : a.samples → Ω) - obtain ⟨p,hp,hseed⟩ := sourceUniformInteriorSeed_proved_allKappa κ hκ hκ8 Ω m P inferInstance γ a h hh hf - have hp1 : p ≤ 1 := by - obtain ⟨K,hK,hKH,hKP⟩ := hseed 1 zero_lt_one - exact hKP.trans (prob_le_one) - refine ⟨p,hp,hp1,?_⟩ - intro s δ hδ A hA - obtain ⟨K,hK,hKH,hKP⟩ := hseed δ hδ - obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian - a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness - let τ : a.samples → ℝ≥0 := fun _ => s - have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s - have hAstop : MeasurableSet[hτ.measurableSpace] A := by - change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A - rw [IsStoppingTime.measurableSpace_const] - exact hA - obtain ⟨hlaw,hident⟩ := hrestart τ hτ - let rp := sourceRestartDriverPath a.B a.continuous_B τ - let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous - let C : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))} - let E : Set a.samples := (fun ω => Φ (rp ω)) ⁻¹' C - have hC : MeasurableSet C := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ - have hrm : Measurable rp := by - apply source_path_measurable - intro t - exact (a.measurable_B (s+t)).sub (a.measurable_B s) - have hem : Measurable (fun ω => Φ (rp ω)) := hΦm.comp hrm - have hEm : MeasurableSet E := hC.preimage hem - have hAm : MeasurableSet A := (a.filtration.le s) A hA - have hprob : Q (A ∩ E) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} := by - have he := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ C) (hlaw A hAstop) - rw [Measure.map_apply hem hC,Measure.smul_apply, - Measure.map_apply a.source_trace_path_measurable_allKappa hC, - Measure.restrict_apply (hC.preimage hem)] at he - change Q (E ∩ A) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} at he - simpa only [inter_comm] using he - have hins : (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0} : Set a.samples) ≤ᵐ[Q] (A \ E : Set a.samples) := by - filter_upwards [hident] with ω hω - intro hx - refine ⟨hx.1,?_⟩ - intro hsuc - have hpositive := source_restart_segment_positive_allKappa κ hk hκ8 h hh hf - (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) - s δ (Φ (rp ω)) hω hK hKH hsuc - rw [hx.2] at hpositive - exact lt_irrefl _ hpositive - have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by - have he : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto - rw [he] - exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _) - calc Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0}) - ≤ Q (A \ E) := measure_mono_ae hins - _ = Q A - Q (A ∩ E) := hdiff - _ ≤ Q A - Q A*p := tsub_le_tsub_left (by rw [hprob]; gcongr) _ - _ = (1-p)*Q A := by - rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul,mul_comm p (Q A)] -end -end SLEExactGauge -end - - -section -open Set Filter MeasureTheory ProbabilityTheory Complex Metric -open scoped Topology NNReal ENNReal -namespace SLEExactGauge -noncomputable section -lemma RegularSLEPresentation.source_positive_fixed_interval_allKappa - {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h) - (hf : HasSmallRadiusFormula h (hFormula κ)) (s t : ℝ≥0) (hst : s < t) : - ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), 0 < hausdorffGauge h (γ ω '' Icc s t) := by - let Q := P.comap (Subtype.val : a.samples → Ω) - obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_null_slot_bound_allKappa hκ hκ8 h hh hf - let c : ℝ≥0∞ := 1-p - have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne' - have hsteps (δ : ℝ≥0) (hδ : 0 < δ) (k : ℕ) : - Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0} ≤ c^k := by - induction k with - | zero => simpa using (prob_le_one (μ := Q) - (s := {ω | hausdorffGauge h (γ ω '' Icc s (s+(0:ℝ≥0)*δ)) = 0})) - | succ k ih => - have he : s+((k+1:ℕ):ℝ≥0)*δ = (s+(k:ℝ≥0)*δ)+δ := by push_cast; ring - rw [he,pow_succ,mul_comm (c^k) c] - let A : Set a.samples := {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0} - have hA : MeasurableSet[a.filtration (s+(k:ℝ≥0)*δ)] A := - a.source_segment_null_measurable_at_allKappa h hh s _ - calc Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ+δ)) = 0} - ≤ Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc (s+(k:ℝ≥0)*δ) (s+(k:ℝ≥0)*δ+δ)) = 0}) := by - apply measure_mono - intro ω hω - constructor - · exact measure_mono_null (image_mono (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))) hω - · exact measure_mono_null (image_mono (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)) hω - _ ≤ c * Q A := hslot _ δ hδ A hA - _ ≤ c * c^k := by gcongr - have hbound : ∀ᶠ n : ℕ in atTop, - Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} ≤ c^n := by - filter_upwards [eventually_ge_atTop 1] with n hn - have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega) - let δ : ℝ≥0 := (t-s)/(n:ℝ≥0) - have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0 - have he : s+(n:ℝ≥0)*δ = t := by - dsimp [δ] - rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le] - simpa only [he] using hsteps δ hδ n - have hzero : Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} = 0 := - le_antisymm (ge_of_tendsto (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc) hbound) zero_le - apply ae_iff.mpr - simpa only [not_lt,nonpos_iff_eq_zero] using hzero -end -end SLEExactGauge -end - - -section -open Set Filter MeasureTheory ProbabilityTheory Complex Metric -open scoped Topology NNReal ENNReal -namespace SLEExactGauge -noncomputable section -lemma RegularSLEPresentation.source_positive_all_intervals_allKappa - {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h) - (hf : HasSmallRadiusFormula h (hFormula κ)) : - ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), - ∀ s t : ℝ≥0, 0 < s → s < t → 0 < hausdorffGauge h (γ ω '' Icc s t) := by - have hrat : ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), - ∀ r q : ℚ, (r:ℝ).toNNReal < (q:ℝ).toNNReal → - 0 < hausdorffGauge h (γ ω '' Icc (r:ℝ).toNNReal (q:ℝ).toNNReal) := by - apply ae_all_iff.mpr - intro r - apply ae_all_iff.mpr - intro q - apply ae_all_iff.mpr - intro hrq - exact a.source_positive_fixed_interval_allKappa hκ hκ8 h hh hf _ _ hrq - filter_upwards [hrat] with ω hω - intro s t hs hst - obtain ⟨r,hsr,hrt⟩ := exists_rat_btwn (show (s:ℝ) < (t:ℝ) from hst) - obtain ⟨q,hrq,hqt⟩ := exists_rat_btwn hrt - have hq : 0 < (q:ℝ) := (show 0 < (s:ℝ) from hs).trans (hsr.trans hrq) - have hp := hω r q ((Real.toNNReal_lt_toNNReal_iff hq).mpr hrq) - apply hp.trans_le (measure_mono (image_mono (Icc_subset_Icc ?_ ?_))) - · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hsr.le - · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hqt.le - - -def SourceLowerAllIntervalsTarget_allKappa : Prop := - ∀ κ : ℝ, 0 < κ → κ < 8 → - ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P → - ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P → - ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) → - ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → - 0 < hausdorffGauge h (γ ω '' Set.Icc s t) -theorem sourceLowerAllIntervals_proved_allKappa : SourceLowerAllIntervalsTarget_allKappa := by - intro κ hκ hκ8 Ω m P hP γ hSLE h hh hf - let : IsProbabilityMeasure P := hP - obtain ⟨a⟩ := exists_regular_sle_presentation hSLE - have ha := a.source_positive_all_intervals_allKappa hκ hκ8 h hh hf - rw [←a.measurePreserving_val.map_eq] - exact (MeasurableEmbedding.subtype_coe a.measurable_samples).ae_map_iff.mpr ha -end -end SLEExactGauge -end - - -section -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal -namespace SLEExactGauge - - -def SourceLowerAllKappaTarget : Prop := - ∀ κ : ℝ, 0 < κ → κ < 8 → - ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P → - ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P → - ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) → - ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → - 0 < hausdorffGauge h (γ ω '' Set.Icc s t) -end SLEExactGauge - -namespace SLEExactGauge -theorem sourceLowerAllKappa_proved : SourceLowerAllKappaTarget := sourceLowerAllIntervals_proved_allKappa -end SLEExactGauge -end - - -section -open Set Filter MeasureTheory ProbabilityTheory Complex -open scoped Topology NNReal ENNReal -namespace SLEExactGauge -noncomputable section -local instance lowerUniverseRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance lowerUniverseComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance lowerUniverseRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ -local instance lowerUniverseComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ -lemma source_lower_universe {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) - (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) : - ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → - 0 < hausdorffGauge h (γ ω '' Icc s t) := by - obtain ⟨a⟩ := exists_regular_sle_presentation hSLE - let μ := P.comap (Subtype.val : a.samples → Ω) - let : IsProbabilityMeasure μ := a.probability - let bp := sourceDriverPath a.B a.continuous_B - let gp := sourceTracePath (γ ∘ Subtype.val) a.trace_continuous - have hbm : Measurable bp := source_path_measurable bp a.measurable_B - have hgm : Measurable gp := source_path_measurable gp (fun t => (hSLE.1 t).comp measurable_subtype_coe) - let pair : a.samples → C(ℝ≥0,ℝ) × C(ℝ≥0,ℂ) := fun ω => (bp ω, gp ω) - have hpair : Measurable pair := hbm.prodMk hgm - let X := range pair - let f : a.samples → X := fun ω => ⟨pair ω, mem_range_self ω⟩ - have hfm : Measurable f := hpair.subtype_mk - let Q := μ.map f - let : IsProbabilityMeasure Q := inferInstance - let B : ℝ≥0 → X → ℝ := fun t x => x.val.1 t - let η : X → ℝ≥0 → ℂ := fun x t => x.val.2 t - have hBm (t : ℝ≥0) : Measurable (B t) := - (continuous_eval_const t).measurable.comp (measurable_fst.comp measurable_subtype_coe) - have hηm (t : ℝ≥0) : Measurable (fun x => η x t) := - (continuous_eval_const t).measurable.comp (measurable_snd.comp measurable_subtype_coe) - have hBrownian : IsBrownianReal B Q := by - constructor - · constructor - intro I - have hmI : Measurable (fun x : X => I.restrict (B · x)) := - Measurable.of_eval (fun index => hBm index) - refine ⟨hmI.aemeasurable, ?_⟩ - rw [show Q = μ.map f from rfl, Measure.map_map hmI hfm] - exact (a.brownian.hasLaw I).map_eq - · exact Eventually.of_forall (fun x => x.val.1.continuous) - have hη : IsOrdinaryChordalSLE κ η Q := by - refine ⟨hηm, B, hBrownian, Eventually.of_forall ?_⟩ - intro x - obtain ⟨ω, hω⟩ := x.property - refine ⟨a.G ω, a.F ω, ?_⟩ - have hp : pair ω = x.val := hω - have heB : ∀ t, B t x = a.B t ω := fun t => congrArg (fun z => z.1 t) hp.symm - have heη : η x = γ ω := by - exact congrArg (fun z => (z.2 : ℝ≥0 → ℂ)) hp.symm - simpa only [heB,heη] using a.witness ω - have hlow := sourceLowerAllKappa_proved κ hκ hκ8 X inferInstance Q inferInstance η hη h hh hf - have hpull := ae_of_ae_map hfm.aemeasurable hlow - rw [←a.measurePreserving_val.map_eq] - apply (MeasurableEmbedding.subtype_coe a.measurable_samples).ae_map_iff.mpr - exact hpull -end -end SLEExactGauge -end - - -end -end -end - -end OAI +import Mathlib +import OAI.Probability.SLE.SimplePastMesh + +namespace OAI + +/-! Lower Restart Conditional All Kappa for chordal SLE. -/ + +section +section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +open Set Filter MeasureTheory +open scoped Topology NNReal +open Set Filter MeasureTheory Complex +open MeasureTheory ProbabilityTheory Set Filter +open Set Filter Metric Complex InnerProductSpace +open scoped Topology Laplacian +open Set Filter Metric Complex MeasureTheory +open Set Filter Metric Complex MeasureTheory InnerProductSpace +open scoped Topology Laplacian ENNReal +open scoped Topology ENNReal +open scoped Topology Laplacian RealInnerProductSpace +open Set Filter Metric Complex +open scoped Topology +open Set Filter Metric +open Set Filter MeasureTheory Metric +open Set Filter Complex +open scoped Topology ComplexConjugate NNReal +open scoped Topology ENNReal NNReal ComplexConjugate +open Set Filter Metric MeasureTheory Complex +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped Topology ENNReal NNReal +open Set Filter MeasureTheory ProbabilityTheory Metric Classical +open Set Filter +open scoped Topology NNReal ENNReal ComplexConjugate +open scoped ENNReal NNReal +noncomputable section +open Set MeasureTheory ProbabilityTheory Filter +open scoped ENNReal NNReal Topology +section +open Set Filter MeasureTheory ProbabilityTheory Complex Metric +open scoped Topology NNReal ENNReal +namespace SLEExactGauge +noncomputable section +local instance lowerRestartAllKappaRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance lowerRestartAllKappaComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance lowerRestartAllKappaRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ +local instance lowerRestartAllKappaComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ +lemma source_translation_gauge_positive_allKappa (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) + (b : ℂ) {S : Set ℂ} (hS : IsCompact S) + (hp : 0 < hausdorffGauge h S) : 0 < hausdorffGauge h ((fun z => z+b) '' S) := by + by_contra hn + have hz : hausdorffGauge h ((fun z => z+b) '' S) = 0 := le_antisymm (le_of_not_gt hn) zero_le + have he := source_gauge_holomorphic_compact_null κ hκ hκ8 h hh hf + (D := univ) (f := fun z => z-b) isOpen_univ (hS.image (continuous_id.add_const b)) + (subset_univ _) (by fun_prop) hz + have hi : (fun z => z-b) '' ((fun z => z+b) '' S) = S := by + rw [←image_comp] + simp only [Function.comp_def,add_sub_cancel_right,image_id'] + change hausdorffGauge h ((fun z => z-b) '' ((fun z => z+b) '' S)) = 0 at he + rw [hi] at he + exact hp.ne' he + +lemma source_restart_segment_positive_allKappa (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) + (U : ℝ≥0 → ℝ) (γ : ℝ≥0 → ℂ) (G F : ℝ≥0 → ℂ → ℂ) + (hw : IsCapacityTwoWitness U γ G F) (s δ : ℝ≥0) (η : C(ℝ≥0,ℂ)) + (hid : ∀ t : ℝ≥0, η t ∈ upperHalfPlane → γ (s+t) = F s (η t + (U s:ℂ))) + {K : Set ℂ} (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) + (hp : 0 < hausdorffGauge h (K ∩ (η '' Icc 0 δ))) : + 0 < hausdorffGauge h (γ '' Icc s (s+δ)) := by + let L := K ∩ (η '' Icc 0 δ) + have hL : IsCompact L := hK.inter_right (isCompact_Icc.image η.continuous).isClosed + have hLH : L ⊆ upperHalfPlane := inter_subset_left.trans hKH + have htrans := source_translation_gauge_positive_allKappa κ hκ hκ8 h hh hf (U s:ℂ) hL hp + have htrK : IsCompact ((fun z => z+(U s:ℂ)) '' L) := hL.image (continuous_id.add_const _) + have htrH : (fun z => z+(U s:ℂ)) '' L ⊆ upperHalfPlane := by + rintro _ ⟨z,hz,rfl⟩ + simpa only [upperHalfPlane,mem_ofPred_eq,add_im,ofReal_im,add_zero] using hLH hz + have hpositive := source_inverse_chart_gauge_positive κ hκ hκ8 h hh hf U γ G F hw s htrK htrH htrans + apply hpositive.trans_le (measure_mono ?_) + rintro _ ⟨_,⟨z,hz,rfl⟩,rfl⟩ + obtain ⟨u,hu,rfl⟩ := hz.2 + refine ⟨s+u,⟨le_add_of_nonneg_right hu.1,add_le_add le_rfl hu.2⟩,?_⟩ + exact hid u (hKH hz.1) + +lemma RegularSLEPresentation.source_segment_positive_measurable_at_allKappa + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} + (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h) + (s t : ℝ≥0) : + MeasurableSet[a.filtration t] {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} := by + let f : a.samples → C(ℝ≥0,ℂ) := fun ω => + ⟨fun u => γ ω (min u t), (a.trace_continuous ω).comp (continuous_id.min continuous_const)⟩ + have hm : @Measurable a.samples C(ℝ≥0,ℂ) (a.filtration t) (borel _) f := by + let : MeasurableSpace a.samples := a.filtration t + apply source_path_measurable + intro u + exact (a.trace_adapted (min u t)).mono (a.filtration.mono (min_le_right u t)) le_rfl + have heq : {ω : a.samples | 0 < hausdorffGauge h (γ ω '' Icc s t)} = + f ⁻¹' {g : C(ℝ≥0,ℂ) | 0 < hausdorffGauge h (univ ∩ (g '' Icc s t))} := by + ext ω + simp only [mem_preimage,mem_ofPred_eq,univ_inter] + have hi : γ ω '' Icc s t = f ω '' Icc s t := by + apply image_congr + intro u hu + simp only [f,ContinuousMap.coe_mk,min_eq_left hu.2] + rw [hi] + rw [heq] + exact (sourceTracePositiveMeasurable_proved h hh univ isClosed_univ s t).preimage hm + +lemma RegularSLEPresentation.source_segment_null_measurable_at_allKappa + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} + (a : RegularSLEPresentation P κ γ) (h : ℝ → ℝ) (hh : IsGauge h) (s t : ℝ≥0) : + MeasurableSet[a.filtration t] {ω : a.samples | hausdorffGauge h (γ ω '' Icc s t) = 0} := by + simpa only [compl_ofPred,not_lt,nonpos_iff_eq_zero] using + (a.source_segment_positive_measurable_at_allKappa h hh s t).compl + +lemma RegularSLEPresentation.source_uniform_null_slot_bound_allKappa + {Ω : Type} [m : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h) + (hf : HasSmallRadiusFormula h (hFormula κ)) : + ∃ p : ℝ≥0∞, 0 < p ∧ p ≤ 1 ∧ ∀ (s δ : ℝ≥0), 0 < δ → + ∀ A : Set a.samples, MeasurableSet[a.filtration s] A → + (P.comap Subtype.val) (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0}) ≤ + (1-p)*(P.comap Subtype.val) A := by + have hk : 0 < κ := by linarith + let Q := P.comap (Subtype.val : a.samples → Ω) + obtain ⟨p,hp,hseed⟩ := sourceUniformInteriorSeed_proved_allKappa κ hκ hκ8 Ω m P inferInstance γ a h hh hf + have hp1 : p ≤ 1 := by + obtain ⟨K,hK,hKH,hKP⟩ := hseed 1 zero_lt_one + exact hKP.trans (prob_le_one) + refine ⟨p,hp,hp1,?_⟩ + intro s δ hδ A hA + obtain ⟨K,hK,hKH,hKP⟩ := hseed δ hδ + obtain ⟨Φ,hΦm,hΦe,hrestart⟩ := sourceStoppedTraceLaw κ hk hκ8 a.B a.brownian + a.measurable_B a.continuous_B (fun ω : a.samples => γ ω) a.G a.F a.witness + let τ : a.samples → ℝ≥0 := fun _ => s + have hτ : IsStoppingTime a.filtration (fun ω => (τ ω : WithTop ℝ≥0)) := isStoppingTime_const a.filtration s + have hAstop : MeasurableSet[hτ.measurableSpace] A := by + change MeasurableSet[(isStoppingTime_const a.filtration s).measurableSpace] A + rw [IsStoppingTime.measurableSpace_const] + exact hA + obtain ⟨hlaw,hident⟩ := hrestart τ hτ + let rp := sourceRestartDriverPath a.B a.continuous_B τ + let gp := sourceTracePath (fun ω : a.samples => γ ω) a.trace_continuous + let C : Set C(ℝ≥0,ℂ) := {g | 0 < hausdorffGauge h (K ∩ (g '' Icc 0 δ))} + let E : Set a.samples := (fun ω => Φ (rp ω)) ⁻¹' C + have hC : MeasurableSet C := sourceTracePositiveMeasurable_proved h hh K hK.isClosed 0 δ + have hrm : Measurable rp := by + apply source_path_measurable + intro t + exact (a.measurable_B (s+t)).sub (a.measurable_B s) + have hem : Measurable (fun ω => Φ (rp ω)) := hΦm.comp hrm + have hEm : MeasurableSet E := hC.preimage hem + have hAm : MeasurableSet A := (a.filtration.le s) A hA + have hprob : Q (A ∩ E) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} := by + have he := congrArg (fun μ : Measure C(ℝ≥0,ℂ) => μ C) (hlaw A hAstop) + rw [Measure.map_apply hem hC,Measure.smul_apply, + Measure.map_apply a.source_trace_path_measurable_allKappa hC, + Measure.restrict_apply (hC.preimage hem)] at he + change Q (E ∩ A) = Q A * Q {ω | 0 < hausdorffGauge h (K ∩ (γ ω '' Icc 0 δ))} at he + simpa only [inter_comm] using he + have hins : (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0} : Set a.samples) ≤ᵐ[Q] (A \ E : Set a.samples) := by + filter_upwards [hident] with ω hω + intro hx + refine ⟨hx.1,?_⟩ + intro hsuc + have hpositive := source_restart_segment_positive_allKappa κ hk hκ8 h hh hf + (fun t => Real.sqrt κ*a.B t ω) (γ ω) (a.G ω) (a.F ω) (a.witness ω) + s δ (Φ (rp ω)) hω hK hKH hsuc + rw [hx.2] at hpositive + exact lt_irrefl _ hpositive + have hdiff : Q (A \ E) = Q A - Q (A ∩ E) := by + have he : A \ E = A \ (A ∩ E) := by ext ω; simp only [Set.mem_sdiff,mem_inter_iff]; tauto + rw [he] + exact measure_sdiff inter_subset_left (hAm.inter hEm).nullMeasurableSet (measure_ne_top Q _) + calc Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc s (s+δ)) = 0}) + ≤ Q (A \ E) := measure_mono_ae hins + _ = Q A - Q (A ∩ E) := hdiff + _ ≤ Q A - Q A*p := tsub_le_tsub_left (by rw [hprob]; gcongr) _ + _ = (1-p)*Q A := by + rw [ENNReal.sub_mul (fun _ _ => measure_ne_top Q A),one_mul,mul_comm p (Q A)] +end +end SLEExactGauge +end + + +section +open Set Filter MeasureTheory ProbabilityTheory Complex Metric +open scoped Topology NNReal ENNReal +namespace SLEExactGauge +noncomputable section +lemma RegularSLEPresentation.source_positive_fixed_interval_allKappa + {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h) + (hf : HasSmallRadiusFormula h (hFormula κ)) (s t : ℝ≥0) (hst : s < t) : + ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), 0 < hausdorffGauge h (γ ω '' Icc s t) := by + let Q := P.comap (Subtype.val : a.samples → Ω) + obtain ⟨p,hp,hp1,hslot⟩ := a.source_uniform_null_slot_bound_allKappa hκ hκ8 h hh hf + let c : ℝ≥0∞ := 1-p + have hc : c < 1 := ENNReal.sub_lt_self (by simp) (by simp) hp.ne' + have hsteps (δ : ℝ≥0) (hδ : 0 < δ) (k : ℕ) : + Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0} ≤ c^k := by + induction k with + | zero => simpa using (prob_le_one (μ := Q) + (s := {ω | hausdorffGauge h (γ ω '' Icc s (s+(0:ℝ≥0)*δ)) = 0})) + | succ k ih => + have he : s+((k+1:ℕ):ℝ≥0)*δ = (s+(k:ℝ≥0)*δ)+δ := by push_cast; ring + rw [he,pow_succ,mul_comm (c^k) c] + let A : Set a.samples := {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ)) = 0} + have hA : MeasurableSet[a.filtration (s+(k:ℝ≥0)*δ)] A := + a.source_segment_null_measurable_at_allKappa h hh s _ + calc Q {ω | hausdorffGauge h (γ ω '' Icc s (s+(k:ℝ≥0)*δ+δ)) = 0} + ≤ Q (A ∩ {ω | hausdorffGauge h (γ ω '' Icc (s+(k:ℝ≥0)*δ) (s+(k:ℝ≥0)*δ+δ)) = 0}) := by + apply measure_mono + intro ω hω + constructor + · exact measure_mono_null (image_mono (Icc_subset_Icc le_rfl (le_add_of_nonneg_right (show 0 ≤ δ from bot_le)))) hω + · exact measure_mono_null (image_mono (Icc_subset_Icc (le_add_of_nonneg_right (by positivity)) le_rfl)) hω + _ ≤ c * Q A := hslot _ δ hδ A hA + _ ≤ c * c^k := by gcongr + have hbound : ∀ᶠ n : ℕ in atTop, + Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} ≤ c^n := by + filter_upwards [eventually_ge_atTop 1] with n hn + have hn0 : 0 < (n:ℝ≥0) := by exact_mod_cast (show 0 < n by omega) + let δ : ℝ≥0 := (t-s)/(n:ℝ≥0) + have hδ : 0 < δ := div_pos (tsub_pos_iff_lt.mpr hst) hn0 + have he : s+(n:ℝ≥0)*δ = t := by + dsimp [δ] + rw [mul_div_cancel₀ _ hn0.ne',add_tsub_cancel_of_le hst.le] + simpa only [he] using hsteps δ hδ n + have hzero : Q {ω | hausdorffGauge h (γ ω '' Icc s t) = 0} = 0 := + le_antisymm (ge_of_tendsto (ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hc) hbound) zero_le + apply ae_iff.mpr + simpa only [not_lt,nonpos_iff_eq_zero] using hzero +end +end SLEExactGauge +end + + +section +open Set Filter MeasureTheory ProbabilityTheory Complex Metric +open scoped Topology NNReal ENNReal +namespace SLEExactGauge +noncomputable section +lemma RegularSLEPresentation.source_positive_all_intervals_allKappa + {Ω : Type} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (hκ8 : κ < 8) (h : ℝ → ℝ) (hh : IsGauge h) + (hf : HasSmallRadiusFormula h (hFormula κ)) : + ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), + ∀ s t : ℝ≥0, 0 < s → s < t → 0 < hausdorffGauge h (γ ω '' Icc s t) := by + have hrat : ∀ᵐ (ω : a.samples) ∂P.comap (Subtype.val : a.samples → Ω), + ∀ r q : ℚ, (r:ℝ).toNNReal < (q:ℝ).toNNReal → + 0 < hausdorffGauge h (γ ω '' Icc (r:ℝ).toNNReal (q:ℝ).toNNReal) := by + apply ae_all_iff.mpr + intro r + apply ae_all_iff.mpr + intro q + apply ae_all_iff.mpr + intro hrq + exact a.source_positive_fixed_interval_allKappa hκ hκ8 h hh hf _ _ hrq + filter_upwards [hrat] with ω hω + intro s t hs hst + obtain ⟨r,hsr,hrt⟩ := exists_rat_btwn (show (s:ℝ) < (t:ℝ) from hst) + obtain ⟨q,hrq,hqt⟩ := exists_rat_btwn hrt + have hq : 0 < (q:ℝ) := (show 0 < (s:ℝ) from hs).trans (hsr.trans hrq) + have hp := hω r q ((Real.toNNReal_lt_toNNReal_iff hq).mpr hrq) + apply hp.trans_le (measure_mono (image_mono (Icc_subset_Icc ?_ ?_))) + · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hsr.le + · simpa only [Real.toNNReal_coe] using Real.toNNReal_le_toNNReal hqt.le + + +def SourceLowerAllIntervalsTarget_allKappa : Prop := + ∀ κ : ℝ, 0 < κ → κ < 8 → + ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P → + ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P → + ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) → + ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → + 0 < hausdorffGauge h (γ ω '' Set.Icc s t) +theorem sourceLowerAllIntervals_proved_allKappa : SourceLowerAllIntervalsTarget_allKappa := by + intro κ hκ hκ8 Ω m P hP γ hSLE h hh hf + let : IsProbabilityMeasure P := hP + obtain ⟨a⟩ := exists_regular_sle_presentation hSLE + have ha := a.source_positive_all_intervals_allKappa hκ hκ8 h hh hf + rw [←a.measurePreserving_val.map_eq] + exact (MeasurableEmbedding.subtype_coe a.measurable_samples).ae_map_iff.mpr ha +end +end SLEExactGauge +end + + +section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal +namespace SLEExactGauge + + +def SourceLowerAllKappaTarget : Prop := + ∀ κ : ℝ, 0 < κ → κ < 8 → + ∀ (Ω : Type) (_ : MeasurableSpace Ω) (P : Measure Ω), IsProbabilityMeasure P → + ∀ (γ : Ω → ℝ≥0 → ℂ), IsOrdinaryChordalSLE κ γ P → + ∀ h : ℝ → ℝ, IsGauge h → HasSmallRadiusFormula h (hFormula κ) → + ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → + 0 < hausdorffGauge h (γ ω '' Set.Icc s t) +end SLEExactGauge + +namespace SLEExactGauge +theorem sourceLowerAllKappa_proved : SourceLowerAllKappaTarget := sourceLowerAllIntervals_proved_allKappa +end SLEExactGauge +end + + +section +open Set Filter MeasureTheory ProbabilityTheory Complex +open scoped Topology NNReal ENNReal +namespace SLEExactGauge +noncomputable section +local instance lowerUniverseRealPathMeasurable : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance lowerUniverseComplexPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance lowerUniverseRealPathBorel : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ +local instance lowerUniverseComplexPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ +lemma source_lower_universe {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} (hSLE : IsOrdinaryChordalSLE κ γ P) + (h : ℝ → ℝ) (hh : IsGauge h) (hf : HasSmallRadiusFormula h (hFormula κ)) : + ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → + 0 < hausdorffGauge h (γ ω '' Icc s t) := by + obtain ⟨a⟩ := exists_regular_sle_presentation hSLE + let μ := P.comap (Subtype.val : a.samples → Ω) + let : IsProbabilityMeasure μ := a.probability + let bp := sourceDriverPath a.B a.continuous_B + let gp := sourceTracePath (γ ∘ Subtype.val) a.trace_continuous + have hbm : Measurable bp := source_path_measurable bp a.measurable_B + have hgm : Measurable gp := source_path_measurable gp (fun t => (hSLE.1 t).comp measurable_subtype_coe) + let pair : a.samples → C(ℝ≥0,ℝ) × C(ℝ≥0,ℂ) := fun ω => (bp ω, gp ω) + have hpair : Measurable pair := hbm.prodMk hgm + let X := range pair + let f : a.samples → X := fun ω => ⟨pair ω, mem_range_self ω⟩ + have hfm : Measurable f := hpair.subtype_mk + let Q := μ.map f + let : IsProbabilityMeasure Q := inferInstance + let B : ℝ≥0 → X → ℝ := fun t x => x.val.1 t + let η : X → ℝ≥0 → ℂ := fun x t => x.val.2 t + have hBm (t : ℝ≥0) : Measurable (B t) := + (continuous_eval_const t).measurable.comp (measurable_fst.comp measurable_subtype_coe) + have hηm (t : ℝ≥0) : Measurable (fun x => η x t) := + (continuous_eval_const t).measurable.comp (measurable_snd.comp measurable_subtype_coe) + have hBrownian : IsBrownianReal B Q := by + constructor + · constructor + intro I + have hmI : Measurable (fun x : X => I.restrict (B · x)) := + Measurable.of_eval (fun index => hBm index) + refine ⟨hmI.aemeasurable, ?_⟩ + rw [show Q = μ.map f from rfl, Measure.map_map hmI hfm] + exact (a.brownian.hasLaw I).map_eq + · exact Eventually.of_forall (fun x => x.val.1.continuous) + have hη : IsOrdinaryChordalSLE κ η Q := by + refine ⟨hηm, B, hBrownian, Eventually.of_forall ?_⟩ + intro x + obtain ⟨ω, hω⟩ := x.property + refine ⟨a.G ω, a.F ω, ?_⟩ + have hp : pair ω = x.val := hω + have heB : ∀ t, B t x = a.B t ω := fun t => congrArg (fun z => z.1 t) hp.symm + have heη : η x = γ ω := by + exact congrArg (fun z => (z.2 : ℝ≥0 → ℂ)) hp.symm + simpa only [heB,heη] using a.witness ω + have hlow := sourceLowerAllKappa_proved κ hκ hκ8 X inferInstance Q inferInstance η hη h hh hf + have hpull := ae_of_ae_map hfm.aemeasurable hlow + rw [←a.measurePreserving_val.map_eq] + apply (MeasurableEmbedding.subtype_coe a.measurable_samples).ae_map_iff.mpr + exact hpull +end +end SLEExactGauge +end + + +end +end +end + +end OAI diff --git a/lean/OAI/Probability/SLE/MainReduction.lean b/lean/OAI/Probability/SLE/MainReduction.lean index 3fa2e16e73b480ec1fb6aac000925fa522130f44..4287667cc36934f06ff0f727e38988a5aed03d2d 100644 --- a/lean/OAI/Probability/SLE/MainReduction.lean +++ b/lean/OAI/Probability/SLE/MainReduction.lean @@ -1,206 +1,206 @@ -import Mathlib -import OAI.Probability.SLE.TraceMeasurability - -namespace OAI - -/-! Main Reduction for chordal SLE. -/ - -section -section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -open Set Filter MeasureTheory -open scoped Topology NNReal -open Set Filter MeasureTheory Complex -open MeasureTheory ProbabilityTheory Set Filter -open Set Filter Metric Complex InnerProductSpace -open scoped Topology Laplacian -open Set Filter Metric Complex MeasureTheory -open Set Filter Metric Complex MeasureTheory InnerProductSpace -open scoped Topology Laplacian ENNReal -open scoped Topology ENNReal -open scoped Topology Laplacian RealInnerProductSpace -open Set Filter Metric Complex -open scoped Topology -open Set Filter Metric -open Set Filter MeasureTheory Metric -open Set Filter Complex -open scoped Topology ComplexConjugate NNReal -open scoped Topology ENNReal NNReal ComplexConjugate -open Set Filter Metric MeasureTheory Complex -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped Topology ENNReal NNReal -open Set Filter MeasureTheory ProbabilityTheory Metric Classical -open Set Filter -open scoped Topology NNReal ENNReal ComplexConjugate -open scoped ENNReal NNReal -noncomputable section -open Set MeasureTheory ProbabilityTheory Filter -open scoped ENNReal NNReal Topology -open Set Filter MeasureTheory -open scoped Topology ENNReal NNReal -namespace SLEExactGauge - - - -def FiniteExpectedWholeTrace {Ω : Type*} [MeasurableSpace Ω] - (P : Measure Ω) (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : Prop := - ∀ R : ℝ, 0 < R → (∫⁻ ω, hausdorffGauge h - (wholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P) < ∞ - -lemma integerRectangle_subset_closedBall (m : ℕ) : - integerRectangle m ⊆ Metric.closedBall (0 : ℂ) (2 * (m : ℝ)) := by - intro z hz - have hre : |z.re| ≤ (m : ℝ) := abs_le.mpr ⟨hz.1, hz.2.1⟩ - have him : |z.im| ≤ (m : ℝ) := by - rw [abs_of_nonneg hz.2.2.1] - exact hz.2.2.2 - have hn : ‖z‖ ≤ 2 * (m : ℝ) := - (Complex.norm_le_abs_re_add_abs_im z).trans (by linarith) - simpa only [Metric.mem_closedBall, dist_zero_right] using hn - - - -lemma trace_inter_bounded_lt_top_of_integer_balls - (h : ℝ → ℝ) (γ : ℝ≥0 → ℂ) - (hballs : ∀ n : ℕ, hausdorffGauge h - (wholeTrace γ ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞) - (K : Set ℂ) (hK : Bornology.IsBounded K) : - hausdorffGauge h (wholeTrace γ ∩ K) < ∞ := by - obtain ⟨R, hR⟩ := hK.subset_closedBall (0 : ℂ) - obtain ⟨n, hn⟩ := exists_nat_ge R - have hsub : K ⊆ Metric.closedBall (0 : ℂ) ((n : ℝ) + 1) := - hR.trans (Metric.closedBall_subset_closedBall (by linarith)) - exact (measure_mono (inter_subset_inter_right _ hsub)).trans_lt (hballs n) - - - -lemma segment_lt_top_of_integer_balls - (h : ℝ → ℝ) (γ : ℝ≥0 → ℂ) (hc : Continuous γ) - (hballs : ∀ n : ℕ, hausdorffGauge h - (wholeTrace γ ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞) - (s t : ℝ≥0) : hausdorffGauge h (segment γ s t) < ∞ := by - have hK : Bornology.IsBounded (segment γ s t) := - (isCompact_Icc.image hc).isBounded - have hf := trace_inter_bounded_lt_top_of_integer_balls h γ hballs _ hK - have hsub : segment γ s t ⊆ wholeTrace γ := image_subset_range _ _ - simpa only [inter_eq_right.mpr hsub] using hf - - - -lemma mainEvent_of_integer_balls {Ω : Type*} - (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) (ω : Ω) - (hc : Continuous (γ ω)) - (hlo : ∀ s t : ℝ≥0, 0 < s → s < t → - 0 < hausdorffGauge h (segment (γ ω) s t)) - (hballs : ∀ n : ℕ, hausdorffGauge h - (wholeTrace (γ ω) ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞) : - ω ∈ MainEvent h γ := by - constructor - · intro s t hs hst - exact ⟨hlo s t hs hst, segment_lt_top_of_integer_balls h (γ ω) hc hballs s t⟩ - · intro m _hm - apply trace_inter_bounded_lt_top_of_integer_balls h (γ ω) hballs - exact Metric.isBounded_closedBall.subset (integerRectangle_subset_closedBall m) - - - -lemma ae_integer_balls_lt_top_of_finiteExpectation - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) - (hm : ∀ R : ℝ, 0 < R → AEMeasurable (fun ω => hausdorffGauge h - (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P) - (he : FiniteExpectedWholeTrace P h γ) : - ∀ᵐ ω ∂P, ∀ n : ℕ, hausdorffGauge h - (wholeTrace (γ ω) ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞ := by - apply ae_all_iff.mpr - intro n - have hn : 0 < (n : ℝ) + 1 := by positivity - exact ae_lt_top' (hm _ hn) (he _ hn).ne - - - -theorem mainConclusion_of_lower_and_finiteExpectation - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) - (hc : ∀ᵐ ω ∂P, Continuous (γ ω)) - (hlo : ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → - 0 < hausdorffGauge h (segment (γ ω) s t)) - (hm : ∀ R : ℝ, 0 < R → AEMeasurable (fun ω => hausdorffGauge h - (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P) - (he : FiniteExpectedWholeTrace P h γ) : MainConclusion P h γ := by - constructor - · have hb := ae_integer_balls_lt_top_of_finiteExpectation P h γ hm he - filter_upwards [hc, hlo, hb] with ω hcω hloω hbω - exact mainEvent_of_integer_balls h γ ω hcω hloω hbω - · intro R hR - refine ⟨hm R hR, ?_⟩ - refine ⟨(∫⁻ ω, hausdorffGauge h - (wholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P).toNNReal, ?_⟩ - rw [ENNReal.coe_toNNReal (he R hR).ne] - -lemma finiteExpectedWholeTrace_of_mainConclusion - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) (hc : MainConclusion P h γ) : - FiniteExpectedWholeTrace P h γ := by - intro R hR - obtain ⟨C, hC⟩ := (hc.2 R hR).2 - exact hC.trans_lt ENNReal.coe_lt_top - - - -lemma finiteExpectedWholeTrace_iff_integer_balls - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : - FiniteExpectedWholeTrace P h γ ↔ - ∀ n : ℕ, (∫⁻ ω, hausdorffGauge h - (wholeTrace (γ ω) ∩ Metric.closedBall 0 ((n : ℝ) + 1)) ∂P) < ∞ := by - constructor - · intro he n - exact he _ (by positivity) - · intro he R _hR - obtain ⟨n, hn⟩ := exists_nat_ge R - apply lt_of_le_of_lt _ (he n) - apply lintegral_mono - intro ω - apply measure_mono - exact inter_subset_inter_right _ (Metric.closedBall_subset_closedBall (by linarith)) - -end SLEExactGauge - - - -open MeasureTheory -open scoped ENNReal NNReal -namespace SLEExactGauge - - - -theorem mainConclusion_iff_finiteExpectedWholeTrace - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (h : ℝ → ℝ) (hh : IsGauge h) - (hf : HasSmallRadiusFormula h (hFormula κ)) : - MainConclusion P h γ ↔ FiniteExpectedWholeTrace P h γ := by - constructor - · exact finiteExpectedWholeTrace_of_mainConclusion P h γ - · intro he - have hc : ∀ᵐ ω ∂P, Continuous (γ ω) := by - obtain ⟨B, _hB, hcap⟩ := hSLE.2 - filter_upwards [hcap] with ω hω - obtain ⟨G, F, hGF⟩ := hω - exact hGF.1 - apply mainConclusion_of_lower_and_finiteExpectation P h γ hc - (sourceLowerMain_proved Ω inferInstance P inferInstance κ hκ hκ8 γ hSLE h hh hf) - _ he - intro R _hR - exact source_sle_whole_trace_aemeasurable P κ γ hSLE h hh _ Metric.isClosed_closedBall - -end SLEExactGauge - -end -end -end - -end OAI +import Mathlib +import OAI.Probability.SLE.TraceMeasurability + +namespace OAI + +/-! Main Reduction for chordal SLE. -/ + +section +section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal +open Set Filter MeasureTheory +open scoped Topology NNReal +open Set Filter MeasureTheory Complex +open MeasureTheory ProbabilityTheory Set Filter +open Set Filter Metric Complex InnerProductSpace +open scoped Topology Laplacian +open Set Filter Metric Complex MeasureTheory +open Set Filter Metric Complex MeasureTheory InnerProductSpace +open scoped Topology Laplacian ENNReal +open scoped Topology ENNReal +open scoped Topology Laplacian RealInnerProductSpace +open Set Filter Metric Complex +open scoped Topology +open Set Filter Metric +open Set Filter MeasureTheory Metric +open Set Filter Complex +open scoped Topology ComplexConjugate NNReal +open scoped Topology ENNReal NNReal ComplexConjugate +open Set Filter Metric MeasureTheory Complex +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped Topology ENNReal NNReal +open Set Filter MeasureTheory ProbabilityTheory Metric Classical +open Set Filter +open scoped Topology NNReal ENNReal ComplexConjugate +open scoped ENNReal NNReal +noncomputable section +open Set MeasureTheory ProbabilityTheory Filter +open scoped ENNReal NNReal Topology +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal +namespace SLEExactGauge + + + +def FiniteExpectedWholeTrace {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : Prop := + ∀ R : ℝ, 0 < R → (∫⁻ ω, hausdorffGauge h + (wholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P) < ∞ + +lemma integerRectangle_subset_closedBall (m : ℕ) : + integerRectangle m ⊆ Metric.closedBall (0 : ℂ) (2 * (m : ℝ)) := by + intro z hz + have hre : |z.re| ≤ (m : ℝ) := abs_le.mpr ⟨hz.1, hz.2.1⟩ + have him : |z.im| ≤ (m : ℝ) := by + rw [abs_of_nonneg hz.2.2.1] + exact hz.2.2.2 + have hn : ‖z‖ ≤ 2 * (m : ℝ) := + (Complex.norm_le_abs_re_add_abs_im z).trans (by linarith) + simpa only [Metric.mem_closedBall, dist_zero_right] using hn + + + +lemma trace_inter_bounded_lt_top_of_integer_balls + (h : ℝ → ℝ) (γ : ℝ≥0 → ℂ) + (hballs : ∀ n : ℕ, hausdorffGauge h + (wholeTrace γ ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞) + (K : Set ℂ) (hK : Bornology.IsBounded K) : + hausdorffGauge h (wholeTrace γ ∩ K) < ∞ := by + obtain ⟨R, hR⟩ := hK.subset_closedBall (0 : ℂ) + obtain ⟨n, hn⟩ := exists_nat_ge R + have hsub : K ⊆ Metric.closedBall (0 : ℂ) ((n : ℝ) + 1) := + hR.trans (Metric.closedBall_subset_closedBall (by linarith)) + exact (measure_mono (inter_subset_inter_right _ hsub)).trans_lt (hballs n) + + + +lemma segment_lt_top_of_integer_balls + (h : ℝ → ℝ) (γ : ℝ≥0 → ℂ) (hc : Continuous γ) + (hballs : ∀ n : ℕ, hausdorffGauge h + (wholeTrace γ ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞) + (s t : ℝ≥0) : hausdorffGauge h (segment γ s t) < ∞ := by + have hK : Bornology.IsBounded (segment γ s t) := + (isCompact_Icc.image hc).isBounded + have hf := trace_inter_bounded_lt_top_of_integer_balls h γ hballs _ hK + have hsub : segment γ s t ⊆ wholeTrace γ := image_subset_range _ _ + simpa only [inter_eq_right.mpr hsub] using hf + + + +lemma mainEvent_of_integer_balls {Ω : Type*} + (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) (ω : Ω) + (hc : Continuous (γ ω)) + (hlo : ∀ s t : ℝ≥0, 0 < s → s < t → + 0 < hausdorffGauge h (segment (γ ω) s t)) + (hballs : ∀ n : ℕ, hausdorffGauge h + (wholeTrace (γ ω) ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞) : + ω ∈ MainEvent h γ := by + constructor + · intro s t hs hst + exact ⟨hlo s t hs hst, segment_lt_top_of_integer_balls h (γ ω) hc hballs s t⟩ + · intro m _hm + apply trace_inter_bounded_lt_top_of_integer_balls h (γ ω) hballs + exact Metric.isBounded_closedBall.subset (integerRectangle_subset_closedBall m) + + + +lemma ae_integer_balls_lt_top_of_finiteExpectation + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) + (hm : ∀ R : ℝ, 0 < R → AEMeasurable (fun ω => hausdorffGauge h + (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P) + (he : FiniteExpectedWholeTrace P h γ) : + ∀ᵐ ω ∂P, ∀ n : ℕ, hausdorffGauge h + (wholeTrace (γ ω) ∩ Metric.closedBall 0 ((n : ℝ) + 1)) < ∞ := by + apply ae_all_iff.mpr + intro n + have hn : 0 < (n : ℝ) + 1 := by positivity + exact ae_lt_top' (hm _ hn) (he _ hn).ne + + + +theorem mainConclusion_of_lower_and_finiteExpectation + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) + (hc : ∀ᵐ ω ∂P, Continuous (γ ω)) + (hlo : ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, 0 < s → s < t → + 0 < hausdorffGauge h (segment (γ ω) s t)) + (hm : ∀ R : ℝ, 0 < R → AEMeasurable (fun ω => hausdorffGauge h + (wholeTrace (γ ω) ∩ Metric.closedBall 0 R)) P) + (he : FiniteExpectedWholeTrace P h γ) : MainConclusion P h γ := by + constructor + · have hb := ae_integer_balls_lt_top_of_finiteExpectation P h γ hm he + filter_upwards [hc, hlo, hb] with ω hcω hloω hbω + exact mainEvent_of_integer_balls h γ ω hcω hloω hbω + · intro R hR + refine ⟨hm R hR, ?_⟩ + refine ⟨(∫⁻ ω, hausdorffGauge h + (wholeTrace (γ ω) ∩ Metric.closedBall 0 R) ∂P).toNNReal, ?_⟩ + rw [ENNReal.coe_toNNReal (he R hR).ne] + +lemma finiteExpectedWholeTrace_of_mainConclusion + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) (hc : MainConclusion P h γ) : + FiniteExpectedWholeTrace P h γ := by + intro R hR + obtain ⟨C, hC⟩ := (hc.2 R hR).2 + exact hC.trans_lt ENNReal.coe_lt_top + + + +lemma finiteExpectedWholeTrace_iff_integer_balls + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (h : ℝ → ℝ) (γ : Ω → ℝ≥0 → ℂ) : + FiniteExpectedWholeTrace P h γ ↔ + ∀ n : ℕ, (∫⁻ ω, hausdorffGauge h + (wholeTrace (γ ω) ∩ Metric.closedBall 0 ((n : ℝ) + 1)) ∂P) < ∞ := by + constructor + · intro he n + exact he _ (by positivity) + · intro he R _hR + obtain ⟨n, hn⟩ := exists_nat_ge R + apply lt_of_le_of_lt _ (he n) + apply lintegral_mono + intro ω + apply measure_mono + exact inter_subset_inter_right _ (Metric.closedBall_subset_closedBall (by linarith)) + +end SLEExactGauge + + + +open MeasureTheory +open scoped ENNReal NNReal +namespace SLEExactGauge + + + +theorem mainConclusion_iff_finiteExpectedWholeTrace + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (h : ℝ → ℝ) (hh : IsGauge h) + (hf : HasSmallRadiusFormula h (hFormula κ)) : + MainConclusion P h γ ↔ FiniteExpectedWholeTrace P h γ := by + constructor + · exact finiteExpectedWholeTrace_of_mainConclusion P h γ + · intro he + have hc : ∀ᵐ ω ∂P, Continuous (γ ω) := by + obtain ⟨B, _hB, hcap⟩ := hSLE.2 + filter_upwards [hcap] with ω hω + obtain ⟨G, F, hGF⟩ := hω + exact hGF.1 + apply mainConclusion_of_lower_and_finiteExpectation P h γ hc + (sourceLowerMain_proved Ω inferInstance P inferInstance κ hκ hκ8 γ hSLE h hh hf) + _ he + intro R _hR + exact source_sle_whole_trace_aemeasurable P κ γ hSLE h hh _ Metric.isClosed_closedBall + +end SLEExactGauge + +end +end +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalImageMass.lean b/lean/OAI/Probability/SLE/Mass/CanonicalImageMass.lean index 14bc0da51e60ec082a26fc70eb33612b7c22c73d..5c10c3bbcb3bcd03fac92533e07e05c087e1c8b5 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalImageMass.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalImageMass.lean @@ -1,71 +1,71 @@ -import OAI.Probability.SLE.Mass.ConformalMassTransport - -namespace OAI - -/-! The canonical process in a fixed conformal image. The pathwise limit -event is common to all image maps and compact tests. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) {D : Set ℂ} - -def imageProcess (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) - (t : ℝ≥0) (ω : Ω) : Measure D := conformalMass J w (C.process t ω) - -lemma imageProcess_progressive (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) : - IsProgressive (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) - (C.imageProcess J w) := by - intro t - exact (conformalMass_measurable J w).comp (C.progressive t) - -lemma imageProcess_radon (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) - (t : ℝ≥0) (ω : Ω) : - (C.imageProcess J w t ω).Regular ∧ IsFiniteMeasureOnCompacts (C.imageProcess J w t ω) := by - let := (C.radon t ω).2 - exact conformalMass_radon J w (C.process t ω) - -lemma imageProcess_paths (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) : - ∀ᵐ ω ∂P, C.imageProcess J w 0 ω = 0 ∧ - Monotone (fun t => C.imageProcess J w t ω) ∧ - (∀ t, C.imageProcess J w t ω ≤ conformalMass J w (C.rational.terminal ω)) ∧ - (∀ f : C_c(D,ℝ), Continuous (fun t => ∫ z, f z ∂C.imageProcess J w t ω)) ∧ - NullSingletonClass (conformalMass J w (C.rational.terminal ω)) := by - filter_upwards [C.paths,C.rational.nonatomic] with ω hp hn - let := hn - refine ⟨?_,?_,?_,?_,conformalMass_nullSingleton J w _⟩ - · simp only [imageProcess,hp.zero,conformalMass_zero] - · intro s t hst - exact conformalMass_mono J w (hp.monotone hst) - · intro t - exact conformalMass_mono J w (hp.dominated t) - · exact conformalMass_test_continuous J w (fun t => C.process t ω) hp.continuous_tests - -/-- No image map changes the extraction: one event works for every map, -every continuous nonnegative weight, all extended times, and all tests. -/ -theorem all_image_limits - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) : - ∀ᵐ ω ∂P, ∀ (D : Set ℂ) (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane,ℝ≥0)) (t : ℝ≥0∞) (f : C_c(D,ℝ)), - Tendsto (fun n => ∫ z, f z ∂conformalMass J w - (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) C.rational.supports C.rational.weights n t ω)) - atTop (𝓝 (∫ z, f z ∂conformalMass J w (C.atTime t ω))) := by - filter_upwards [C.rational.all_extended_times_limit hBc hκ hκ8 F hw - C.process C.radon C.paths] with ω hω - intro D J w t f - exact conformalMass_test_tendsto J w _ _ (hω t) f - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Mass.ConformalMassTransport + +namespace OAI + +/-! The canonical process in a fixed conformal image. The pathwise limit +event is common to all image maps and compact tests. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) {D : Set ℂ} + +def imageProcess (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) + (t : ℝ≥0) (ω : Ω) : Measure D := conformalMass J w (C.process t ω) + +lemma imageProcess_progressive (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) : + IsProgressive (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) + (C.imageProcess J w) := by + intro t + exact (conformalMass_measurable J w).comp (C.progressive t) + +lemma imageProcess_radon (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) + (t : ℝ≥0) (ω : Ω) : + (C.imageProcess J w t ω).Regular ∧ IsFiniteMeasureOnCompacts (C.imageProcess J w t ω) := by + let := (C.radon t ω).2 + exact conformalMass_radon J w (C.process t ω) + +lemma imageProcess_paths (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane,ℝ≥0)) : + ∀ᵐ ω ∂P, C.imageProcess J w 0 ω = 0 ∧ + Monotone (fun t => C.imageProcess J w t ω) ∧ + (∀ t, C.imageProcess J w t ω ≤ conformalMass J w (C.rational.terminal ω)) ∧ + (∀ f : C_c(D,ℝ), Continuous (fun t => ∫ z, f z ∂C.imageProcess J w t ω)) ∧ + NullSingletonClass (conformalMass J w (C.rational.terminal ω)) := by + filter_upwards [C.paths,C.rational.nonatomic] with ω hp hn + let := hn + refine ⟨?_,?_,?_,?_,conformalMass_nullSingleton J w _⟩ + · simp only [imageProcess,hp.zero,conformalMass_zero] + · intro s t hst + exact conformalMass_mono J w (hp.monotone hst) + · intro t + exact conformalMass_mono J w (hp.dominated t) + · exact conformalMass_test_continuous J w (fun t => C.process t ω) hp.continuous_tests + +/-- No image map changes the extraction: one event works for every map, +every continuous nonnegative weight, all extended times, and all tests. -/ +theorem all_image_limits + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) : + ∀ᵐ ω ∂P, ∀ (D : Set ℂ) (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane,ℝ≥0)) (t : ℝ≥0∞) (f : C_c(D,ℝ)), + Tendsto (fun n => ∫ z, f z ∂conformalMass J w + (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) C.rational.supports C.rational.weights n t ω)) + atTop (𝓝 (∫ z, f z ∂conformalMass J w (C.atTime t ω))) := by + filter_upwards [C.rational.all_extended_times_limit hBc hκ hκ8 F hw + C.process C.radon C.paths] with ω hω + intro D J w t f + exact conformalMass_test_tendsto J w _ _ (hω t) f + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassAllTimes.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassAllTimes.lean index 4fcbc87215e078d676236203b3b826bee45fff45..e2f6766d700f4447b53ef4b5342545275045b128 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassAllTimes.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassAllTimes.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Mass.CanonicalMassContinuity -import OAI.Probability.SLE.Geometry.MonotoneRationalLimit - -namespace OAI - -/-! One common convex sequence converges at every finite time, on one -event of full probability. In particular it can be evaluated at a random time. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem RationalCanonicalMass.all_finite_times_limit - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (C : RationalCanonicalMass P κ B hBm γ G) - (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) - (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) - (C.terminal ω) (fun t => N t ω)) : - ∀ᵐ ω ∂P, ∀ (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n t ω) atTop (𝓝 (∫ z, f z ∂N t ω)) := by - filter_upwards [C.mass_limit,hpath] with ω hl hp - let M (n : ℕ) (t : ℝ≥0) := convexLevelMeasure κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n t ω - let (n : ℕ) (t : ℝ≥0) : IsFiniteMeasureOnCompacts (M n t) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) - _ C.supports C.weights n t - let (t : ℝ≥0) : IsFiniteMeasureOnCompacts (N t ω) := (hNr t ω).2 - have hpositive (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (t : ℝ≥0) : - Tendsto (fun n => ∫ z, f z ∂M n t) atTop (𝓝 (∫ z, f z ∂N t ω)) := by - have hrat (q : ℚ) : - Tendsto (fun n => ∫ z, f z ∂M n (q:ℝ).toNNReal) atTop - (𝓝 (∫ z, f z ∂N (q:ℝ).toNNReal ω)) := by - by_cases hq : 0 ≤ q - · rw [hp.rational q hq] - exact hl q f - · have he : (q:ℝ).toNNReal = 0 := - Real.toNNReal_eq_zero.mpr (Rat.cast_nonpos.mpr (le_of_not_ge hq)) - have hzero : N 0 ω = C.mass 0 ω := by - simpa only [Rat.cast_zero,Real.toNNReal_zero] using hp.rational 0 le_rfl - simpa only [he,hzero,Rat.cast_zero,Real.toNNReal_zero] using hl 0 f - have hlim := tendsto_of_monotone_rational_limits - (fun n r => ∫ z, f z ∂M n r.toNNReal) - (fun r => ∫ z, f z ∂N r.toNNReal ω) (fun n r s hrs => - integral_mono_measure (convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n ω (ENNReal.coe_le_coe.mpr (Real.toNNReal_mono hrs))) - (ae_of_all _ hf) f.integrable) - ((hp.continuous_tests f).comp continuous_real_toNNReal) hrat (t:ℝ) - simpa only [Real.toNNReal_coe] using hlim - intro t f - let fp := f ⊔ 0 - let fm := (-f) ⊔ 0 - have hsplit (ρ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts ρ] : - (∫ z, f z ∂ρ) = (∫ z, fp z ∂ρ)-(∫ z, fm z ∂ρ) := by - rw [← integral_sub fp.integrable fm.integrable] - apply integral_congr_ae - filter_upwards [] with z - change f z = max (f z) 0-max (-f z) 0 - exact (max_zero_sub_eq_self (f z)).symm - simpa only [← hsplit] using - (hpositive fp (fun z => le_sup_of_le_right (by rfl)) t).sub - (hpositive fm (fun z => le_sup_of_le_right (by rfl)) t) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassContinuity +import OAI.Probability.SLE.Geometry.MonotoneRationalLimit + +namespace OAI + +/-! One common convex sequence converges at every finite time, on one +event of full probability. In particular it can be evaluated at a random time. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem RationalCanonicalMass.all_finite_times_limit + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (C : RationalCanonicalMass P κ B hBm γ G) + (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) + (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) + (C.terminal ω) (fun t => N t ω)) : + ∀ᵐ ω ∂P, ∀ (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n t ω) atTop (𝓝 (∫ z, f z ∂N t ω)) := by + filter_upwards [C.mass_limit,hpath] with ω hl hp + let M (n : ℕ) (t : ℝ≥0) := convexLevelMeasure κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n t ω + let (n : ℕ) (t : ℝ≥0) : IsFiniteMeasureOnCompacts (M n t) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) + _ C.supports C.weights n t + let (t : ℝ≥0) : IsFiniteMeasureOnCompacts (N t ω) := (hNr t ω).2 + have hpositive (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (t : ℝ≥0) : + Tendsto (fun n => ∫ z, f z ∂M n t) atTop (𝓝 (∫ z, f z ∂N t ω)) := by + have hrat (q : ℚ) : + Tendsto (fun n => ∫ z, f z ∂M n (q:ℝ).toNNReal) atTop + (𝓝 (∫ z, f z ∂N (q:ℝ).toNNReal ω)) := by + by_cases hq : 0 ≤ q + · rw [hp.rational q hq] + exact hl q f + · have he : (q:ℝ).toNNReal = 0 := + Real.toNNReal_eq_zero.mpr (Rat.cast_nonpos.mpr (le_of_not_ge hq)) + have hzero : N 0 ω = C.mass 0 ω := by + simpa only [Rat.cast_zero,Real.toNNReal_zero] using hp.rational 0 le_rfl + simpa only [he,hzero,Rat.cast_zero,Real.toNNReal_zero] using hl 0 f + have hlim := tendsto_of_monotone_rational_limits + (fun n r => ∫ z, f z ∂M n r.toNNReal) + (fun r => ∫ z, f z ∂N r.toNNReal ω) (fun n r s hrs => + integral_mono_measure (convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n ω (ENNReal.coe_le_coe.mpr (Real.toNNReal_mono hrs))) + (ae_of_all _ hf) f.integrable) + ((hp.continuous_tests f).comp continuous_real_toNNReal) hrat (t:ℝ) + simpa only [Real.toNNReal_coe] using hlim + intro t f + let fp := f ⊔ 0 + let fm := (-f) ⊔ 0 + have hsplit (ρ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts ρ] : + (∫ z, f z ∂ρ) = (∫ z, fp z ∂ρ)-(∫ z, fm z ∂ρ) := by + rw [← integral_sub fp.integrable fm.integrable] + apply integral_congr_ae + filter_upwards [] with z + change f z = max (f z) 0-max (-f z) 0 + exact (max_zero_sub_eq_self (f z)).symm + simpa only [← hsplit] using + (hpositive fp (fun z => le_sup_of_le_right (by rfl)) t).sub + (hpositive fm (fun z => le_sup_of_le_right (by rfl)) t) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassAllVisits.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassAllVisits.lean index 120b7b532cc8f530194716f0966c5a534a9e1e91..e592e052586d5be3345be19f7a2746bc3883578c 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassAllVisits.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassAllVisits.lean @@ -1,47 +1,47 @@ -import OAI.Probability.SLE.Mass.CanonicalMassFirstVisit -import OAI.Probability.SLE.Loewner.TraceRestrictionContinuity - -namespace OAI - -/-! The first-visit identity holds on one event simultaneously at every -finite time. In particular later returns carry no previously assigned mass. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem CanonicalMass.all_times_eq_restrict_past - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (C : CanonicalMass P κ B hBm γ G) : - ∀ᵐ ω ∂P, ∀ t : ℝ≥0, C.process t ω = (C.rational.terminal ω).restrict - {z : upperHalfPlane | (z:ℂ) ∈ γ ω '' Icc 0 t} := by - have hq (q : ℚ) := C.atTime_eq_restrict_past hB hBm hBc hB0 hκ hκ8 F hw (q:ℝ).toNNReal - have hlim := C.rational.all_finite_times_limit hBc hκ hκ8 F hw C.process C.radon C.paths - filter_upwards [ae_all_iff.mpr hq,C.paths,hlim] with ω hq hp hl - let : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 - have hpast (t : ℝ≥0) : C.process t ω {z : upperHalfPlane | (z:ℂ) ∉ γ ω '' Icc 0 t} = 0 := by - let : (C.process t ω).Regular := (C.radon t ω).1 - exact convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) t (fun n => (n:ℝ)) - tendsto_natCast_atTop_atTop C.rational.supports C.rational.weights (C.process t ω) (hl t) - apply trace_restriction_of_rational_times (γ ω) (hw ω).1 (fun q => C.rational.mass q ω) - (C.rational.terminal ω) (fun t => C.process t ω) hp hpast - intro q _ - simpa only [CanonicalMass.atTime,extendedCanonicalMass,ENNReal.coe_lt_top, - ite_true,ENNReal.toNNReal_coe] using hq q - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassFirstVisit +import OAI.Probability.SLE.Loewner.TraceRestrictionContinuity + +namespace OAI + +/-! The first-visit identity holds on one event simultaneously at every +finite time. In particular later returns carry no previously assigned mass. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem CanonicalMass.all_times_eq_restrict_past + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (C : CanonicalMass P κ B hBm γ G) : + ∀ᵐ ω ∂P, ∀ t : ℝ≥0, C.process t ω = (C.rational.terminal ω).restrict + {z : upperHalfPlane | (z:ℂ) ∈ γ ω '' Icc 0 t} := by + have hq (q : ℚ) := C.atTime_eq_restrict_past hB hBm hBc hB0 hκ hκ8 F hw (q:ℝ).toNNReal + have hlim := C.rational.all_finite_times_limit hBc hκ hκ8 F hw C.process C.radon C.paths + filter_upwards [ae_all_iff.mpr hq,C.paths,hlim] with ω hq hp hl + let : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 + have hpast (t : ℝ≥0) : C.process t ω {z : upperHalfPlane | (z:ℂ) ∉ γ ω '' Icc 0 t} = 0 := by + let : (C.process t ω).Regular := (C.radon t ω).1 + exact convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) t (fun n => (n:ℝ)) + tendsto_natCast_atTop_atTop C.rational.supports C.rational.weights (C.process t ω) (hl t) + apply trace_restriction_of_rational_times (γ ω) (hw ω).1 (fun q => C.rational.mass q ω) + (C.rational.terminal ω) (fun t => C.process t ω) hp hpast + intro q _ + simpa only [CanonicalMass.atTime,extendedCanonicalMass,ENNReal.coe_lt_top, + ite_true,ENNReal.toNNReal_coe] using hq q + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassCompensator.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassCompensator.lean index 6e71f7a953a409e41196a84027e40771184513f6..d8eb8b77f7e523ad0d16f1b01b781bd72d2480a3 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassCompensator.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassCompensator.lean @@ -1,53 +1,53 @@ -import OAI.Probability.SLE.Geometry.WeightedCompactTerminal - -namespace OAI - -/-! Predictable dyadic recovery for the actual compact weighted canonical -mass. The time argument is the one used by the given filtration. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -def compensatorMeshFiltration {Ω : Type*} [mΩ : MeasurableSpace Ω] - (ℱ : Filtration ℝ≥0 mΩ) (n : ℕ) : Filtration ℕ mΩ where - seq i := ℱ (compensatorMeshTime n i) - mono' := fun _ _ h => ℱ.mono (compensatorMeshTime_mono n h) - le' _i := ℱ.le _ - -namespace Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -theorem compactWeightedMass_compensator_recovery - (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) - (T : Ω → ℝ≥0∞) (hT : Measurable T) : - TendstoInMeasure P - (fun n ω => meshMassCompensator P - (compensatorMeshFiltration (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable))) - (fun t ω => compactWeightedMass w K (C.process t ω)) n - (compensatorStopIndex n (T ω)) ω) atTop - (fun ω => massAtExtendedTime (fun t => compactWeightedMass w K (C.process t ω)) - (compactWeightedMass w K (C.rational.terminal ω)) (T ω)) := by - apply meshMassCompensator_stopped_tendstoInMeasure P _ _ _ _ _ - (C.compactWeightedMass_terminal_memLp w K hK) (C.compactWeightedMass_paths w K hK) T hT - · intro t - exact (compactWeightedMass_measurable w K hK.measurableSet).comp - (C.progressive.adapted.measurable (i := t)) - · intro n i - exact ((compactWeightedMass_measurable w K hK.measurableSet).comp - (C.progressive.adapted (compensatorMeshTime n i))).stronglyMeasurable - -end Multipoint.CanonicalMass -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.WeightedCompactTerminal + +namespace OAI + +/-! Predictable dyadic recovery for the actual compact weighted canonical +mass. The time argument is the one used by the given filtration. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +def compensatorMeshFiltration {Ω : Type*} [mΩ : MeasurableSpace Ω] + (ℱ : Filtration ℝ≥0 mΩ) (n : ℕ) : Filtration ℕ mΩ where + seq i := ℱ (compensatorMeshTime n i) + mono' := fun _ _ h => ℱ.mono (compensatorMeshTime_mono n h) + le' _i := ℱ.le _ + +namespace Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +theorem compactWeightedMass_compensator_recovery + (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) + (T : Ω → ℝ≥0∞) (hT : Measurable T) : + TendstoInMeasure P + (fun n ω => meshMassCompensator P + (compensatorMeshFiltration (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable))) + (fun t ω => compactWeightedMass w K (C.process t ω)) n + (compensatorStopIndex n (T ω)) ω) atTop + (fun ω => massAtExtendedTime (fun t => compactWeightedMass w K (C.process t ω)) + (compactWeightedMass w K (C.rational.terminal ω)) (T ω)) := by + apply meshMassCompensator_stopped_tendstoInMeasure P _ _ _ _ _ + (C.compactWeightedMass_terminal_memLp w K hK) (C.compactWeightedMass_paths w K hK) T hT + · intro t + exact (compactWeightedMass_measurable w K hK.measurableSet).comp + (C.progressive.adapted.measurable (i := t)) + · intro n i + exact ((compactWeightedMass_measurable w K hK.measurableSet).comp + (C.progressive.adapted (compensatorMeshTime n i))).stronglyMeasurable + +end Multipoint.CanonicalMass +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassConstruction.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassConstruction.lean index c9b1c2ce06ccae3e5329fb9af1e18ca80e0e55cd..23e647b025255414c370b001104c3998a539b2c7 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassConstruction.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassConstruction.lean @@ -1,81 +1,81 @@ -import OAI.Probability.SLE.Mass.CanonicalMassStoppingKernel -import OAI.Probability.SLE.Mass.CanonicalMassExhaustion - -namespace OAI - -/-! The first paper's canonical mass construction, including the exact -conditional kernel at arbitrary stopping times and terminal exhaustion. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -structure CanonicalMass - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ) - (B : ℝ≥0 → Ω → ℝ) (hBm : Measurable (Function.uncurry B)) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) where - rational : RationalCanonicalMass P κ B hBm γ G - process : ℝ≥0 → Ω → Measure upperHalfPlane - progressive : IsProgressive (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) process - radon : ∀ t ω, (process t ω).Regular ∧ IsFiniteMeasureOnCompacts (process t ω) - paths : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => rational.mass q ω) - (rational.terminal ω) (fun t => process t ω) - exhaustion : ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (rational.terminal ω-process t ω) K) atTop (𝓝 0) - kernel : ∀ S : Ω → WithTop ℝ≥0, - ∀ hS : IsStoppingTime (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S, - ∀ E : Set Ω, MeasurableSet[hS.measurableSpace] E → - (P.restrict E).bind rational.terminal = - (P.restrict E).bind (fun ω => extendedCanonicalMass process rational.terminal (S ω) ω) + - (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) - -/-- Proposition `mass:construction` of the first manuscript. The sole -published input is the disk-growth consequence of Duren (1983), §2.3; -all stochastic estimates and limiting arguments are proved here. -/ -theorem exists_canonicalMass - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) : - Nonempty (CanonicalMass P κ B hBm γ G) := by - obtain ⟨C⟩ := exists_rationalCanonicalMass hB - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - obtain ⟨N,hNp,hNr,hpath⟩ := C.exists_continuous_progressive_extension (fun ω => (hw ω).1) - exact ⟨{ - rational := C - process := N - progressive := hNp - radon := hNr - paths := hpath - exhaustion := C.finite_times_exhaust_terminal hB hBm hBc hB0 hκ hκ8 γ G F hw N hpath - kernel := C.stopping_conditional_kernel hB hBm hBc hB0 hκ hκ8 F hw hgrowth N hNp hNr hpath }⟩ - -end SLEExactGauge.Multipoint - -namespace SLEExactGauge - -/-- The canonical mass main construction for an ordinary SLE in its -regular Brownian presentation, on its full-measure sample space. -/ -theorem RegularSLEPresentation.canonical_mass_main - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) - (hκ : 0 < κ) (hκ8 : κ < 8) (hgrowth : PublishedDiskUnivalentGrowth) : - Nonempty (Multipoint.CanonicalMass (P.comap (Subtype.val : a.samples → Ω)) κ a.B - (measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B) - (fun ω => γ ω) a.G) := - Multipoint.exists_canonicalMass a.brownian - (measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B) - a.continuous_B a.zero_B hκ hκ8 (fun ω => γ ω) a.G a.F a.witness hgrowth - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassStoppingKernel +import OAI.Probability.SLE.Mass.CanonicalMassExhaustion + +namespace OAI + +/-! The first paper's canonical mass construction, including the exact +conditional kernel at arbitrary stopping times and terminal exhaustion. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +structure CanonicalMass + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (κ : ℝ) + (B : ℝ≥0 → Ω → ℝ) (hBm : Measurable (Function.uncurry B)) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) where + rational : RationalCanonicalMass P κ B hBm γ G + process : ℝ≥0 → Ω → Measure upperHalfPlane + progressive : IsProgressive (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) process + radon : ∀ t ω, (process t ω).Regular ∧ IsFiniteMeasureOnCompacts (process t ω) + paths : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => rational.mass q ω) + (rational.terminal ω) (fun t => process t ω) + exhaustion : ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (rational.terminal ω-process t ω) K) atTop (𝓝 0) + kernel : ∀ S : Ω → WithTop ℝ≥0, + ∀ hS : IsStoppingTime (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S, + ∀ E : Set Ω, MeasurableSet[hS.measurableSpace] E → + (P.restrict E).bind rational.terminal = + (P.restrict E).bind (fun ω => extendedCanonicalMass process rational.terminal (S ω) ω) + + (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) + +/-- Proposition `mass:construction` of the first manuscript. The sole +published input is the disk-growth consequence of Duren (1983), §2.3; +all stochastic estimates and limiting arguments are proved here. -/ +theorem exists_canonicalMass + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) : + Nonempty (CanonicalMass P κ B hBm γ G) := by + obtain ⟨C⟩ := exists_rationalCanonicalMass hB + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + obtain ⟨N,hNp,hNr,hpath⟩ := C.exists_continuous_progressive_extension (fun ω => (hw ω).1) + exact ⟨{ + rational := C + process := N + progressive := hNp + radon := hNr + paths := hpath + exhaustion := C.finite_times_exhaust_terminal hB hBm hBc hB0 hκ hκ8 γ G F hw N hpath + kernel := C.stopping_conditional_kernel hB hBm hBc hB0 hκ hκ8 F hw hgrowth N hNp hNr hpath }⟩ + +end SLEExactGauge.Multipoint + +namespace SLEExactGauge + +/-- The canonical mass main construction for an ordinary SLE in its +regular Brownian presentation, on its full-measure sample space. -/ +theorem RegularSLEPresentation.canonical_mass_main + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} (a : RegularSLEPresentation P κ γ) + (hκ : 0 < κ) (hκ8 : κ < 8) (hgrowth : PublishedDiskUnivalentGrowth) : + Nonempty (Multipoint.CanonicalMass (P.comap (Subtype.val : a.samples → Ω)) κ a.B + (measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B) + (fun ω => γ ω) a.G) := + Multipoint.exists_canonicalMass a.brownian + (measurable_uncurry_of_continuous_of_measurable a.continuous_B a.measurable_B) + a.continuous_B a.zero_B hκ hκ8 (fun ω => γ ω) a.G a.F a.witness hgrowth + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassContinuity.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassContinuity.lean index ad6a4d91cfd56f76e761266df83c9feb58312127..f9a68e2dc155cd3cf78388feaf19584cdb8a9707 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassContinuity.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassContinuity.lean @@ -1,115 +1,115 @@ -import OAI.Probability.SLE.Mass.RationalCanonicalMass -import OAI.Probability.SLE.Mass.RationalTraceMassSupport -import OAI.Probability.SLE.Mass.RationalTraceMassLocalVariation -import OAI.Probability.SLE.Mass.RationalTraceMassProgressiveVersion - -namespace OAI - -/-! The actual rational construction extends to an adapted, locally -continuous family, with simultaneous support of every finite increment. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -structure ContinuousMassPath (γ : ℝ≥0 → ℂ) (μ : ℚ → Measure upperHalfPlane) - (ν : Measure upperHalfPlane) (N : ℝ≥0 → Measure upperHalfPlane) : Prop where - zero : N 0 = 0 - dominated : ∀ t, N t ≤ ν - monotone : Monotone N - rational : ∀ q : ℚ, 0 ≤ q → N (q:ℝ).toNNReal = μ q - continuous_tests : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t) - increment_support : ∀ s t, s ≤ t → - (N t-N s) {z : upperHalfPlane | (z:ℂ) ∉ - (fun r : ℝ => γ r.toNNReal) '' Icc (s:ℝ) t} = 0 - local_variation : ∀ K : Set upperHalfPlane, IsCompact K → - ∀ t₀ : ℝ≥0, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧ - ∀ s t : ℝ≥0, dist (s:ℝ) t₀ < δ → dist (t:ℝ) t₀ < δ → - ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K → - |(N s A).toReal-(N t A).toReal| < ε - -theorem RationalCanonicalMass.exists_continuous_progressive_extension - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : RationalCanonicalMass P κ B hBm γ G) (hγ : ∀ ω, Continuous (γ ω)) : - ∃ N : ℝ≥0 → Ω → Measure upperHalfPlane, - IsProgressive (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) N ∧ - (∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) ∧ - ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) (C.terminal ω) (fun t => N t ω) := by - let Good (ω : Ω) : Prop := - HalfPlaneTraceIncrements (fun q => C.mass q ω) (C.terminal ω) - (fun r : ℝ => γ ω r.toNNReal) ∧ NullSingletonClass (C.terminal ω) - have hex (ω : Ω) : ∃ M : ℝ → Measure upperHalfPlane, - (∀ t, (M t).Regular) ∧ - (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t)) ∧ - (Good ω → (∀ t, M t ≤ C.terminal ω) ∧ (∀ q : ℚ, M q = C.mass q ω) ∧ - Monotone M ∧ - (∀ K : Set upperHalfPlane, IsCompact K → - ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧ - ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ → - ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K → - |(M s A).toReal-(M t A).toReal| < ε)) := by - by_cases h : Good ω - · let : (C.terminal ω).Regular := (C.terminal_radon ω).1 - let : NullSingletonClass (C.terminal ω) := h.2 - obtain ⟨M,hr,hd,hq,hm,hc,hv⟩ := h.1.exists_locally_tv_continuous_extension - ((hγ ω).comp continuous_real_toNNReal) - exact ⟨M,hr,hc,fun _ => ⟨hd,hq,hm,hv⟩⟩ - · refine ⟨fun _ => 0,fun _ => inferInstance,?_,fun hh => (h hh).elim⟩ - intro f - simpa only [integral_zero_measure] using (continuous_const : Continuous (fun _ : ℝ => (0:ℝ))) - choose M hMr hMc hM using hex - have hrat : ∀ᵐ ω ∂P, ∀ q : ℚ, M ω q = C.mass q ω := by - filter_upwards [C.increments,C.nonatomic] with ω hi hn - exact (hM ω ⟨hi,hn⟩).2.1 - let N := rationalRadonExtension C.mass - have hNp : IsProgressive (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) N := - rationalRadonExtension_progressive _ C.mass (fun q ω => (C.mass_radon q ω).1) - (fun q _ => C.mass_measurable q) - have hNr := rationalRadonExtension_regular C.mass - have hNe := rationalRadonExtension_common_version P C.mass (fun t ω => M ω t) - (fun t ω => hMr ω t) (Eventually.of_forall hMc) hrat - refine ⟨N,hNp,hNr,?_⟩ - filter_upwards [C.increments,C.nonatomic,C.zero,hNe] with ω hi hn hz he - change ∀ t : ℝ≥0, N t ω = M ω t at he - obtain ⟨hd,hq,hm,hv⟩ := hM ω ⟨hi,hn⟩ - refine ⟨?_,?_,?_,?_,?_,?_,?_⟩ - · rw [he 0] - simpa only [NNReal.coe_zero,Rat.cast_zero] using (hq 0).trans hz - · intro t - rw [he t] - exact hd t - · intro s t hst - change N s ω ≤ N t ω - rw [he s,he t] - exact hm (NNReal.coe_le_coe.mpr hst) - · intro q hq0 - rw [he _ ,Real.coe_toNNReal _ (Rat.cast_nonneg.mpr hq0)] - exact hq q - · intro f - have hc := (hMc ω f).comp NNReal.continuous_coe - simpa only [Function.comp_def,← he] using hc - · intro s t hst - rw [he s,he t] - let : (C.terminal ω).Regular := (C.terminal_radon ω).1 - exact hi.extension_increment_carried ((hγ ω).comp continuous_real_toNNReal) - (M ω) (hMr ω) hq hm (NNReal.coe_le_coe.mpr hst) - · intro K hK t₀ ε hε - obtain ⟨δ,hδ,hδprop⟩ := hv K hK t₀ ε hε - refine ⟨δ,hδ,?_⟩ - intro s t hs ht A hA hAK - rw [he s,he t] - exact hδprop s t hs ht A hA hAK - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.RationalCanonicalMass +import OAI.Probability.SLE.Mass.RationalTraceMassSupport +import OAI.Probability.SLE.Mass.RationalTraceMassLocalVariation +import OAI.Probability.SLE.Mass.RationalTraceMassProgressiveVersion + +namespace OAI + +/-! The actual rational construction extends to an adapted, locally +continuous family, with simultaneous support of every finite increment. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +structure ContinuousMassPath (γ : ℝ≥0 → ℂ) (μ : ℚ → Measure upperHalfPlane) + (ν : Measure upperHalfPlane) (N : ℝ≥0 → Measure upperHalfPlane) : Prop where + zero : N 0 = 0 + dominated : ∀ t, N t ≤ ν + monotone : Monotone N + rational : ∀ q : ℚ, 0 ≤ q → N (q:ℝ).toNNReal = μ q + continuous_tests : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t) + increment_support : ∀ s t, s ≤ t → + (N t-N s) {z : upperHalfPlane | (z:ℂ) ∉ + (fun r : ℝ => γ r.toNNReal) '' Icc (s:ℝ) t} = 0 + local_variation : ∀ K : Set upperHalfPlane, IsCompact K → + ∀ t₀ : ℝ≥0, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧ + ∀ s t : ℝ≥0, dist (s:ℝ) t₀ < δ → dist (t:ℝ) t₀ < δ → + ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K → + |(N s A).toReal-(N t A).toReal| < ε + +theorem RationalCanonicalMass.exists_continuous_progressive_extension + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : RationalCanonicalMass P κ B hBm γ G) (hγ : ∀ ω, Continuous (γ ω)) : + ∃ N : ℝ≥0 → Ω → Measure upperHalfPlane, + IsProgressive (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) N ∧ + (∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) ∧ + ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) (C.terminal ω) (fun t => N t ω) := by + let Good (ω : Ω) : Prop := + HalfPlaneTraceIncrements (fun q => C.mass q ω) (C.terminal ω) + (fun r : ℝ => γ ω r.toNNReal) ∧ NullSingletonClass (C.terminal ω) + have hex (ω : Ω) : ∃ M : ℝ → Measure upperHalfPlane, + (∀ t, (M t).Regular) ∧ + (∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂M t)) ∧ + (Good ω → (∀ t, M t ≤ C.terminal ω) ∧ (∀ q : ℚ, M q = C.mass q ω) ∧ + Monotone M ∧ + (∀ K : Set upperHalfPlane, IsCompact K → + ∀ t₀ : ℝ, ∀ ε : ℝ, 0 < ε → ∃ δ : ℝ, 0 < δ ∧ + ∀ s t : ℝ, dist s t₀ < δ → dist t t₀ < δ → + ∀ A : Set upperHalfPlane, MeasurableSet A → A ⊆ K → + |(M s A).toReal-(M t A).toReal| < ε)) := by + by_cases h : Good ω + · let : (C.terminal ω).Regular := (C.terminal_radon ω).1 + let : NullSingletonClass (C.terminal ω) := h.2 + obtain ⟨M,hr,hd,hq,hm,hc,hv⟩ := h.1.exists_locally_tv_continuous_extension + ((hγ ω).comp continuous_real_toNNReal) + exact ⟨M,hr,hc,fun _ => ⟨hd,hq,hm,hv⟩⟩ + · refine ⟨fun _ => 0,fun _ => inferInstance,?_,fun hh => (h hh).elim⟩ + intro f + simpa only [integral_zero_measure] using (continuous_const : Continuous (fun _ : ℝ => (0:ℝ))) + choose M hMr hMc hM using hex + have hrat : ∀ᵐ ω ∂P, ∀ q : ℚ, M ω q = C.mass q ω := by + filter_upwards [C.increments,C.nonatomic] with ω hi hn + exact (hM ω ⟨hi,hn⟩).2.1 + let N := rationalRadonExtension C.mass + have hNp : IsProgressive (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) N := + rationalRadonExtension_progressive _ C.mass (fun q ω => (C.mass_radon q ω).1) + (fun q _ => C.mass_measurable q) + have hNr := rationalRadonExtension_regular C.mass + have hNe := rationalRadonExtension_common_version P C.mass (fun t ω => M ω t) + (fun t ω => hMr ω t) (Eventually.of_forall hMc) hrat + refine ⟨N,hNp,hNr,?_⟩ + filter_upwards [C.increments,C.nonatomic,C.zero,hNe] with ω hi hn hz he + change ∀ t : ℝ≥0, N t ω = M ω t at he + obtain ⟨hd,hq,hm,hv⟩ := hM ω ⟨hi,hn⟩ + refine ⟨?_,?_,?_,?_,?_,?_,?_⟩ + · rw [he 0] + simpa only [NNReal.coe_zero,Rat.cast_zero] using (hq 0).trans hz + · intro t + rw [he t] + exact hd t + · intro s t hst + change N s ω ≤ N t ω + rw [he s,he t] + exact hm (NNReal.coe_le_coe.mpr hst) + · intro q hq0 + rw [he _ ,Real.coe_toNNReal _ (Rat.cast_nonneg.mpr hq0)] + exact hq q + · intro f + have hc := (hMc ω f).comp NNReal.continuous_coe + simpa only [Function.comp_def,← he] using hc + · intro s t hst + rw [he s,he t] + let : (C.terminal ω).Regular := (C.terminal_radon ω).1 + exact hi.extension_increment_carried ((hγ ω).comp continuous_real_toNNReal) + (M ω) (hMr ω) hq hm (NNReal.coe_le_coe.mpr hst) + · intro K hK t₀ ε hε + obtain ⟨δ,hδ,hδprop⟩ := hv K hK t₀ ε hε + refine ⟨δ,hδ,?_⟩ + intro s t hs ht A hA hAK + rw [he s,he t] + exact hδprop s t hs ht A hA hAK + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassExhaustion.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassExhaustion.lean index 624b0b1bec0516150b0702d9929b4ba8f5cef4a0..e342ad5bba244738197ce96f2a969ad44110633c 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassExhaustion.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassExhaustion.lean @@ -1,104 +1,104 @@ -import OAI.Probability.SLE.Mass.CanonicalMassContinuity -import OAI.Probability.SLE.Geometry.GreenCompactDecay -import OAI.Probability.SLE.Mass.MonotoneMassExhaustion -import Mathlib.MeasureTheory.Integral.Lebesgue.Sub - -namespace OAI - -/-! The actual canonical terminal measure is exhausted by finite times, -locally in total variation. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma RationalCanonicalMass.mass_ambient_measurable - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : RationalCanonicalMass P κ B hBm γ G) (q : ℚ) : Measurable (C.mass q) := - (C.mass_measurable q).mono (Filtration.le _ _) le_rfl - -lemma RationalCanonicalMass.integer_remaining_mean - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : RationalCanonicalMass P κ B hBm γ G) (K : Set upperHalfPlane) (hK : IsCompact K) - (n : ℕ) : - (∫⁻ ω, C.terminal ω K-C.mass n ω K ∂P) = - P.bind (halfPlaneGreenMeasure κ B γ G n) K := by - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hmm : Measurable (fun ω => C.mass n ω K) := - (Measure.measurable_coe hK.measurableSet).comp (C.mass_ambient_measurable n) - have hk := congrArg (fun ρ : Measure upperHalfPlane => ρ K) - (C.conditional n univ MeasurableSet.univ) - simp only [Measure.restrict_univ,Measure.add_apply,Rat.cast_natCast, - Real.toNNReal_natCast] at hk - rw [Measure.bind_apply hK.measurableSet C.terminal_measurable.aemeasurable, - Measure.bind_apply hK.measurableSet (C.mass_ambient_measurable n).aemeasurable] at hk - have htf : (∫⁻ ω, C.terminal ω K ∂P) < ∞ := by - rw [← Measure.bind_apply hK.measurableSet C.terminal_measurable.aemeasurable,C.mean] - exact hK.measure_lt_top - have hmf : (∫⁻ ω, C.mass n ω K ∂P) ≠ ∞ := - (lt_of_le_of_lt (by rw [hk]; exact le_add_right le_rfl) htf).ne - have hle : (fun ω => C.mass n ω K) ≤ᵐ[P] (fun ω => C.terminal ω K) := - C.increments.mono (fun _ hω => hω.dominated n K) - rw [lintegral_sub hmm hmf hle,hk,ENNReal.add_sub_cancel_left hmf] - -theorem RationalCanonicalMass.finite_times_exhaust_terminal - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun s => B s ω)) - (hB0 : ∀ ω, B 0 ω = 0) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (C : RationalCanonicalMass P κ B hBm γ G) - (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) - (C.terminal ω) (fun t => N t ω)) : - ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (C.terminal ω-N t ω) K) atTop (𝓝 0) := by - let D := localTestFamily upperHalfPlane - have hm (n j : ℕ) : Measurable (fun ω => C.terminal ω (D.compactSet j)-C.mass n ω (D.compactSet j)) := - ((Measure.measurable_coe (D.isCompact j).measurableSet).comp C.terminal_measurable).sub - ((Measure.measurable_coe (D.isCompact j).measurableSet).comp (C.mass_ambient_measurable n)) - have ha (j : ℕ) : ∀ᵐ ω ∂P, Antitone (fun n : ℕ => C.terminal ω (D.compactSet j)-C.mass n ω (D.compactSet j)) := by - filter_upwards [C.increments] with ω hω - intro n m hnm - exact tsub_le_tsub_left (hω.monotone (Nat.cast_le.mpr hnm) (D.compactSet j)) _ - have hmean (j : ℕ) : Tendsto (fun n : ℕ => ∫⁻ ω, - C.terminal ω (D.compactSet j)-C.mass n ω (D.compactSet j) ∂P) atTop (𝓝 0) := by - simp_rw [C.integer_remaining_mean _ (D.isCompact j)] - exact halfPlaneGreenMeasure_compact_mean_tendsto_zero hB hBm hBc hB0 hκ hκ8 γ G F hw _ (D.isCompact j) - have hzero (j : ℕ) := ae_tendsto_zero_of_antitone_lintegral P _ (fun n => hm n j) (ha j) (hmean j) - filter_upwards [ae_all_iff.mpr hzero,hpath] with ω hz hp - let : (C.terminal ω).Regular := (C.terminal_radon ω).1 - have hD (j : ℕ) : Tendsto (fun t : ℝ≥0 => (C.terminal ω-N t ω) (D.compactSet j)) - atTop (𝓝 0) := by - apply ENNReal.tendsto_atTop_zero.mpr - intro ε hε - obtain ⟨n,hn⟩ := ENNReal.tendsto_atTop_zero.mp (hz j) ε hε - refine ⟨n,fun t ht => ?_⟩ - have hnat : N n ω = C.mass n ω := by - simpa only [Rat.cast_natCast,Real.toNNReal_natCast] using hp.rational n (Nat.cast_nonneg n) - have horder : C.mass n ω ≤ N t ω := by rw [← hnat]; exact hp.monotone ht - have hdom := hp.dominated t - let : IsFiniteMeasureOnCompacts (N t ω) := - ⟨fun K hK => lt_of_le_of_lt (hdom K) hK.measure_lt_top⟩ - rw [sub_apply_of_finite_restriction hdom (D.isCompact j).measurableSet - (D.isCompact j).measure_lt_top.ne] - exact (tsub_le_tsub_left (horder _) _).trans (hn n le_rfl) - intro K hK - obtain ⟨j,hj⟩ := D.exhaustive K hK - exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (hD j) - (fun _ => zero_le) (fun t => measure_mono hj) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassContinuity +import OAI.Probability.SLE.Geometry.GreenCompactDecay +import OAI.Probability.SLE.Mass.MonotoneMassExhaustion +import Mathlib.MeasureTheory.Integral.Lebesgue.Sub + +namespace OAI + +/-! The actual canonical terminal measure is exhausted by finite times, +locally in total variation. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma RationalCanonicalMass.mass_ambient_measurable + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : RationalCanonicalMass P κ B hBm γ G) (q : ℚ) : Measurable (C.mass q) := + (C.mass_measurable q).mono (Filtration.le _ _) le_rfl + +lemma RationalCanonicalMass.integer_remaining_mean + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : RationalCanonicalMass P κ B hBm γ G) (K : Set upperHalfPlane) (hK : IsCompact K) + (n : ℕ) : + (∫⁻ ω, C.terminal ω K-C.mass n ω K ∂P) = + P.bind (halfPlaneGreenMeasure κ B γ G n) K := by + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hmm : Measurable (fun ω => C.mass n ω K) := + (Measure.measurable_coe hK.measurableSet).comp (C.mass_ambient_measurable n) + have hk := congrArg (fun ρ : Measure upperHalfPlane => ρ K) + (C.conditional n univ MeasurableSet.univ) + simp only [Measure.restrict_univ,Measure.add_apply,Rat.cast_natCast, + Real.toNNReal_natCast] at hk + rw [Measure.bind_apply hK.measurableSet C.terminal_measurable.aemeasurable, + Measure.bind_apply hK.measurableSet (C.mass_ambient_measurable n).aemeasurable] at hk + have htf : (∫⁻ ω, C.terminal ω K ∂P) < ∞ := by + rw [← Measure.bind_apply hK.measurableSet C.terminal_measurable.aemeasurable,C.mean] + exact hK.measure_lt_top + have hmf : (∫⁻ ω, C.mass n ω K ∂P) ≠ ∞ := + (lt_of_le_of_lt (by rw [hk]; exact le_add_right le_rfl) htf).ne + have hle : (fun ω => C.mass n ω K) ≤ᵐ[P] (fun ω => C.terminal ω K) := + C.increments.mono (fun _ hω => hω.dominated n K) + rw [lintegral_sub hmm hmf hle,hk,ENNReal.add_sub_cancel_left hmf] + +theorem RationalCanonicalMass.finite_times_exhaust_terminal + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun s => B s ω)) + (hB0 : ∀ ω, B 0 ω = 0) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (C : RationalCanonicalMass P κ B hBm γ G) + (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) + (C.terminal ω) (fun t => N t ω)) : + ∀ᵐ ω ∂P, ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (C.terminal ω-N t ω) K) atTop (𝓝 0) := by + let D := localTestFamily upperHalfPlane + have hm (n j : ℕ) : Measurable (fun ω => C.terminal ω (D.compactSet j)-C.mass n ω (D.compactSet j)) := + ((Measure.measurable_coe (D.isCompact j).measurableSet).comp C.terminal_measurable).sub + ((Measure.measurable_coe (D.isCompact j).measurableSet).comp (C.mass_ambient_measurable n)) + have ha (j : ℕ) : ∀ᵐ ω ∂P, Antitone (fun n : ℕ => C.terminal ω (D.compactSet j)-C.mass n ω (D.compactSet j)) := by + filter_upwards [C.increments] with ω hω + intro n m hnm + exact tsub_le_tsub_left (hω.monotone (Nat.cast_le.mpr hnm) (D.compactSet j)) _ + have hmean (j : ℕ) : Tendsto (fun n : ℕ => ∫⁻ ω, + C.terminal ω (D.compactSet j)-C.mass n ω (D.compactSet j) ∂P) atTop (𝓝 0) := by + simp_rw [C.integer_remaining_mean _ (D.isCompact j)] + exact halfPlaneGreenMeasure_compact_mean_tendsto_zero hB hBm hBc hB0 hκ hκ8 γ G F hw _ (D.isCompact j) + have hzero (j : ℕ) := ae_tendsto_zero_of_antitone_lintegral P _ (fun n => hm n j) (ha j) (hmean j) + filter_upwards [ae_all_iff.mpr hzero,hpath] with ω hz hp + let : (C.terminal ω).Regular := (C.terminal_radon ω).1 + have hD (j : ℕ) : Tendsto (fun t : ℝ≥0 => (C.terminal ω-N t ω) (D.compactSet j)) + atTop (𝓝 0) := by + apply ENNReal.tendsto_atTop_zero.mpr + intro ε hε + obtain ⟨n,hn⟩ := ENNReal.tendsto_atTop_zero.mp (hz j) ε hε + refine ⟨n,fun t ht => ?_⟩ + have hnat : N n ω = C.mass n ω := by + simpa only [Rat.cast_natCast,Real.toNNReal_natCast] using hp.rational n (Nat.cast_nonneg n) + have horder : C.mass n ω ≤ N t ω := by rw [← hnat]; exact hp.monotone ht + have hdom := hp.dominated t + let : IsFiniteMeasureOnCompacts (N t ω) := + ⟨fun K hK => lt_of_le_of_lt (hdom K) hK.measure_lt_top⟩ + rw [sub_apply_of_finite_restriction hdom (D.isCompact j).measurableSet + (D.isCompact j).measure_lt_top.ne] + exact (tsub_le_tsub_left (horder _) _).trans (hn n le_rfl) + intro K hK + obtain ⟨j,hj⟩ := D.exhaustive K hK + exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (hD j) + (fun _ => zero_le) (fun t => measure_mono hj) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassExtendedRestart.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassExtendedRestart.lean index de5f178005a37c33942df5b2c2e5a94178c05bea..09d18d0ad7dcf66c1baf51b27d831971e341ea87 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassExtendedRestart.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassExtendedRestart.lean @@ -1,102 +1,102 @@ -import OAI.Probability.SLE.Mass.CanonicalMassRestart -import OAI.Probability.SLE.Geometry.ExtendedStopLocalization - -namespace OAI - -/-! Pathwise canonical restart at every possibly infinite stopping time, -on its finite event, simultaneously for all finite and terminal later times. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance extendedRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance extendedRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -/-- One raw functional works at every stopping time. The equality is of -whole measures, hence permits random later times and nonnegative Borel tests. -/ -theorem CanonicalMass.exists_raw_all_stopping_restart - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : - ∃ R : RawCanonicalMass, - (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ - ∀ (S : Ω → WithTop ℝ≥0) - (_hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), - ∀ᵐ ω ∂P, S ω < ⊤ → ∀ u : ℝ≥0∞, - extendedCanonicalMass C.process C.rational.terminal - ((((S ω).untopD (0 : ℝ≥0) : ℝ≥0) : ℝ≥0∞)+u) ω-C.process ((S ω).untopD (0 : ℝ≥0)) ω = - weightedImageMass - (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) - (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) - (extendedCanonicalMass R.process R.terminal u - (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0 : ℝ≥0)) ω)) := by - obtain ⟨R,he,hr⟩ := C.exists_raw_pathwise_restart hB hc hκ F hw hpublished - refine ⟨R,he,?_⟩ - intro S hS - let ℱ := Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) - let Q : Ω → ℝ≥0 → Prop := fun ω T => ∀ u : ℝ≥0∞, - extendedCanonicalMass C.process C.rational.terminal ((T:ℝ≥0∞)+u) ω-C.process T ω = - weightedImageMass (stoppedMassPoint B κ γ G F hw (fun _ => T) ω) - (stoppedMassWeight B κ F (fun _ => T) ω) - (extendedCanonicalMass R.process R.terminal u - (sourceRestartDriverPath B hc (fun _ => T) ω)) - have hQ (n : ℕ) : ∀ᵐ ω ∂P, Q ω (cutoffStop S n ω) := by - have hτ := cutoffStop_stopping ℱ S hS (n:ℝ≥0) - filter_upwards [hr (cutoffStop S n) hτ] with ω hω - have hp : stoppedMassPoint B κ γ G F hw (cutoffStop S n) ω = - stoppedMassPoint B κ γ G F hw (fun _ => cutoffStop S n ω) ω := by - funext z - apply Subtype.ext - rfl - have hv : stoppedMassWeight B κ F (cutoffStop S n) ω = - stoppedMassWeight B κ F (fun _ => cutoffStop S n ω) ω := rfl - have hy : sourceRestartDriverPath B hc (cutoffStop S n) ω = - sourceRestartDriverPath B hc (fun _ => cutoffStop S n ω) ω := by - ext t - rfl - intro u - by_cases hu : u=∞ - · subst u - have ht := hω.2 - unfold stoppedFreshTerminalMass at ht - rw [hp,hv,hy] at ht - simpa only [Q,add_top,extendedCanonicalMass,lt_self_iff_false,ite_false] using ht - · lift u to ℝ≥0 using hu with t - have ht := hω.1 t - unfold stoppedFreshMass at ht - rw [hp,hv,hy] at ht - simpa only [Q,← ENNReal.coe_add,extendedCanonicalMass,ENNReal.coe_lt_top, - ite_true,ENNReal.toNNReal_coe] using ht - filter_upwards [ae_of_cutoffStop S P Q hQ] with ω hω - intro hf u - have hp : stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0:ℝ≥0)) ω = - stoppedMassPoint B κ γ G F hw (fun _ => (S ω).untopD (0:ℝ≥0)) ω := by - funext z - apply Subtype.ext - rfl - have hv : stoppedMassWeight B κ F (fun ω => (S ω).untopD (0:ℝ≥0)) ω = - stoppedMassWeight B κ F (fun _ => (S ω).untopD (0:ℝ≥0)) ω := rfl - have hy : sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0:ℝ≥0)) ω = - sourceRestartDriverPath B hc (fun _ => (S ω).untopD (0:ℝ≥0)) ω := by - ext t - rfl - rw [hp,hv,hy] - exact hω hf u - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassRestart +import OAI.Probability.SLE.Geometry.ExtendedStopLocalization + +namespace OAI + +/-! Pathwise canonical restart at every possibly infinite stopping time, +on its finite event, simultaneously for all finite and terminal later times. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance extendedRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance extendedRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +/-- One raw functional works at every stopping time. The equality is of +whole measures, hence permits random later times and nonnegative Borel tests. -/ +theorem CanonicalMass.exists_raw_all_stopping_restart + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : + ∃ R : RawCanonicalMass, + (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ + ∀ (S : Ω → WithTop ℝ≥0) + (_hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), + ∀ᵐ ω ∂P, S ω < ⊤ → ∀ u : ℝ≥0∞, + extendedCanonicalMass C.process C.rational.terminal + ((((S ω).untopD (0 : ℝ≥0) : ℝ≥0) : ℝ≥0∞)+u) ω-C.process ((S ω).untopD (0 : ℝ≥0)) ω = + weightedImageMass + (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) + (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0 : ℝ≥0)) ω) + (extendedCanonicalMass R.process R.terminal u + (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0 : ℝ≥0)) ω)) := by + obtain ⟨R,he,hr⟩ := C.exists_raw_pathwise_restart hB hc hκ F hw hpublished + refine ⟨R,he,?_⟩ + intro S hS + let ℱ := Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) + let Q : Ω → ℝ≥0 → Prop := fun ω T => ∀ u : ℝ≥0∞, + extendedCanonicalMass C.process C.rational.terminal ((T:ℝ≥0∞)+u) ω-C.process T ω = + weightedImageMass (stoppedMassPoint B κ γ G F hw (fun _ => T) ω) + (stoppedMassWeight B κ F (fun _ => T) ω) + (extendedCanonicalMass R.process R.terminal u + (sourceRestartDriverPath B hc (fun _ => T) ω)) + have hQ (n : ℕ) : ∀ᵐ ω ∂P, Q ω (cutoffStop S n ω) := by + have hτ := cutoffStop_stopping ℱ S hS (n:ℝ≥0) + filter_upwards [hr (cutoffStop S n) hτ] with ω hω + have hp : stoppedMassPoint B κ γ G F hw (cutoffStop S n) ω = + stoppedMassPoint B κ γ G F hw (fun _ => cutoffStop S n ω) ω := by + funext z + apply Subtype.ext + rfl + have hv : stoppedMassWeight B κ F (cutoffStop S n) ω = + stoppedMassWeight B κ F (fun _ => cutoffStop S n ω) ω := rfl + have hy : sourceRestartDriverPath B hc (cutoffStop S n) ω = + sourceRestartDriverPath B hc (fun _ => cutoffStop S n ω) ω := by + ext t + rfl + intro u + by_cases hu : u=∞ + · subst u + have ht := hω.2 + unfold stoppedFreshTerminalMass at ht + rw [hp,hv,hy] at ht + simpa only [Q,add_top,extendedCanonicalMass,lt_self_iff_false,ite_false] using ht + · lift u to ℝ≥0 using hu with t + have ht := hω.1 t + unfold stoppedFreshMass at ht + rw [hp,hv,hy] at ht + simpa only [Q,← ENNReal.coe_add,extendedCanonicalMass,ENNReal.coe_lt_top, + ite_true,ENNReal.toNNReal_coe] using ht + filter_upwards [ae_of_cutoffStop S P Q hQ] with ω hω + intro hf u + have hp : stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0:ℝ≥0)) ω = + stoppedMassPoint B κ γ G F hw (fun _ => (S ω).untopD (0:ℝ≥0)) ω := by + funext z + apply Subtype.ext + rfl + have hv : stoppedMassWeight B κ F (fun ω => (S ω).untopD (0:ℝ≥0)) ω = + stoppedMassWeight B κ F (fun _ => (S ω).untopD (0:ℝ≥0)) ω := rfl + have hy : sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0:ℝ≥0)) ω = + sourceRestartDriverPath B hc (fun _ => (S ω).untopD (0:ℝ≥0)) ω := by + ext t + rfl + rw [hp,hv,hy] + exact hω hf u + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassFirstVisit.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassFirstVisit.lean index e6ce47fc61526fdc8f573b80bbd21f1a3afc7ae6..7ed1569ffc68fe02b3089e9ad4ddf788aff3294a 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassFirstVisit.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassFirstVisit.lean @@ -1,80 +1,80 @@ -import OAI.Probability.SLE.Mass.CanonicalMassRemaining - -namespace OAI - -/-! At each fixed time the canonical mass is the terminal measure -restricted to the already visited trace. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma halfPlaneStoppingGreenMeasure_const - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) : - halfPlaneStoppingGreenMeasure κ B γ G (fun _ => (t:ℝ≥0∞)) = - halfPlaneGreenMeasure κ B γ G t := by - funext ω - simp only [halfPlaneStoppingGreenMeasure,stoppingGreen,ENNReal.coe_lt_top, - ite_true,ENNReal.toNNReal_coe,halfPlaneGreenMeasure] - -theorem CanonicalMass.atTime_eq_restrict_past - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (C : CanonicalMass P κ B hBm γ G) (t : ℝ≥0) : - ∀ᵐ ω ∂P, C.atTime t ω = (C.rational.terminal ω).restrict - {z : upperHalfPlane | (z:ℂ) ∈ γ ω '' Icc 0 t} := by - let ℱ := Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) - have hstop : IsStoppingTime ℱ (fun _ => (t:WithTop ℝ≥0)) := isStoppingTime_const ℱ t - obtain ⟨η,hηm,hηr,hηsum,hηkernel⟩ := C.exists_remaining_kernel _ hstop - have hevent (E : Set Ω) (hE : MeasurableSet[ℱ t] E) : - (P.restrict E).bind η = (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) := by - rw [← halfPlaneStoppingGreenMeasure_const κ B γ G t] - apply hηkernel E - change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E - rw [IsStoppingTime.measurableSpace_const] - exact hE - have hfuture := ae_remainder_carried_by_surviving hB hBm hBc hB0 hκ hκ8 γ G F hw - t η hηm (fun ω => (hηr ω).2) hevent - have hlim := C.rational.all_finite_times_limit hBc hκ hκ8 F hw C.process C.radon C.paths - filter_upwards [hηsum,hfuture,hlim] with ω hsum hfuture hlim - let A : Set upperHalfPlane := {z | (z:ℂ) ∈ γ ω '' Icc 0 t} - have hA : MeasurableSet A := - ((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet - let : (C.atTime t ω).Regular := (C.atTime_radon t ω).1 - have hpast : C.atTime t ω Aᶜ = 0 := by - have hh := convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) t (fun n => (n:ℝ)) - tendsto_natCast_atTop_atTop C.rational.supports C.rational.weights - (C.atTime t ω) (by - simpa only [CanonicalMass.atTime,extendedCanonicalMass,ENNReal.coe_lt_top, - ite_true,ENNReal.toNNReal_coe] using hlim t) - exact hh - have hηA : η ω A = 0 := by - apply le_zero_iff.mp - apply (measure_mono (show A ⊆ {z : upperHalfPlane | (z:ℂ) ∉ survivingDomain (γ ω) t} from ?_)).trans_eq hfuture - intro z hz hs - exact hs.1.2 hz - have hrestr : (C.atTime t ω).restrict A = C.atTime t ω := by - apply Measure.restrict_eq_self_of_ae_mem - filter_upwards [measure_eq_zero_iff_ae_notMem.mp hpast] with z hz - exact not_not.mp hz - change C.atTime t ω = (C.rational.terminal ω).restrict A - rw [← hsum,Measure.restrict_add,Measure.restrict_eq_zero.mpr hηA,zero_add] - exact hrestr.symm - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassRemaining + +namespace OAI + +/-! At each fixed time the canonical mass is the terminal measure +restricted to the already visited trace. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma halfPlaneStoppingGreenMeasure_const + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) : + halfPlaneStoppingGreenMeasure κ B γ G (fun _ => (t:ℝ≥0∞)) = + halfPlaneGreenMeasure κ B γ G t := by + funext ω + simp only [halfPlaneStoppingGreenMeasure,stoppingGreen,ENNReal.coe_lt_top, + ite_true,ENNReal.toNNReal_coe,halfPlaneGreenMeasure] + +theorem CanonicalMass.atTime_eq_restrict_past + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (C : CanonicalMass P κ B hBm γ G) (t : ℝ≥0) : + ∀ᵐ ω ∂P, C.atTime t ω = (C.rational.terminal ω).restrict + {z : upperHalfPlane | (z:ℂ) ∈ γ ω '' Icc 0 t} := by + let ℱ := Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) + have hstop : IsStoppingTime ℱ (fun _ => (t:WithTop ℝ≥0)) := isStoppingTime_const ℱ t + obtain ⟨η,hηm,hηr,hηsum,hηkernel⟩ := C.exists_remaining_kernel _ hstop + have hevent (E : Set Ω) (hE : MeasurableSet[ℱ t] E) : + (P.restrict E).bind η = (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) := by + rw [← halfPlaneStoppingGreenMeasure_const κ B γ G t] + apply hηkernel E + change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E + rw [IsStoppingTime.measurableSpace_const] + exact hE + have hfuture := ae_remainder_carried_by_surviving hB hBm hBc hB0 hκ hκ8 γ G F hw + t η hηm (fun ω => (hηr ω).2) hevent + have hlim := C.rational.all_finite_times_limit hBc hκ hκ8 F hw C.process C.radon C.paths + filter_upwards [hηsum,hfuture,hlim] with ω hsum hfuture hlim + let A : Set upperHalfPlane := {z | (z:ℂ) ∈ γ ω '' Icc 0 t} + have hA : MeasurableSet A := + ((isCompact_Icc.image (hw ω).1).isClosed.preimage continuous_subtype_val).measurableSet + let : (C.atTime t ω).Regular := (C.atTime_radon t ω).1 + have hpast : C.atTime t ω Aᶜ = 0 := by + have hh := convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) t (fun n => (n:ℝ)) + tendsto_natCast_atTop_atTop C.rational.supports C.rational.weights + (C.atTime t ω) (by + simpa only [CanonicalMass.atTime,extendedCanonicalMass,ENNReal.coe_lt_top, + ite_true,ENNReal.toNNReal_coe] using hlim t) + exact hh + have hηA : η ω A = 0 := by + apply le_zero_iff.mp + apply (measure_mono (show A ⊆ {z : upperHalfPlane | (z:ℂ) ∉ survivingDomain (γ ω) t} from ?_)).trans_eq hfuture + intro z hz hs + exact hs.1.2 hz + have hrestr : (C.atTime t ω).restrict A = C.atTime t ω := by + apply Measure.restrict_eq_self_of_ae_mem + filter_upwards [measure_eq_zero_iff_ae_notMem.mp hpast] with z hz + exact not_not.mp hz + change C.atTime t ω = (C.rational.terminal ω).restrict A + rw [← hsum,Measure.restrict_add,Measure.restrict_eq_zero.mpr hηA,zero_add] + exact hrestr.symm + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassImageRestart.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassImageRestart.lean index 2e51f22aeddf342667f20ce807877f1b58b6c9c4..685da0a4cdab03985b02191e535b4cc099fb417a 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassImageRestart.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassImageRestart.lean @@ -1,68 +1,68 @@ -import OAI.Probability.SLE.Mass.CanonicalMassExtendedRestart -import OAI.Probability.SLE.Geometry.ConformalRestartTransport - -namespace OAI - -/-! The same raw functional and the same restart event work in every fixed -conformal image. No image-dependent extraction or exceptional set is needed. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance imageRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance imageRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -/-- Every fixed conformal image inherits the pathwise restart identity, -with the same raw functional and one event for all image maps and later times. -/ -theorem CanonicalMass.exists_raw_all_image_restart - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : - ∃ R : RawCanonicalMass, - (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ - ∀ (S : Ω → WithTop ℝ≥0) - (_hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), - ∀ᵐ ω ∂P, S ω < ⊤ → ∀ (D : Set ℂ) (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane,ℝ≥0)) (u : ℝ≥0∞), - conformalMass J w (C.atTime ((((S ω).untopD (0:ℝ≥0):ℝ≥0):ℝ≥0∞)+u) ω)- - conformalMass J w (C.process ((S ω).untopD (0:ℝ≥0)) ω) = - conformalMass J w (weightedImageMass - (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0:ℝ≥0)) ω) - (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0:ℝ≥0)) ω) - (extendedCanonicalMass R.process R.terminal u - (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0:ℝ≥0)) ω))) := by - obtain ⟨R,he,hr⟩ := C.exists_raw_all_stopping_restart hB hc hκ F hw hpublished - refine ⟨R,he,?_⟩ - intro S hS - filter_upwards [hr S hS,C.paths] with ω hω hp - intro hf D J w u - let T := (S ω).untopD (0:ℝ≥0) - have hle : C.process T ω ≤ C.atTime ((T:ℝ≥0∞)+u) ω := by - by_cases hu : u=∞ - · subst u - simpa only [CanonicalMass.atTime,add_top,extendedCanonicalMass,lt_self_iff_false,ite_false] - using hp.dominated T - · lift u to ℝ≥0 using hu with t - simpa only [CanonicalMass.atTime,← ENNReal.coe_add,extendedCanonicalMass, - ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] using - hp.monotone (le_add_of_nonneg_right (show 0 ≤ t from zero_le)) - let := (C.atTime_radon ((T:ℝ≥0∞)+u) ω).2 - rw [← conformalMass_sub J w hle] - exact congrArg (conformalMass J w) (hω hf u) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassExtendedRestart +import OAI.Probability.SLE.Geometry.ConformalRestartTransport + +namespace OAI + +/-! The same raw functional and the same restart event work in every fixed +conformal image. No image-dependent extraction or exceptional set is needed. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance imageRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance imageRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +/-- Every fixed conformal image inherits the pathwise restart identity, +with the same raw functional and one event for all image maps and later times. -/ +theorem CanonicalMass.exists_raw_all_image_restart + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : + ∃ R : RawCanonicalMass, + (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ + ∀ (S : Ω → WithTop ℝ≥0) + (_hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S), + ∀ᵐ ω ∂P, S ω < ⊤ → ∀ (D : Set ℂ) (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane,ℝ≥0)) (u : ℝ≥0∞), + conformalMass J w (C.atTime ((((S ω).untopD (0:ℝ≥0):ℝ≥0):ℝ≥0∞)+u) ω)- + conformalMass J w (C.process ((S ω).untopD (0:ℝ≥0)) ω) = + conformalMass J w (weightedImageMass + (stoppedMassPoint B κ γ G F hw (fun ω => (S ω).untopD (0:ℝ≥0)) ω) + (stoppedMassWeight B κ F (fun ω => (S ω).untopD (0:ℝ≥0)) ω) + (extendedCanonicalMass R.process R.terminal u + (sourceRestartDriverPath B hc (fun ω => (S ω).untopD (0:ℝ≥0)) ω))) := by + obtain ⟨R,he,hr⟩ := C.exists_raw_all_stopping_restart hB hc hκ F hw hpublished + refine ⟨R,he,?_⟩ + intro S hS + filter_upwards [hr S hS,C.paths] with ω hω hp + intro hf D J w u + let T := (S ω).untopD (0:ℝ≥0) + have hle : C.process T ω ≤ C.atTime ((T:ℝ≥0∞)+u) ω := by + by_cases hu : u=∞ + · subst u + simpa only [CanonicalMass.atTime,add_top,extendedCanonicalMass,lt_self_iff_false,ite_false] + using hp.dominated T + · lift u to ℝ≥0 using hu with t + simpa only [CanonicalMass.atTime,← ENNReal.coe_add,extendedCanonicalMass, + ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] using + hp.monotone (le_add_of_nonneg_right (show 0 ≤ t from zero_le)) + let := (C.atTime_radon ((T:ℝ≥0∞)+u) ω).2 + rw [← conformalMass_sub J w hle] + exact congrArg (conformalMass J w) (hω hf u) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassNontrivial.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassNontrivial.lean index 33f2fd9feeda12dac77accfb464ae7153af8d8a4..3de3890626d067159b03f9cb58972745d4b8e6ad 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassNontrivial.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassNontrivial.lean @@ -1,74 +1,74 @@ -import OAI.Probability.SLE.Mass.CanonicalMassConstruction -import OAI.Probability.SLE.Mass.RadonTerminalUniqueness -import OAI.Probability.SLE.Geometry.RestartInitialGreenIntegral - -namespace OAI - -/-! Positive initial Green mass and terminal exhaustion give a positive -probability of nonzero canonical mass at a finite deterministic time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -lemma initialGreenMeasure_univ_pos (κ : ℝ) : 0 < initialGreenMeasure κ univ := by - have hm : Measurable (fun z : ℂ => ENNReal.ofReal (initialGreen κ z)) := by - unfold initialGreen - fun_prop - have he : initialGreenMeasure κ univ = - ∫⁻ z in upperHalfPlane, ENNReal.ofReal (initialGreen κ z) ∂volume := by - simpa only [lintegral_const,Measure.restrict_univ,one_mul] using - initialGreenMeasure_lintegral κ (fun _ => 1) - rw [he,lintegral_pos_iff_support hm,Measure.restrict_apply (measurableSet_support hm)] - have hs : Function.support (fun z : ℂ => ENNReal.ofReal (initialGreen κ z)) ∩ upperHalfPlane = - upperHalfPlane := by - ext z - constructor - · exact fun h => h.2 - · intro hz - exact ⟨(ENNReal.ofReal_pos.mpr (initialGreen_pos κ z hz)).ne',hz⟩ - rw [hs] - exact isOpen_upperHalfPlane.measure_pos (μ:=volume) ⟨Complex.I,by simp [upperHalfPlane]⟩ - -namespace CanonicalMass - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma exists_nonzero_finite_time : ∃ n : ℕ, 0

0) := by - filter_upwards [ht,C.paths,C.exhaustion] with ω hω hp he - let := (C.rational.terminal_radon ω).1 - apply radon_terminal_eq_of_common_exhaustion (C.rational.terminal ω) 0 - (fun t => C.process t ω) hp.dominated (fun t => by rw [hω t]) he - intro K hK - simp only [Measure.zero_sub,Measure.coe_zero,Pi.zero_apply] - exact tendsto_const_nhds - have hmean : initialGreenMeasure κ=0 := by - rw [← C.rational.mean,Measure.bind_congr_right hterminal,Measure.bind_zero_right'] - have hp := initialGreenMeasure_univ_pos κ - simp only [hmean,Measure.coe_zero,Pi.zero_apply,lt_self_iff_false] at hp - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassConstruction +import OAI.Probability.SLE.Mass.RadonTerminalUniqueness +import OAI.Probability.SLE.Geometry.RestartInitialGreenIntegral + +namespace OAI + +/-! Positive initial Green mass and terminal exhaustion give a positive +probability of nonzero canonical mass at a finite deterministic time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +lemma initialGreenMeasure_univ_pos (κ : ℝ) : 0 < initialGreenMeasure κ univ := by + have hm : Measurable (fun z : ℂ => ENNReal.ofReal (initialGreen κ z)) := by + unfold initialGreen + fun_prop + have he : initialGreenMeasure κ univ = + ∫⁻ z in upperHalfPlane, ENNReal.ofReal (initialGreen κ z) ∂volume := by + simpa only [lintegral_const,Measure.restrict_univ,one_mul] using + initialGreenMeasure_lintegral κ (fun _ => 1) + rw [he,lintegral_pos_iff_support hm,Measure.restrict_apply (measurableSet_support hm)] + have hs : Function.support (fun z : ℂ => ENNReal.ofReal (initialGreen κ z)) ∩ upperHalfPlane = + upperHalfPlane := by + ext z + constructor + · exact fun h => h.2 + · intro hz + exact ⟨(ENNReal.ofReal_pos.mpr (initialGreen_pos κ z hz)).ne',hz⟩ + rw [hs] + exact isOpen_upperHalfPlane.measure_pos (μ:=volume) ⟨Complex.I,by simp [upperHalfPlane]⟩ + +namespace CanonicalMass + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma exists_nonzero_finite_time : ∃ n : ℕ, 0

0) := by + filter_upwards [ht,C.paths,C.exhaustion] with ω hω hp he + let := (C.rational.terminal_radon ω).1 + apply radon_terminal_eq_of_common_exhaustion (C.rational.terminal ω) 0 + (fun t => C.process t ω) hp.dominated (fun t => by rw [hω t]) he + intro K hK + simp only [Measure.zero_sub,Measure.coe_zero,Pi.zero_apply] + exact tendsto_const_nhds + have hmean : initialGreenMeasure κ=0 := by + rw [← C.rational.mean,Measure.bind_congr_right hterminal,Measure.bind_zero_right'] + have hp := initialGreenMeasure_univ_pos κ + simp only [hmean,Measure.coe_zero,Pi.zero_apply,lt_self_iff_false] at hp + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassPositiveReduction.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassPositiveReduction.lean index eeb3328bb676434e3e8f983269603bf96d816c34..6b7aac96b1884b6283d7b7389945564082d536fe 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassPositiveReduction.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassPositiveReduction.lean @@ -1,65 +1,65 @@ -import OAI.Probability.SLE.Mass.PositiveMassGerm -import OAI.Probability.SLE.Mass.CanonicalMassNontrivial - -namespace OAI - -/-! The remaining scaling link in positive canonical mass: equal -nonzero-mass probabilities at positive times suffice, with the Green -mean and the Brownian germ argument already proved. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -theorem positive_all_times_of_scaling - (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hscale : ∀ s t : ℝ≥0, 0 1/((j:ℝ≥0)+1) - have hεp (j : ℕ) : 0<ε j := by dsimp [ε]; positivity - have hεa : Antitone ε := by - intro i j hij - exact one_div_le_one_div_of_le (by positivity) (by exact_mod_cast Nat.add_le_add_right hij 1) - have hε : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat - have hZa : Adapted (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) - (fun t ω => C.process t ω univ) := by - intro t - exact (Measure.measurable_coe MeasurableSet.univ).comp (C.progressive.adapted t) - have hZm : ∀ᵐ ω ∂P, Monotone (fun t => C.process t ω univ) := - C.paths.mono (fun _ h => fun _ _ hst => h.monotone hst univ) - have hp := positive_mass_all_times_of_uniform_probability hB - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 - (fun t ω => C.process t ω univ) hZa hZm ε hεa hε - (P {ω | C.process (n+1) ω≠0}) hN (fun j => by - simp only [Measure.measure_univ_pos] - exact le_of_eq (hscale (n+1) (ε j) (by positivity) (hεp j))) - filter_upwards [hp] with ω hω - intro t ht - exact Measure.measure_univ_pos.mp (hω t ht) - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Mass.PositiveMassGerm +import OAI.Probability.SLE.Mass.CanonicalMassNontrivial + +namespace OAI + +/-! The remaining scaling link in positive canonical mass: equal +nonzero-mass probabilities at positive times suffice, with the Green +mean and the Brownian germ argument already proved. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +theorem positive_all_times_of_scaling + (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hscale : ∀ s t : ℝ≥0, 0 1/((j:ℝ≥0)+1) + have hεp (j : ℕ) : 0<ε j := by dsimp [ε]; positivity + have hεa : Antitone ε := by + intro i j hij + exact one_div_le_one_div_of_le (by positivity) (by exact_mod_cast Nat.add_le_add_right hij 1) + have hε : Tendsto ε atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat + have hZa : Adapted (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) + (fun t ω => C.process t ω univ) := by + intro t + exact (Measure.measurable_coe MeasurableSet.univ).comp (C.progressive.adapted t) + have hZm : ∀ᵐ ω ∂P, Monotone (fun t => C.process t ω univ) := + C.paths.mono (fun _ h => fun _ _ hst => h.monotone hst univ) + have hp := positive_mass_all_times_of_uniform_probability hB + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0 + (fun t ω => C.process t ω univ) hZa hZm ε hεa hε + (P {ω | C.process (n+1) ω≠0}) hN (fun j => by + simp only [Measure.measure_univ_pos] + exact le_of_eq (hscale (n+1) (ε j) (by positivity) (hεp j))) + filter_upwards [hp] with ω hω + intro t ht + exact Measure.measure_univ_pos.mp (hω t ht) + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassRandomTime.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassRandomTime.lean index d741ea1858fdef24b202f920c60a9e0506bba38c..76ffa13e54a6e3b05e04e5ebf83744f316f7989e 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassRandomTime.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassRandomTime.lean @@ -1,190 +1,190 @@ -import OAI.Probability.SLE.Mass.CanonicalMassAllTimes -import OAI.Probability.SLE.Mass.LevelStoppingConvex -import OAI.Probability.SLE.Mass.NonnegativeL2Passage - -namespace OAI - -/-! Evaluating the common convex sequence at any measurable extended time -preserves convergence in mean on every event. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -def extendedCanonicalMass {Ω : Type*} (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (ν : Ω → Measure upperHalfPlane) (t : ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := - if t < ∞ then N t.toNNReal ω else ν ω - -lemma extendedCanonicalMass_radon - {Ω : Type*} (N : ℝ≥0 → Ω → Measure upperHalfPlane) (ν : Ω → Measure upperHalfPlane) - (hN : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) - (hν : ∀ ω, (ν ω).Regular ∧ IsFiniteMeasureOnCompacts (ν ω)) (t : ℝ≥0∞) (ω : Ω) : - (extendedCanonicalMass N ν t ω).Regular ∧ - IsFiniteMeasureOnCompacts (extendedCanonicalMass N ν t ω) := by - unfold extendedCanonicalMass - split_ifs - · exact hN _ _ - · exact hν _ - -theorem RationalCanonicalMass.all_extended_times_limit - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (C : RationalCanonicalMass P κ B hBm γ G) - (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) - (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) - (C.terminal ω) (fun t => N t ω)) : - ∀ᵐ ω ∂P, ∀ (t : ℝ≥0∞) (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n t ω) atTop - (𝓝 (∫ z, f z ∂extendedCanonicalMass N C.terminal t ω)) := by - filter_upwards [C.all_finite_times_limit hBc hκ hκ8 F hw N hNr hpath,C.terminal_limit] - with ω hf ht - intro t f - by_cases h : t < ∞ - · simpa only [extendedCanonicalMass,ite_eq_left h,ENNReal.coe_toNNReal h.ne] using hf t.toNNReal f - · have he : t = ∞ := top_le_iff.mp (le_of_not_gt h) - simpa only [he,extendedCanonicalMass,lt_self_iff_false,ite_false] using ht f - -theorem RationalCanonicalMass.random_time_nonnegative_mean_tendsto - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (C : RationalCanonicalMass P κ B hBm γ G) - (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) - (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) - (C.terminal ω) (fun t => N t ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : - Integrable (fun ω => ∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) (P.restrict E) ∧ - Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n (S ω) ω) ∂P) atTop - (𝓝 (∫ ω in E, (∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) ∂P)) := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - let M (n : ℕ) (t : ℝ≥0∞) (ω : Ω) := convexLevelMeasure κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n t ω - let (n : ℕ) (t : ℝ≥0∞) (ω : Ω) : IsFiniteMeasureOnCompacts (M n t ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ C.supports C.weights n t - let X (n : ℕ) (ω : Ω) := ∫ z, f z ∂M n (S ω) ω - let Z (n : ℕ) (ω : Ω) := ∫ z, f z ∂M n ∞ ω - have hXm (n : ℕ) : Measurable (X n) := - measurable_halfPlane_compact_test_integral _ - (convexLevelMeasure_random_time_measurable hm hBc κ γ G F hw _ - (fun n => Nat.cast_nonneg n) C.supports C.weights n S hS) - (fun _ => inferInstance) f - have hZm (n : ℕ) : Measurable (Z n) := - convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw _ - (fun n => Nat.cast_nonneg n) C.supports C.weights C.weights_nonneg n ∞ f - obtain ⟨A,hA,hbound⟩ := uniform_convexLevelMeasure_test_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) - C.supports C.weights C.weights_nonneg C.weights_sum f hf 2 - have hZ (n : ℕ) := memLp_two_of_nonnegative_square_bound P (Z n) (hZm n) - (fun _ => integral_nonneg hf) A hA (hbound n ∞) - have hXZ (n : ℕ) (ω : Ω) : X n ω ≤ Z n ω := - integral_mono_measure (convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n ω le_top) (ae_of_all _ hf) f.integrable - apply integral_tendsto_of_nonnegative_L2_domination (P.restrict E) X Z _ hXm - (fun _ _ => integral_nonneg hf) hXZ (fun n => (hZ n).1.restrict E) A.toReal - · intro n - exact (integral_mono_measure Measure.restrict_le_self - (ae_of_all _ (fun ω => sq_nonneg (Z n ω))) (hZ n).1.integrable_sq).trans (hZ n).2 - · filter_upwards [ae_restrict_of_ae (C.all_extended_times_limit hBc hκ hκ8 F hw N hNr hpath)] - with ω hω - exact hω (S ω) f - -lemma convexLevelMeasure_random_time_test_integrable - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (n : ℕ) (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) - (f : C_c(upperHalfPlane,ℝ)) : - Integrable (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (S ω) ω) - (P.restrict E) := by - let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - have hμm := convexLevelMeasure_random_time_measurable hm hBc κ γ G F hw u hu s w n S hS - have hνm := convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞ - have hle := restricted_bind_le_of_ae_le P _ _ hμm hνm - (ae_of_all _ (fun ω => convexLevelMeasure_mono κ B γ G u s w n ω le_top)) E - rw [convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu s w hw₀ hw₁] at hle - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - exact (integrable_integral_bind_of_sections (P.restrict E) _ hμm f f.continuous.measurable - (fun ω => by - let := convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (S ω) - exact f.integrable) (f.integrable.mono_measure hle)).1 - -theorem RationalCanonicalMass.random_time_test_mean_tendsto - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (C : RationalCanonicalMass P κ B hBm γ G) - (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) - (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) - (C.terminal ω) (fun t => N t ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) - (f : C_c(upperHalfPlane,ℝ)) : - Integrable (fun ω => ∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) (P.restrict E) ∧ - Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n (S ω) ω) ∂P) atTop - (𝓝 (∫ ω in E, (∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) ∂P)) := by - let fp := f ⊔ 0 - let fm := (-f) ⊔ 0 - have hp := C.random_time_nonnegative_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth - N hNr hpath S hS E fp (fun _ => le_sup_of_le_right le_rfl) - have hn := C.random_time_nonnegative_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth - N hNr hpath S hS E fm (fun _ => le_sup_of_le_right le_rfl) - have hsplit (ρ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts ρ] : - (∫ z, f z ∂ρ) = (∫ z, fp z ∂ρ)-(∫ z, fm z ∂ρ) := by - rw [← integral_sub fp.integrable fm.integrable] - apply integral_congr_ae - exact ae_of_all _ (fun z => (max_zero_sub_eq_self (f z)).symm) - let (ω : Ω) : IsFiniteMeasureOnCompacts (extendedCanonicalMass N C.terminal (S ω) ω) := - (extendedCanonicalMass_radon N C.terminal hNr C.terminal_radon _ _).2 - let (n : ℕ) (ω : Ω) : IsFiniteMeasureOnCompacts - (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) C.supports C.weights n (S ω) ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ C.supports C.weights n _ - refine ⟨?_,?_⟩ - · apply (hp.1.sub hn.1).congr - filter_upwards [] with ω - exact (hsplit (extendedCanonicalMass N C.terminal (S ω) ω)).symm - · have hi (g : C_c(upperHalfPlane,ℝ)) (n : ℕ) : Integrable (fun ω => ∫ z, g z - ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) C.supports C.weights n (S ω) ω) - (P.restrict E) := - convexLevelMeasure_random_time_test_integrable hB hBm hBc hB0 hκ hκ8 γ G F hw - (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) C.supports C.weights - C.weights_nonneg C.weights_sum n S hS E g - simpa only [hsplit,integral_sub (hi fp _) (hi fm _),integral_sub hp.1 hn.1] using hp.2.sub hn.2 - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassAllTimes +import OAI.Probability.SLE.Mass.LevelStoppingConvex +import OAI.Probability.SLE.Mass.NonnegativeL2Passage + +namespace OAI + +/-! Evaluating the common convex sequence at any measurable extended time +preserves convergence in mean on every event. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +def extendedCanonicalMass {Ω : Type*} (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (ν : Ω → Measure upperHalfPlane) (t : ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := + if t < ∞ then N t.toNNReal ω else ν ω + +lemma extendedCanonicalMass_radon + {Ω : Type*} (N : ℝ≥0 → Ω → Measure upperHalfPlane) (ν : Ω → Measure upperHalfPlane) + (hN : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) + (hν : ∀ ω, (ν ω).Regular ∧ IsFiniteMeasureOnCompacts (ν ω)) (t : ℝ≥0∞) (ω : Ω) : + (extendedCanonicalMass N ν t ω).Regular ∧ + IsFiniteMeasureOnCompacts (extendedCanonicalMass N ν t ω) := by + unfold extendedCanonicalMass + split_ifs + · exact hN _ _ + · exact hν _ + +theorem RationalCanonicalMass.all_extended_times_limit + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (C : RationalCanonicalMass P κ B hBm γ G) + (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) + (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) + (C.terminal ω) (fun t => N t ω)) : + ∀ᵐ ω ∂P, ∀ (t : ℝ≥0∞) (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n t ω) atTop + (𝓝 (∫ z, f z ∂extendedCanonicalMass N C.terminal t ω)) := by + filter_upwards [C.all_finite_times_limit hBc hκ hκ8 F hw N hNr hpath,C.terminal_limit] + with ω hf ht + intro t f + by_cases h : t < ∞ + · simpa only [extendedCanonicalMass,ite_eq_left h,ENNReal.coe_toNNReal h.ne] using hf t.toNNReal f + · have he : t = ∞ := top_le_iff.mp (le_of_not_gt h) + simpa only [he,extendedCanonicalMass,lt_self_iff_false,ite_false] using ht f + +theorem RationalCanonicalMass.random_time_nonnegative_mean_tendsto + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (C : RationalCanonicalMass P κ B hBm γ G) + (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) + (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) + (C.terminal ω) (fun t => N t ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : + Integrable (fun ω => ∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) (P.restrict E) ∧ + Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n (S ω) ω) ∂P) atTop + (𝓝 (∫ ω in E, (∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) ∂P)) := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + let M (n : ℕ) (t : ℝ≥0∞) (ω : Ω) := convexLevelMeasure κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n t ω + let (n : ℕ) (t : ℝ≥0∞) (ω : Ω) : IsFiniteMeasureOnCompacts (M n t ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ C.supports C.weights n t + let X (n : ℕ) (ω : Ω) := ∫ z, f z ∂M n (S ω) ω + let Z (n : ℕ) (ω : Ω) := ∫ z, f z ∂M n ∞ ω + have hXm (n : ℕ) : Measurable (X n) := + measurable_halfPlane_compact_test_integral _ + (convexLevelMeasure_random_time_measurable hm hBc κ γ G F hw _ + (fun n => Nat.cast_nonneg n) C.supports C.weights n S hS) + (fun _ => inferInstance) f + have hZm (n : ℕ) : Measurable (Z n) := + convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw _ + (fun n => Nat.cast_nonneg n) C.supports C.weights C.weights_nonneg n ∞ f + obtain ⟨A,hA,hbound⟩ := uniform_convexLevelMeasure_test_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) + C.supports C.weights C.weights_nonneg C.weights_sum f hf 2 + have hZ (n : ℕ) := memLp_two_of_nonnegative_square_bound P (Z n) (hZm n) + (fun _ => integral_nonneg hf) A hA (hbound n ∞) + have hXZ (n : ℕ) (ω : Ω) : X n ω ≤ Z n ω := + integral_mono_measure (convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n ω le_top) (ae_of_all _ hf) f.integrable + apply integral_tendsto_of_nonnegative_L2_domination (P.restrict E) X Z _ hXm + (fun _ _ => integral_nonneg hf) hXZ (fun n => (hZ n).1.restrict E) A.toReal + · intro n + exact (integral_mono_measure Measure.restrict_le_self + (ae_of_all _ (fun ω => sq_nonneg (Z n ω))) (hZ n).1.integrable_sq).trans (hZ n).2 + · filter_upwards [ae_restrict_of_ae (C.all_extended_times_limit hBc hκ hκ8 F hw N hNr hpath)] + with ω hω + exact hω (S ω) f + +lemma convexLevelMeasure_random_time_test_integrable + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (n : ℕ) (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) + (f : C_c(upperHalfPlane,ℝ)) : + Integrable (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (S ω) ω) + (P.restrict E) := by + let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + have hμm := convexLevelMeasure_random_time_measurable hm hBc κ γ G F hw u hu s w n S hS + have hνm := convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞ + have hle := restricted_bind_le_of_ae_le P _ _ hμm hνm + (ae_of_all _ (fun ω => convexLevelMeasure_mono κ B γ G u s w n ω le_top)) E + rw [convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu s w hw₀ hw₁] at hle + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + exact (integrable_integral_bind_of_sections (P.restrict E) _ hμm f f.continuous.measurable + (fun ω => by + let := convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (S ω) + exact f.integrable) (f.integrable.mono_measure hle)).1 + +theorem RationalCanonicalMass.random_time_test_mean_tendsto + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (C : RationalCanonicalMass P κ B hBm γ G) + (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) + (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) + (C.terminal ω) (fun t => N t ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) + (f : C_c(upperHalfPlane,ℝ)) : + Integrable (fun ω => ∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) (P.restrict E) ∧ + Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n (S ω) ω) ∂P) atTop + (𝓝 (∫ ω in E, (∫ z, f z ∂extendedCanonicalMass N C.terminal (S ω) ω) ∂P)) := by + let fp := f ⊔ 0 + let fm := (-f) ⊔ 0 + have hp := C.random_time_nonnegative_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth + N hNr hpath S hS E fp (fun _ => le_sup_of_le_right le_rfl) + have hn := C.random_time_nonnegative_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth + N hNr hpath S hS E fm (fun _ => le_sup_of_le_right le_rfl) + have hsplit (ρ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts ρ] : + (∫ z, f z ∂ρ) = (∫ z, fp z ∂ρ)-(∫ z, fm z ∂ρ) := by + rw [← integral_sub fp.integrable fm.integrable] + apply integral_congr_ae + exact ae_of_all _ (fun z => (max_zero_sub_eq_self (f z)).symm) + let (ω : Ω) : IsFiniteMeasureOnCompacts (extendedCanonicalMass N C.terminal (S ω) ω) := + (extendedCanonicalMass_radon N C.terminal hNr C.terminal_radon _ _).2 + let (n : ℕ) (ω : Ω) : IsFiniteMeasureOnCompacts + (convexLevelMeasure κ B γ G (fun n => (n:ℝ)) C.supports C.weights n (S ω) ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ C.supports C.weights n _ + refine ⟨?_,?_⟩ + · apply (hp.1.sub hn.1).congr + filter_upwards [] with ω + exact (hsplit (extendedCanonicalMass N C.terminal (S ω) ω)).symm + · have hi (g : C_c(upperHalfPlane,ℝ)) (n : ℕ) : Integrable (fun ω => ∫ z, g z + ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) C.supports C.weights n (S ω) ω) + (P.restrict E) := + convexLevelMeasure_random_time_test_integrable hB hBm hBc hB0 hκ hκ8 γ G F hw + (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) C.supports C.weights + C.weights_nonneg C.weights_sum n S hS E g + simpa only [hsplit,integral_sub (hi fp _) (hi fm _),integral_sub hp.1 hn.1] using hp.2.sub hn.2 + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassRemaining.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassRemaining.lean index 349fd1d81ed9aebcedfac161bf21ad76a2ebcaeb..23b8a0e4a82abaf417d04ef87c50f1b9b5718113 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassRemaining.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassRemaining.lean @@ -1,118 +1,118 @@ -import OAI.Probability.SLE.Mass.CanonicalMassConstruction -import OAI.Probability.SLE.Geometry.RandomSpatialTestingAE - -namespace OAI - -/-! The positive remaining mass and its conditional random spatial tests. -The stopping-time sigma algebra is the original, uncompleted one. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -namespace CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -def atTime (t : ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := - extendedCanonicalMass C.process C.rational.terminal t ω - -lemma atTime_radon (t : ℝ≥0∞) (ω : Ω) : - (C.atTime t ω).Regular ∧ IsFiniteMeasureOnCompacts (C.atTime t ω) := - extendedCanonicalMass_radon C.process C.rational.terminal C.radon C.rational.terminal_radon t ω - -lemma atTime_measurable (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun ω => C.atTime (S ω) ω) := - extendedCanonicalMass_measurable _ _ _ C.progressive C.rational.terminal_measurable S hS - -lemma atTime_le_terminal (S : Ω → ℝ≥0∞) : - ∀ᵐ ω ∂P, C.atTime (S ω) ω ≤ C.rational.terminal ω := by - filter_upwards [C.paths] with ω hω - dsimp only [atTime,extendedCanonicalMass] - split_ifs - · exact hω.dominated _ - · exact le_rfl - -lemma atTime_mean_le (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) : - (P.restrict E).bind (fun ω => C.atTime (S ω) ω) ≤ initialGreenMeasure κ := by - have hh := restricted_bind_le_of_ae_le P _ _ (C.atTime_measurable S hS) - C.rational.terminal_measurable (C.atTime_le_terminal S) E - rwa [C.rational.mean] at hh - -theorem exists_remaining_kernel (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) : - ∃ η : Ω → Measure upperHalfPlane, Measurable η ∧ - (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ - (∀ᵐ ω ∂P, η ω + C.atTime (S ω) ω = C.rational.terminal ω) ∧ - ∀ E : Set Ω, MeasurableSet[hS.measurableSpace] E → - (P.restrict E).bind η = (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) := by - obtain ⟨η,hηm,hηr,hηsum⟩ := exists_measurable_halfPlane_difference P C.rational.terminal - (fun ω => C.atTime (S ω) ω) C.rational.terminal_measurable (C.atTime_measurable S hS.measurable') - (fun ω => (C.rational.terminal_radon ω).1) (fun ω => (C.atTime_radon (S ω) ω).1) - (C.atTime_le_terminal S) - refine ⟨η,hηm,hηr,hηsum,?_⟩ - intro E hE - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (fun ω => C.atTime (S ω) ω)) := - ⟨fun K hK => lt_of_le_of_lt (C.atTime_mean_le S hS.measurable' E K) hK.measure_lt_top⟩ - exact bind_remainder_eq_of_sum (P.restrict E) _ _ η (C.atTime_measurable S hS.measurable') - C.rational.terminal_measurable hηm _ (ae_restrict_of_ae hηsum) (C.kernel S hS E hE) - -/-- The kernel permits all nonnegative tests jointly measurable in the -original stopped past and the spatial point, with arbitrary compact support. -/ -theorem exists_remaining_random_tests - (hBc : ∀ ω, Continuous (fun t => B t ω)) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) : - ∃ η : Ω → Measure upperHalfPlane, Measurable η ∧ - (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ - (∀ᵐ ω ∂P, η ω + C.atTime (S ω) ω = C.rational.terminal ω) ∧ - ∀ K : Set upperHalfPlane, IsCompact K → ∀ f : Ω × upperHalfPlane → ℝ≥0∞, - Measurable[hS.measurableSpace.prod inferInstance] f → - (∫⁻ ω, (∫⁻ z in K, f (ω,z) ∂η ω) ∂P) = - ∫⁻ ω, (∫⁻ z in K, f (ω,z) ∂halfPlaneStoppingGreenMeasure κ B γ G S ω) ∂P := by - obtain ⟨η,hηm,hηr,hηsum,hηkernel⟩ := C.exists_remaining_kernel S hS - have hηdom : ∀ᵐ ω ∂P, η ω ≤ C.rational.terminal ω := by - filter_upwards [hηsum] with ω hω - rw [← hω] - apply Measure.le_iff.mpr - intro A _ - simp only [Measure.add_apply] - exact le_add_right le_rfl - have hηmean : P.bind η ≤ initialGreenMeasure κ := by - have hh := restricted_bind_le_of_ae_le P η C.rational.terminal hηm - C.rational.terminal_measurable hηdom univ - simpa only [Measure.restrict_univ,C.rational.mean] using hh - have hGm := halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS.measurable' - have hGeq : P.bind η = P.bind (halfPlaneStoppingGreenMeasure κ B γ G S) := by - simpa only [Measure.restrict_univ] using hηkernel univ MeasurableSet.univ - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - refine ⟨η,hηm,hηr,hηsum,?_⟩ - intro K hK f hf - have hηK : P.bind η K < ∞ := lt_of_le_of_lt (hηmean K) hK.measure_lt_top - have hGK : ∀ᵐ ω ∂P, halfPlaneStoppingGreenMeasure κ B γ G S ω K < ∞ := by - apply ae_lt_top (μ := P) ((Measure.measurable_coe hK.measurableSet).comp hGm) - change (∫⁻ ω, halfPlaneStoppingGreenMeasure κ B γ G S ω K ∂P) ≠ ∞ - rw [← Measure.bind_apply hK.measurableSet hGm.aemeasurable,← hGeq] - exact hηK.ne - exact conditional_random_spatial_test_on_ae hS.measurableSpace hS.measurableSpace_le P - η (halfPlaneStoppingGreenMeasure κ B γ G S) hηm hGm K hK.measurableSet - (ae_of_all _ (fun ω => by let := (hηr ω).2; exact hK.measure_lt_top)) hGK hηK - hηkernel f hf - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassConstruction +import OAI.Probability.SLE.Geometry.RandomSpatialTestingAE + +namespace OAI + +/-! The positive remaining mass and its conditional random spatial tests. +The stopping-time sigma algebra is the original, uncompleted one. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +namespace CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +def atTime (t : ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := + extendedCanonicalMass C.process C.rational.terminal t ω + +lemma atTime_radon (t : ℝ≥0∞) (ω : Ω) : + (C.atTime t ω).Regular ∧ IsFiniteMeasureOnCompacts (C.atTime t ω) := + extendedCanonicalMass_radon C.process C.rational.terminal C.radon C.rational.terminal_radon t ω + +lemma atTime_measurable (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun ω => C.atTime (S ω) ω) := + extendedCanonicalMass_measurable _ _ _ C.progressive C.rational.terminal_measurable S hS + +lemma atTime_le_terminal (S : Ω → ℝ≥0∞) : + ∀ᵐ ω ∂P, C.atTime (S ω) ω ≤ C.rational.terminal ω := by + filter_upwards [C.paths] with ω hω + dsimp only [atTime,extendedCanonicalMass] + split_ifs + · exact hω.dominated _ + · exact le_rfl + +lemma atTime_mean_le (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) : + (P.restrict E).bind (fun ω => C.atTime (S ω) ω) ≤ initialGreenMeasure κ := by + have hh := restricted_bind_le_of_ae_le P _ _ (C.atTime_measurable S hS) + C.rational.terminal_measurable (C.atTime_le_terminal S) E + rwa [C.rational.mean] at hh + +theorem exists_remaining_kernel (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) : + ∃ η : Ω → Measure upperHalfPlane, Measurable η ∧ + (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ + (∀ᵐ ω ∂P, η ω + C.atTime (S ω) ω = C.rational.terminal ω) ∧ + ∀ E : Set Ω, MeasurableSet[hS.measurableSpace] E → + (P.restrict E).bind η = (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) := by + obtain ⟨η,hηm,hηr,hηsum⟩ := exists_measurable_halfPlane_difference P C.rational.terminal + (fun ω => C.atTime (S ω) ω) C.rational.terminal_measurable (C.atTime_measurable S hS.measurable') + (fun ω => (C.rational.terminal_radon ω).1) (fun ω => (C.atTime_radon (S ω) ω).1) + (C.atTime_le_terminal S) + refine ⟨η,hηm,hηr,hηsum,?_⟩ + intro E hE + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (fun ω => C.atTime (S ω) ω)) := + ⟨fun K hK => lt_of_le_of_lt (C.atTime_mean_le S hS.measurable' E K) hK.measure_lt_top⟩ + exact bind_remainder_eq_of_sum (P.restrict E) _ _ η (C.atTime_measurable S hS.measurable') + C.rational.terminal_measurable hηm _ (ae_restrict_of_ae hηsum) (C.kernel S hS E hE) + +/-- The kernel permits all nonnegative tests jointly measurable in the +original stopped past and the spatial point, with arbitrary compact support. -/ +theorem exists_remaining_random_tests + (hBc : ∀ ω, Continuous (fun t => B t ω)) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) : + ∃ η : Ω → Measure upperHalfPlane, Measurable η ∧ + (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ + (∀ᵐ ω ∂P, η ω + C.atTime (S ω) ω = C.rational.terminal ω) ∧ + ∀ K : Set upperHalfPlane, IsCompact K → ∀ f : Ω × upperHalfPlane → ℝ≥0∞, + Measurable[hS.measurableSpace.prod inferInstance] f → + (∫⁻ ω, (∫⁻ z in K, f (ω,z) ∂η ω) ∂P) = + ∫⁻ ω, (∫⁻ z in K, f (ω,z) ∂halfPlaneStoppingGreenMeasure κ B γ G S ω) ∂P := by + obtain ⟨η,hηm,hηr,hηsum,hηkernel⟩ := C.exists_remaining_kernel S hS + have hηdom : ∀ᵐ ω ∂P, η ω ≤ C.rational.terminal ω := by + filter_upwards [hηsum] with ω hω + rw [← hω] + apply Measure.le_iff.mpr + intro A _ + simp only [Measure.add_apply] + exact le_add_right le_rfl + have hηmean : P.bind η ≤ initialGreenMeasure κ := by + have hh := restricted_bind_le_of_ae_le P η C.rational.terminal hηm + C.rational.terminal_measurable hηdom univ + simpa only [Measure.restrict_univ,C.rational.mean] using hh + have hGm := halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS.measurable' + have hGeq : P.bind η = P.bind (halfPlaneStoppingGreenMeasure κ B γ G S) := by + simpa only [Measure.restrict_univ] using hηkernel univ MeasurableSet.univ + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + refine ⟨η,hηm,hηr,hηsum,?_⟩ + intro K hK f hf + have hηK : P.bind η K < ∞ := lt_of_le_of_lt (hηmean K) hK.measure_lt_top + have hGK : ∀ᵐ ω ∂P, halfPlaneStoppingGreenMeasure κ B γ G S ω K < ∞ := by + apply ae_lt_top (μ := P) ((Measure.measurable_coe hK.measurableSet).comp hGm) + change (∫⁻ ω, halfPlaneStoppingGreenMeasure κ B γ G S ω K ∂P) ≠ ∞ + rw [← Measure.bind_apply hK.measurableSet hGm.aemeasurable,← hGeq] + exact hηK.ne + exact conditional_random_spatial_test_on_ae hS.measurableSpace hS.measurableSpace_le P + η (halfPlaneStoppingGreenMeasure κ B γ G S) hηm hGm K hK.measurableSet + (ae_of_all _ (fun ω => by let := (hηr ω).2; exact hK.measure_lt_top)) hGK hηK + hηkernel f hf + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassRestart.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassRestart.lean index b19707460e574ba1724a909d93ac5e250b445a7a..0e6937b63555a365f7839a7f1eeca37ef8aa8039 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassRestart.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassRestart.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Loewner.StoppedInverseTerminalUniqueness - -namespace OAI - -/-! The canonical raw driver functional restarts through the actual inverse -Loewner chart. The trace-generation input is precisely the published one -already used for ordinary SLE, and strong Markov provides each fresh path. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance canonicalRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance canonicalRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -/-- One raw functional realizes the original mass and satisfies the finite -and terminal pathwise restart identities at every finite stopping time. -/ -theorem CanonicalMass.exists_raw_pathwise_restart - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : - ∃ R : RawCanonicalMass, - (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ - ∀ (τ : Ω → ℝ≥0) - (_hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))), - ∀ᵐ ω ∂P, - (∀ t : ℝ≥0, C.process (τ ω+t) ω-C.process (τ ω) ω = - stoppedFreshMass B κ γ G F hw τ hc R t ω) ∧ - C.rational.terminal ω-C.process (τ ω) ω = - stoppedFreshTerminalMass B κ γ G F hw τ hc R ω := by - obtain ⟨R,he,hr⟩ := C.exists_raw_restarted_canonical hB hc - refine ⟨R,he,?_⟩ - intro τ hτ - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hFresh := fresh_capacity_witness_exists hB hm hc hpublished τ hτ - have hfin := C.stoppedFreshTerminalMass_ae_finite_on_compacts hB hc hκ F hw τ hτ R - (he.mono fun _ h => h.2) hpublished - have hraw := (hr τ hτ).2 - have hp := stoppedFreshMass_paths B κ γ G F hw τ hc R hm hτ hκ - (hraw.mono fun _ h => ⟨h.1,h.2.1,h.2.2.1,h.2.2.2.1⟩) hfin - have hx := stoppedFreshMass_terminal_exhaustion B κ γ G F hw τ hc R hm hτ hκ - (hraw.mono fun _ h => h.2.2.2.1) (hraw.mono fun _ h => h.2.2.2.2) hfin - filter_upwards [C.stoppedFreshMass_indistinguishable hB hc hκ F hw R he τ hτ hFresh hp, - C.stoppedFreshTerminalMass_eq_remainder hB hc hκ F hw R he τ hτ hFresh hp hfin hx] - with ω hf ht - exact ⟨hf,ht⟩ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Loewner.StoppedInverseTerminalUniqueness + +namespace OAI + +/-! The canonical raw driver functional restarts through the actual inverse +Loewner chart. The trace-generation input is precisely the published one +already used for ordinary SLE, and strong Markov provides each fresh path. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance canonicalRestartDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance canonicalRestartDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +/-- One raw functional realizes the original mass and satisfies the finite +and terminal pathwise restart identities at every finite stopping time. -/ +theorem CanonicalMass.exists_raw_pathwise_restart + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : + ∃ R : RawCanonicalMass, + (∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) ∧ + ∀ (τ : Ω → ℝ≥0) + (_hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))), + ∀ᵐ ω ∂P, + (∀ t : ℝ≥0, C.process (τ ω+t) ω-C.process (τ ω) ω = + stoppedFreshMass B κ γ G F hw τ hc R t ω) ∧ + C.rational.terminal ω-C.process (τ ω) ω = + stoppedFreshTerminalMass B κ γ G F hw τ hc R ω := by + obtain ⟨R,he,hr⟩ := C.exists_raw_restarted_canonical hB hc + refine ⟨R,he,?_⟩ + intro τ hτ + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hFresh := fresh_capacity_witness_exists hB hm hc hpublished τ hτ + have hfin := C.stoppedFreshTerminalMass_ae_finite_on_compacts hB hc hκ F hw τ hτ R + (he.mono fun _ h => h.2) hpublished + have hraw := (hr τ hτ).2 + have hp := stoppedFreshMass_paths B κ γ G F hw τ hc R hm hτ hκ + (hraw.mono fun _ h => ⟨h.1,h.2.1,h.2.2.1,h.2.2.2.1⟩) hfin + have hx := stoppedFreshMass_terminal_exhaustion B κ γ G F hw τ hc R hm hτ hκ + (hraw.mono fun _ h => h.2.2.2.1) (hraw.mono fun _ h => h.2.2.2.2) hfin + filter_upwards [C.stoppedFreshMass_indistinguishable hB hc hκ F hw R he τ hτ hFresh hp, + C.stoppedFreshTerminalMass_eq_remainder hB hc hκ F hw R he τ hτ hFresh hp hfin hx] + with ω hf ht + exact ⟨hf,ht⟩ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassScaling.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassScaling.lean index d6b1826dee2f0d02e530855470c45fe7ebfce538..b400dc24920a262f9beeba957b499ee4d056f586 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassScaling.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassScaling.lean @@ -1,129 +1,129 @@ -import OAI.Probability.SLE.Mass.RawScaledMassMeans -import OAI.Probability.SLE.Mass.CanonicalMassPositiveReduction - -namespace OAI - -/-! Canonical mass is identified with its dimension-weighted dilation -by the proved Green kernel and continuous increasing-path uniqueness. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance canonicalScalingDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance canonicalScalingDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ -local instance canonicalScalingLocallyCompact : LocallyCompactSpace upperHalfPlane := - isOpen_upperHalfPlane.locallyCompactSpace - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma CanonicalMass.scaled_raw_process_eq - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ - R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) - (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) - (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E) - (hEp : ∀ p ∈ E, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) - (c : ℝ≥0) (hcp : 0 < c) : - ∀ᵐ ω ∂P, ∀ t, dilationMass κ c hcp (C.process t ω)=scaledRawMass R c B hc t ω := by - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hd : Measurable (dilationMass κ c hcp) := conformalMass_measurable _ _ - have hA : Adapted (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) - (fun t ω => dilationMass κ c hcp (C.process t ω)) := - fun t => hd.comp (C.progressive.adapted t) - have hZ : Measurable (scaledRawTerminal R c B hc) := R.terminal_measurable.comp - (source_path_measurable _ (scaledBrownianDriver_measurable c B hm)) - have hAb (t : ℝ≥0) : P.bind (fun ω => dilationMass κ c hcp (C.process t ω)) ≤ - initialGreenMeasure κ := by - rw [dilationMass_bind P κ c hcp _ (C.progressive.adapted.measurable (i:=t))] - apply (dilationMass_mono κ c hcp ?_).trans_eq (dilationMass_initialGreen κ c hcp) - simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le t univ - have hAT : P.bind (fun ω => dilationMass κ c hcp (C.rational.terminal ω)) ≤ - initialGreenMeasure κ := by - rw [dilationMass_bind P κ c hcp _ C.rational.terminal_measurable,C.rational.mean, - dilationMass_initialGreen] - have hpaths : ∀ᵐ ω ∂P, PositiveMassPaths - (fun t => dilationMass κ c hcp (C.process t ω)) - (dilationMass κ c hcp (C.rational.terminal ω)) := by - filter_upwards [C.paths] with ω hp - refine ⟨?_,?_,?_,?_,?_⟩ - · simp only [hp.zero,dilationMass,conformalMass_zero] - · intro s t hst - exact dilationMass_mono κ c hcp (hp.monotone hst) - · intro t - exact dilationMass_mono κ c hcp (hp.dominated t) - · intro t - let := (C.radon t ω).2 - exact conformalMass_radon _ _ _ - · intro f _hf - exact conformalMass_test_continuous _ _ _ hp.continuous_tests f - exact positive_mass_process_eq_of_cross_kernel _ _ _ _ hA - (scaledRawMass_adapted R c hcp B hm hc) (hd.comp C.rational.terminal_measurable) hZ - hAb (C.scaledRawMass_mean_le hB hc R he c hcp) - hAT (le_of_eq (C.scaledRawTerminal_mean hB hc R he c hcp)) hpaths - (scaledRawMass_positive_paths R E hE B hB hm hc hEf hEp c hcp) - (C.scaledRawMass_cross_kernel hB hc R he c hcp F hw) - -lemma CanonicalMass.nonzero_probability_scaled - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (c : ℝ≥0) (hcp : 0 < c) (t : ℝ≥0) : - P {ω | C.process t ω ≠ 0}=P {ω | C.process (c^2*t) ω ≠ 0} := by - obtain ⟨R,he,E,hEm,hEf,hEp⟩ := C.exists_raw_positive_paths hc - have hs := C.scaled_raw_process_eq hB hc F hw R he E hEm hEf hEp c hcp - have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - let A : Set C(ℝ≥0,ℝ) := {p | 0 < R.process (c^2*t) p univ} - have hAm : MeasurableSet A := measurableSet_lt measurable_const - ((Measure.measurable_coe MeasurableSet.univ).comp (R.progressive.adapted.measurable (i:=c^2*t))) - let X := sourceDriverPath B hc - let Y := sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) - have hXm : Measurable X := source_path_measurable _ hm - have hYm : Measurable Y := source_path_measurable _ (scaledBrownianDriver_measurable c B hm) - calc - _ = P (Y ⁻¹' A) := by - apply measure_congr - filter_upwards [hs] with ω hω - apply propext - change (C.process t ω ≠ 0) ↔ 0 < scaledRawMass R c B hc t ω univ - rw [← hω t,Measure.measure_univ_pos,dilationMass_ne_zero_iff] - _ = (P.map Y) A := (Measure.map_apply hYm hAm).symm - _ = (P.map X) A := by rw [scaledBrownianDriver_path_law c hcp B hB hm hc] - _ = P (X ⁻¹' A) := Measure.map_apply hXm hAm - _ = _ := by - apply measure_congr - filter_upwards [he] with ω hω - apply propext - change (0 < R.process (c^2*t) (X ω) univ) ↔ C.process (c^2*t) ω ≠ 0 - rw [hω.1 (c^2*t),Measure.measure_univ_pos] - -lemma CanonicalMass.nonzero_probability_eq - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (s t : ℝ≥0) (hs : 0 < s) (ht : 0 < t) : - P {ω | C.process s ω ≠ 0}=P {ω | C.process t ω ≠ 0} := by - have hscale : 0 < NNReal.sqrt (t/s) := NNReal.sqrt_pos.mpr (div_pos ht hs) - have h := C.nonzero_probability_scaled hB hc F hw (NNReal.sqrt (t/s)) hscale s - simpa only [NNReal.sq_sqrt,div_mul_cancel₀ _ hs.ne'] using h - -lemma CanonicalMass.positive_all_times - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) - (h0 : ∀ ω, B 0 ω=0) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) : - ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 < t → C.process t ω ≠ 0 := - C.positive_all_times_of_scaling hB hc h0 (C.nonzero_probability_eq hB hc F hw) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.RawScaledMassMeans +import OAI.Probability.SLE.Mass.CanonicalMassPositiveReduction + +namespace OAI + +/-! Canonical mass is identified with its dimension-weighted dilation +by the proved Green kernel and continuous increasing-path uniqueness. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance canonicalScalingDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance canonicalScalingDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ +local instance canonicalScalingLocallyCompact : LocallyCompactSpace upperHalfPlane := + isOpen_upperHalfPlane.locallyCompactSpace + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma CanonicalMass.scaled_raw_process_eq + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, (∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) ∧ + R.terminal (sourceDriverPath B hc ω)=C.rational.terminal ω) + (E : Set C(ℝ≥0,ℝ)) (hE : MeasurableSet E) + (hEf : ∀ᵐ p ∂P.map (sourceDriverPath B hc), p ∈ E) + (hEp : ∀ p ∈ E, PositiveMassPaths (fun t => R.process t p) (R.terminal p)) + (c : ℝ≥0) (hcp : 0 < c) : + ∀ᵐ ω ∂P, ∀ t, dilationMass κ c hcp (C.process t ω)=scaledRawMass R c B hc t ω := by + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hd : Measurable (dilationMass κ c hcp) := conformalMass_measurable _ _ + have hA : Adapted (Filtration.natural B (fun s => (hm s).stronglyMeasurable)) + (fun t ω => dilationMass κ c hcp (C.process t ω)) := + fun t => hd.comp (C.progressive.adapted t) + have hZ : Measurable (scaledRawTerminal R c B hc) := R.terminal_measurable.comp + (source_path_measurable _ (scaledBrownianDriver_measurable c B hm)) + have hAb (t : ℝ≥0) : P.bind (fun ω => dilationMass κ c hcp (C.process t ω)) ≤ + initialGreenMeasure κ := by + rw [dilationMass_bind P κ c hcp _ (C.progressive.adapted.measurable (i:=t))] + apply (dilationMass_mono κ c hcp ?_).trans_eq (dilationMass_initialGreen κ c hcp) + simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le t univ + have hAT : P.bind (fun ω => dilationMass κ c hcp (C.rational.terminal ω)) ≤ + initialGreenMeasure κ := by + rw [dilationMass_bind P κ c hcp _ C.rational.terminal_measurable,C.rational.mean, + dilationMass_initialGreen] + have hpaths : ∀ᵐ ω ∂P, PositiveMassPaths + (fun t => dilationMass κ c hcp (C.process t ω)) + (dilationMass κ c hcp (C.rational.terminal ω)) := by + filter_upwards [C.paths] with ω hp + refine ⟨?_,?_,?_,?_,?_⟩ + · simp only [hp.zero,dilationMass,conformalMass_zero] + · intro s t hst + exact dilationMass_mono κ c hcp (hp.monotone hst) + · intro t + exact dilationMass_mono κ c hcp (hp.dominated t) + · intro t + let := (C.radon t ω).2 + exact conformalMass_radon _ _ _ + · intro f _hf + exact conformalMass_test_continuous _ _ _ hp.continuous_tests f + exact positive_mass_process_eq_of_cross_kernel _ _ _ _ hA + (scaledRawMass_adapted R c hcp B hm hc) (hd.comp C.rational.terminal_measurable) hZ + hAb (C.scaledRawMass_mean_le hB hc R he c hcp) + hAT (le_of_eq (C.scaledRawTerminal_mean hB hc R he c hcp)) hpaths + (scaledRawMass_positive_paths R E hE B hB hm hc hEf hEp c hcp) + (C.scaledRawMass_cross_kernel hB hc R he c hcp F hw) + +lemma CanonicalMass.nonzero_probability_scaled + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (c : ℝ≥0) (hcp : 0 < c) (t : ℝ≥0) : + P {ω | C.process t ω ≠ 0}=P {ω | C.process (c^2*t) ω ≠ 0} := by + obtain ⟨R,he,E,hEm,hEf,hEp⟩ := C.exists_raw_positive_paths hc + have hs := C.scaled_raw_process_eq hB hc F hw R he E hEm hEf hEp c hcp + have hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + let A : Set C(ℝ≥0,ℝ) := {p | 0 < R.process (c^2*t) p univ} + have hAm : MeasurableSet A := measurableSet_lt measurable_const + ((Measure.measurable_coe MeasurableSet.univ).comp (R.progressive.adapted.measurable (i:=c^2*t))) + let X := sourceDriverPath B hc + let Y := sourceDriverPath (scaledBrownianDriver c B) (scaledBrownianDriver_continuous c B hc) + have hXm : Measurable X := source_path_measurable _ hm + have hYm : Measurable Y := source_path_measurable _ (scaledBrownianDriver_measurable c B hm) + calc + _ = P (Y ⁻¹' A) := by + apply measure_congr + filter_upwards [hs] with ω hω + apply propext + change (C.process t ω ≠ 0) ↔ 0 < scaledRawMass R c B hc t ω univ + rw [← hω t,Measure.measure_univ_pos,dilationMass_ne_zero_iff] + _ = (P.map Y) A := (Measure.map_apply hYm hAm).symm + _ = (P.map X) A := by rw [scaledBrownianDriver_path_law c hcp B hB hm hc] + _ = P (X ⁻¹' A) := Measure.map_apply hXm hAm + _ = _ := by + apply measure_congr + filter_upwards [he] with ω hω + apply propext + change (0 < R.process (c^2*t) (X ω) univ) ↔ C.process (c^2*t) ω ≠ 0 + rw [hω.1 (c^2*t),Measure.measure_univ_pos] + +lemma CanonicalMass.nonzero_probability_eq + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (s t : ℝ≥0) (hs : 0 < s) (ht : 0 < t) : + P {ω | C.process s ω ≠ 0}=P {ω | C.process t ω ≠ 0} := by + have hscale : 0 < NNReal.sqrt (t/s) := NNReal.sqrt_pos.mpr (div_pos ht hs) + have h := C.nonzero_probability_scaled hB hc F hw (NNReal.sqrt (t/s)) hscale s + simpa only [NNReal.sq_sqrt,div_mul_cancel₀ _ hs.ne'] using h + +lemma CanonicalMass.positive_all_times + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) + (h0 : ∀ ω, B 0 ω=0) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) : + ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 < t → C.process t ω ≠ 0 := + C.positive_all_times_of_scaling hB hc h0 (C.nonzero_probability_eq hB hc F hw) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassStoppingKernel.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassStoppingKernel.lean index 3b37d08ef0368af9b6e9a6b7b500f892446f8840..46894a7b25cd80d6e855d4db166e66dc3d748898 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassStoppingKernel.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassStoppingKernel.lean @@ -1,144 +1,144 @@ -import OAI.Probability.SLE.Mass.CanonicalMassRandomTime -import OAI.Probability.SLE.Stochastic.ProgressiveRandomTime - -namespace OAI - -/-! The canonical mass has the exact conditional Green kernel at every -possibly infinite stopping time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma extendedCanonicalMass_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (N : ℝ≥0 → Ω → Measure upperHalfPlane) (ν : Ω → Measure upperHalfPlane) - (hN : IsProgressive ℱ N) (hν : Measurable ν) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun ω => extendedCanonicalMass N ν (S ω) ω) := - Measurable.ite (measurableSet_lt hS measurable_const) - (_root_.OAI.MeasureTheory.IsProgressive.extended_random_time_measurable hN S hS) hν - -theorem RationalCanonicalMass.stopping_conditional_kernel - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - (hκ : 0 < κ) (hκ8 : κ < 8) - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (C : RationalCanonicalMass P κ B hBm γ G) - (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (hNp : IsProgressive (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) N) - (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) - (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) - (C.terminal ω) (fun t => N t ω)) - (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) - (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : - (P.restrict E).bind C.terminal = - (P.restrict E).bind (fun ω => extendedCanonicalMass N C.terminal (S ω) ω) + - (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) := by - let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - let V (ω : Ω) := extendedCanonicalMass N C.terminal (S ω) ω - have hVm : Measurable V := extendedCanonicalMass_measurable _ N C.terminal hNp - C.terminal_measurable S hS.measurable' - have hVr (ω : Ω) := extendedCanonicalMass_radon N C.terminal hNr C.terminal_radon (S ω) ω - have hVdom : ∀ᵐ ω ∂P, V ω ≤ C.terminal ω := by - filter_upwards [hpath] with ω hω - dsimp only [V,extendedCanonicalMass] - split_ifs - · exact hω.dominated _ - · exact le_rfl - have hVmean := restricted_bind_le_of_ae_le P V C.terminal hVm C.terminal_measurable hVdom E - rw [C.mean] at hVmean - have hTmean := restricted_bind_le_of_ae_le P C.terminal C.terminal - C.terminal_measurable C.terminal_measurable (ae_of_all _ (fun _ => le_rfl)) E - rw [C.mean] at hTmean - have hGmean := halfPlaneStoppingGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw S hS E - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind V) := - ⟨fun K hK => lt_of_le_of_lt (hVmean K) hK.measure_lt_top⟩ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind C.terminal) := - ⟨fun K hK => lt_of_le_of_lt (hTmean K) hK.measure_lt_top⟩ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S)) := - ⟨fun K hK => lt_of_le_of_lt (hGmean K) hK.measure_lt_top⟩ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind V + - (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S)) := by - constructor - intro K hK - rw [Measure.add_apply] - exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩ - let M (n : ℕ) (t : ℝ≥0∞) := convexLevelMeasure κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n t - have hMm (n : ℕ) (T : Ω → ℝ≥0∞) (hT : Measurable T) := - convexLevelMeasure_random_time_measurable hm hBc κ γ G F hw (fun n => (n:ℝ)) - (fun n => Nat.cast_nonneg n) C.supports C.weights n T hT - have hMmean (n : ℕ) (T : Ω → ℝ≥0∞) (hT : Measurable T) : - (P.restrict E).bind (fun ω => M n (T ω) ω) ≤ initialGreenMeasure κ := by - have hh := restricted_bind_le_of_ae_le P _ _ (hMm n T hT) - (convexLevelMeasure_measurable hm hBc κ γ G F hw _ (fun n => Nat.cast_nonneg n) - C.supports C.weights n ∞) - (ae_of_all _ (fun ω => convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) - C.supports C.weights n ω le_top)) E - rwa [convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw _ - (fun n => Nat.cast_nonneg n) C.supports C.weights C.weights_nonneg C.weights_sum] at hh - let (ω : Ω) : IsFiniteMeasureOnCompacts (V ω) := (hVr ω).2 - let (ω : Ω) : IsFiniteMeasureOnCompacts (C.terminal ω) := (C.terminal_radon ω).2 - let (n : ℕ) (t : ℝ≥0∞) (ω : Ω) : IsFiniteMeasureOnCompacts (M n t ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ C.supports C.weights n t - apply (localTestFamily upperHalfPlane).measure_eq_of_test_integrals - intro k - let f := (localTestFamily upperHalfPlane).test k - have heM (n : ℕ) (T : Ω → ℝ≥0∞) (hT : Measurable T) : - (∫ ω in E, (∫ z, f z ∂M n (T ω) ω) ∂P) = - ∫ z, f z ∂(P.restrict E).bind (fun ω => M n (T ω) ω) := - integral_bind_of_integrable_sections (P.restrict E) _ (hMm n T hT) - f f.continuous.measurable (fun _ => f.integrable) (f.integrable.mono_measure (hMmean n T hT)) - have heV : (∫ ω in E, (∫ z, f z ∂V ω) ∂P) = ∫ z, f z ∂(P.restrict E).bind V := - integral_bind_of_integrable_sections (P.restrict E) V hVm f f.continuous.measurable - (fun _ => f.integrable) f.integrable - have heT : (∫ ω in E, (∫ z, f z ∂C.terminal ω) ∂P) = ∫ z, f z ∂(P.restrict E).bind C.terminal := - integral_bind_of_integrable_sections (P.restrict E) C.terminal C.terminal_measurable - f f.continuous.measurable (fun _ => f.integrable) f.integrable - have ht := (C.random_time_test_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth N hNr hpath - (fun _ => ∞) measurable_const E f).2 - simp only [extendedCanonicalMass,lt_self_iff_false,ite_false,heT] at ht - change Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂M n ∞ ω) ∂P) atTop - (𝓝 (∫ z, f z ∂(P.restrict E).bind C.terminal)) at ht - have heInfinity (n : ℕ) := heM n (fun _ => ∞) measurable_const - simp only [heInfinity] at ht - have hv := (C.random_time_test_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth N hNr hpath - S hS.measurable' E f).2 - change Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂M n (S ω) ω) ∂P) atTop - (𝓝 (∫ ω in E, (∫ z, f z ∂V ω) ∂P)) at hv - have heS (n : ℕ) := heM n S hS.measurable' - simp only [heV,heS] at hv - have hg := convexStoppingCurrentMean_test_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw - (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) (fun _ _ h => Nat.cast_le.mpr h) - tendsto_natCast_atTop_atTop C.supports C.weights C.weights_nonneg C.weights_sum S hS E f - have he (n : ℕ) : (∫ z, f z ∂(P.restrict E).bind (M n ∞)) = - (∫ z, f z ∂(P.restrict E).bind (fun ω => M n (S ω) ω)) + - ∫ z, f z ∂convexStoppingCurrentMean (P.restrict E) κ B γ G - (fun n => (n:ℝ)) C.supports C.weights n S := by - rw [convexLevelMeasure_stopping_conditional_split hB hBm hBc hB0 hκ hκ8 γ G F hw - _ (fun n => Nat.cast_nonneg n) C.supports C.weights n S hS E hE] - exact integral_add_measure (f.integrable.mono_measure (hMmean n S hS.measurable')) - (f.integrable.mono_measure (convexStoppingCurrentMean_le hB hBm hBc hB0 hκ hκ8 γ G F hw - _ (fun n => Nat.cast_nonneg n) C.supports C.weights C.weights_nonneg C.weights_sum n S hS E)) - rw [integral_add_measure f.integrable f.integrable] - exact tendsto_nhds_unique ht ((hv.add hg).congr' (Eventually.of_forall (fun n => (he n).symm))) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassRandomTime +import OAI.Probability.SLE.Stochastic.ProgressiveRandomTime + +namespace OAI + +/-! The canonical mass has the exact conditional Green kernel at every +possibly infinite stopping time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma extendedCanonicalMass_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (N : ℝ≥0 → Ω → Measure upperHalfPlane) (ν : Ω → Measure upperHalfPlane) + (hN : IsProgressive ℱ N) (hν : Measurable ν) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun ω => extendedCanonicalMass N ν (S ω) ω) := + Measurable.ite (measurableSet_lt hS measurable_const) + (_root_.OAI.MeasureTheory.IsProgressive.extended_random_time_measurable hN S hS) hν + +theorem RationalCanonicalMass.stopping_conditional_kernel + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + (hκ : 0 < κ) (hκ8 : κ < 8) + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (C : RationalCanonicalMass P κ B hBm γ G) + (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (hNp : IsProgressive (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) N) + (hNr : ∀ t ω, (N t ω).Regular ∧ IsFiniteMeasureOnCompacts (N t ω)) + (hpath : ∀ᵐ ω ∂P, ContinuousMassPath (γ ω) (fun q => C.mass q ω) + (C.terminal ω) (fun t => N t ω)) + (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) + (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : + (P.restrict E).bind C.terminal = + (P.restrict E).bind (fun ω => extendedCanonicalMass N C.terminal (S ω) ω) + + (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) := by + let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + let V (ω : Ω) := extendedCanonicalMass N C.terminal (S ω) ω + have hVm : Measurable V := extendedCanonicalMass_measurable _ N C.terminal hNp + C.terminal_measurable S hS.measurable' + have hVr (ω : Ω) := extendedCanonicalMass_radon N C.terminal hNr C.terminal_radon (S ω) ω + have hVdom : ∀ᵐ ω ∂P, V ω ≤ C.terminal ω := by + filter_upwards [hpath] with ω hω + dsimp only [V,extendedCanonicalMass] + split_ifs + · exact hω.dominated _ + · exact le_rfl + have hVmean := restricted_bind_le_of_ae_le P V C.terminal hVm C.terminal_measurable hVdom E + rw [C.mean] at hVmean + have hTmean := restricted_bind_le_of_ae_le P C.terminal C.terminal + C.terminal_measurable C.terminal_measurable (ae_of_all _ (fun _ => le_rfl)) E + rw [C.mean] at hTmean + have hGmean := halfPlaneStoppingGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw S hS E + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind V) := + ⟨fun K hK => lt_of_le_of_lt (hVmean K) hK.measure_lt_top⟩ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind C.terminal) := + ⟨fun K hK => lt_of_le_of_lt (hTmean K) hK.measure_lt_top⟩ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S)) := + ⟨fun K hK => lt_of_le_of_lt (hGmean K) hK.measure_lt_top⟩ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind V + + (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S)) := by + constructor + intro K hK + rw [Measure.add_apply] + exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩ + let M (n : ℕ) (t : ℝ≥0∞) := convexLevelMeasure κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n t + have hMm (n : ℕ) (T : Ω → ℝ≥0∞) (hT : Measurable T) := + convexLevelMeasure_random_time_measurable hm hBc κ γ G F hw (fun n => (n:ℝ)) + (fun n => Nat.cast_nonneg n) C.supports C.weights n T hT + have hMmean (n : ℕ) (T : Ω → ℝ≥0∞) (hT : Measurable T) : + (P.restrict E).bind (fun ω => M n (T ω) ω) ≤ initialGreenMeasure κ := by + have hh := restricted_bind_le_of_ae_le P _ _ (hMm n T hT) + (convexLevelMeasure_measurable hm hBc κ γ G F hw _ (fun n => Nat.cast_nonneg n) + C.supports C.weights n ∞) + (ae_of_all _ (fun ω => convexLevelMeasure_mono κ B γ G (fun n => (n:ℝ)) + C.supports C.weights n ω le_top)) E + rwa [convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw _ + (fun n => Nat.cast_nonneg n) C.supports C.weights C.weights_nonneg C.weights_sum] at hh + let (ω : Ω) : IsFiniteMeasureOnCompacts (V ω) := (hVr ω).2 + let (ω : Ω) : IsFiniteMeasureOnCompacts (C.terminal ω) := (C.terminal_radon ω).2 + let (n : ℕ) (t : ℝ≥0∞) (ω : Ω) : IsFiniteMeasureOnCompacts (M n t ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) _ C.supports C.weights n t + apply (localTestFamily upperHalfPlane).measure_eq_of_test_integrals + intro k + let f := (localTestFamily upperHalfPlane).test k + have heM (n : ℕ) (T : Ω → ℝ≥0∞) (hT : Measurable T) : + (∫ ω in E, (∫ z, f z ∂M n (T ω) ω) ∂P) = + ∫ z, f z ∂(P.restrict E).bind (fun ω => M n (T ω) ω) := + integral_bind_of_integrable_sections (P.restrict E) _ (hMm n T hT) + f f.continuous.measurable (fun _ => f.integrable) (f.integrable.mono_measure (hMmean n T hT)) + have heV : (∫ ω in E, (∫ z, f z ∂V ω) ∂P) = ∫ z, f z ∂(P.restrict E).bind V := + integral_bind_of_integrable_sections (P.restrict E) V hVm f f.continuous.measurable + (fun _ => f.integrable) f.integrable + have heT : (∫ ω in E, (∫ z, f z ∂C.terminal ω) ∂P) = ∫ z, f z ∂(P.restrict E).bind C.terminal := + integral_bind_of_integrable_sections (P.restrict E) C.terminal C.terminal_measurable + f f.continuous.measurable (fun _ => f.integrable) f.integrable + have ht := (C.random_time_test_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth N hNr hpath + (fun _ => ∞) measurable_const E f).2 + simp only [extendedCanonicalMass,lt_self_iff_false,ite_false,heT] at ht + change Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂M n ∞ ω) ∂P) atTop + (𝓝 (∫ z, f z ∂(P.restrict E).bind C.terminal)) at ht + have heInfinity (n : ℕ) := heM n (fun _ => ∞) measurable_const + simp only [heInfinity] at ht + have hv := (C.random_time_test_mean_tendsto hB hBm hBc hB0 hκ hκ8 F hw hgrowth N hNr hpath + S hS.measurable' E f).2 + change Tendsto (fun n => ∫ ω in E, (∫ z, f z ∂M n (S ω) ω) ∂P) atTop + (𝓝 (∫ ω in E, (∫ z, f z ∂V ω) ∂P)) at hv + have heS (n : ℕ) := heM n S hS.measurable' + simp only [heV,heS] at hv + have hg := convexStoppingCurrentMean_test_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw + (fun n => (n:ℝ)) (fun n => Nat.cast_nonneg n) (fun _ _ h => Nat.cast_le.mpr h) + tendsto_natCast_atTop_atTop C.supports C.weights C.weights_nonneg C.weights_sum S hS E f + have he (n : ℕ) : (∫ z, f z ∂(P.restrict E).bind (M n ∞)) = + (∫ z, f z ∂(P.restrict E).bind (fun ω => M n (S ω) ω)) + + ∫ z, f z ∂convexStoppingCurrentMean (P.restrict E) κ B γ G + (fun n => (n:ℝ)) C.supports C.weights n S := by + rw [convexLevelMeasure_stopping_conditional_split hB hBm hBc hB0 hκ hκ8 γ G F hw + _ (fun n => Nat.cast_nonneg n) C.supports C.weights n S hS E hE] + exact integral_add_measure (f.integrable.mono_measure (hMmean n S hS.measurable')) + (f.integrable.mono_measure (convexStoppingCurrentMean_le hB hBm hBc hB0 hκ hκ8 γ G F hw + _ (fun n => Nat.cast_nonneg n) C.supports C.weights C.weights_nonneg C.weights_sum n S hS E)) + rw [integral_add_measure f.integrable f.integrable] + exact tendsto_nhds_unique ht ((hv.add hg).congr' (Eventually.of_forall (fun n => (he n).symm))) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalMassUniqueness.lean b/lean/OAI/Probability/SLE/Mass/CanonicalMassUniqueness.lean index ffddbcd1de1aef7297f7d1074c74938c723c0c98..b207163ae3d8c3f583b0e6eda2a4da7eb84b1c81 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalMassUniqueness.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalMassUniqueness.lean @@ -1,184 +1,184 @@ -import OAI.Probability.SLE.Mass.CanonicalMassConstruction -import OAI.Probability.SLE.Mass.RadonTerminalUniqueness - -namespace OAI - -/-! The constructed canonical mass does not depend on the convex extraction. -The uniqueness argument uses only integrable terminal compact tests and the -conditional remainder kernel, rather than second moments of a competitor. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -namespace CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma uniqueness_process_mean_le (t : ℝ≥0) (E : Set Ω) : - (P.restrict E).bind (C.process t) ≤ initialGreenMeasure κ := by - have hdom : ∀ᵐ ω ∂P, C.process t ω ≤ C.rational.terminal ω := - C.paths.mono (fun _ h => h.dominated t) - have h := restricted_bind_le_of_ae_le P (C.process t) C.rational.terminal - (C.progressive.adapted.measurable (i := t)) C.rational.terminal_measurable hdom E - rwa [C.rational.mean] at h - -lemma uniqueness_terminal_mean_le (E : Set Ω) : - (P.restrict E).bind C.rational.terminal ≤ initialGreenMeasure κ := by - have h := restricted_bind_le_of_ae_le P C.rational.terminal C.rational.terminal - C.rational.terminal_measurable C.rational.terminal_measurable (ae_of_all _ (fun _ => le_rfl)) E - rwa [C.rational.mean] at h - -lemma uniqueness_test_integrable (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)) : - Integrable (fun ω => ∫ z, f z ∂C.process t ω) P := by - let (ω : Ω) : (C.process t ω).Regular := (C.radon t ω).1 - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - apply (integrable_integral_bind_of_sections P (C.process t) - (C.progressive.adapted.measurable (i := t)) f f.continuous.measurable - (fun _ => f.integrable) ?_).1 - have hfI : Integrable f (initialGreenMeasure κ) := f.integrable - exact hfI.mono_measure (by - simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le t univ) - -lemma uniqueness_terminal_test_integrable (f : C_c(upperHalfPlane,ℝ)) : - Integrable (fun ω => ∫ z, f z ∂C.rational.terminal ω) P := by - let (ω : Ω) : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - apply (integrable_integral_bind_of_sections P C.rational.terminal C.rational.terminal_measurable - f f.continuous.measurable (fun _ => f.integrable) ?_).1 - rw [C.rational.mean] - exact f.integrable - -lemma uniqueness_test_adapted (f : C_c(upperHalfPlane,ℝ)) : - StronglyAdapted (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) - (fun t ω => ∫ z, f z ∂C.process t ω) := by - intro t - exact (@measurable_halfPlane_compact_test_integral Ω - ((Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) t) - (C.process t) - (C.progressive.adapted t) (fun ω => (C.radon t ω).1) f).stronglyMeasurable - -lemma uniqueness_cross_mean (D : CanonicalMass P κ B hBm γ G) - (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) t] E) : - (P.restrict E).bind C.rational.terminal+(P.restrict E).bind (D.process t) = - (P.restrict E).bind D.rational.terminal+(P.restrict E).bind (C.process t) := by - let ℱ := Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) - have hS : IsStoppingTime ℱ (fun _ : Ω => (t:WithTop ℝ≥0)) := isStoppingTime_const ℱ t - have hES : MeasurableSet[hS.measurableSpace] E := by - change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E - rw [IsStoppingTime.measurableSpace_const] - exact hE - have hc := C.kernel (fun _ => (t:WithTop ℝ≥0)) hS E hES - have hd := D.kernel (fun _ => (t:WithTop ℝ≥0)) hS E hES - have hfinite : (t:WithTop ℝ≥0) < ∞ := WithTop.coe_lt_top t - have heC : (fun ω => extendedCanonicalMass C.process C.rational.terminal - ((t:WithTop ℝ≥0):ℝ≥0∞) ω) = C.process t := by - funext ω - unfold extendedCanonicalMass - split_ifs with ht - · congr 1 - · exact (ht hfinite).elim - have heD : (fun ω => extendedCanonicalMass D.process D.rational.terminal - ((t:WithTop ℝ≥0):ℝ≥0∞) ω) = D.process t := by - funext ω - unfold extendedCanonicalMass - split_ifs with ht - · congr 1 - · exact (ht hfinite).elim - rw [heC] at hc - rw [heD] at hd - rw [hc,hd] - ac_rfl - -lemma uniqueness_test_terminal_mean (D : CanonicalMass P κ B hBm γ G) - (f : C_c(upperHalfPlane,ℝ)) (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) t] E) : - (∫ ω in E, (∫ z, f z ∂C.process t ω)-(∫ z, f z ∂D.process t ω) ∂P) = - ∫ ω in E, (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂D.rational.terminal ω) ∂P := by - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - let (ω : Ω) : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 - let (ω : Ω) : (D.rational.terminal ω).Regular := (D.rational.terminal_radon ω).1 - let (ω : Ω) : (C.process t ω).Regular := (C.radon t ω).1 - let (ω : Ω) : (D.process t ω).Regular := (D.radon t ω).1 - have hiC := f.integrable.mono_measure (C.uniqueness_process_mean_le t E) - have hiD := f.integrable.mono_measure (D.uniqueness_process_mean_le t E) - have hiCT := f.integrable.mono_measure (C.uniqueness_terminal_mean_le E) - have hiDT := f.integrable.mono_measure (D.uniqueness_terminal_mean_le E) - have hcross := congrArg (fun μ : Measure upperHalfPlane => ∫ z, f z ∂μ) - (C.uniqueness_cross_mean D t E hE) - rw [integral_add_measure hiCT hiD,integral_add_measure hiDT hiC] at hcross - have heC := integral_bind_of_integrable_sections (P.restrict E) (C.process t) - (C.progressive.adapted.measurable (i := t)) f f.continuous.measurable (fun _ => f.integrable) hiC - have heD := integral_bind_of_integrable_sections (P.restrict E) (D.process t) - (D.progressive.adapted.measurable (i := t)) f f.continuous.measurable (fun _ => f.integrable) hiD - have heCT := integral_bind_of_integrable_sections (P.restrict E) C.rational.terminal - C.rational.terminal_measurable f f.continuous.measurable (fun _ => f.integrable) hiCT - have heDT := integral_bind_of_integrable_sections (P.restrict E) D.rational.terminal - D.rational.terminal_measurable f f.continuous.measurable (fun _ => f.integrable) hiDT - rw [integral_sub (C.uniqueness_test_integrable t f).integrableOn - (D.uniqueness_test_integrable t f).integrableOn, - integral_sub (C.uniqueness_terminal_test_integrable f).integrableOn - (D.uniqueness_terminal_test_integrable f).integrableOn,heC,heD,heCT,heDT] - linarith - -variable [IsFiniteMeasure P] - -/-- Two canonical mass constructions for the same trace have identical finite -time measures on one full-measure event. -/ -theorem process_indistinguishable (D : CanonicalMass P κ B hBm γ G) : - ∀ᵐ ω ∂P, ∀ t, C.process t ω = D.process t ω := by - apply random_radon_process_eq_of_positive_tests P (localTestFamily upperHalfPlane) - C.process D.process (fun t ω => (C.radon t ω).1) (fun t ω => (D.radon t ω).1) - intro f hf - apply increasing_mass_unique_of_terminal_mean - (fun t ω => ∫ z, f z ∂C.process t ω) (fun t ω => ∫ z, f z ∂D.process t ω) - (fun ω => (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂D.rational.terminal ω)) - ((C.uniqueness_terminal_test_integrable f).sub (D.uniqueness_terminal_test_integrable f)) - (C.uniqueness_test_adapted f) (D.uniqueness_test_adapted f) - (fun t => C.uniqueness_test_integrable t f) (fun t => D.uniqueness_test_integrable t f) - (C.uniqueness_test_terminal_mean D f) - · filter_upwards [C.paths] with ω hω - intro s t hst - let : (C.process t ω).Regular := (C.radon t ω).1 - exact integral_mono_measure (hω.monotone hst) (ae_of_all _ hf) f.integrable - · filter_upwards [D.paths] with ω hω - intro s t hst - let : (D.process t ω).Regular := (D.radon t ω).1 - exact integral_mono_measure (hω.monotone hst) (ae_of_all _ hf) f.integrable - · filter_upwards [C.paths] with ω hω - simp only [hω.zero,integral_zero_measure,Pi.zero_apply] - · filter_upwards [D.paths] with ω hω - simp only [hω.zero,integral_zero_measure,Pi.zero_apply] - · filter_upwards [C.paths,D.paths] with ω hc hd - exact (hc.continuous_tests f).sub (hd.continuous_tests f) - -/-- The locally exhausted terminal measures also agree. -/ -theorem terminal_ae_eq (D : CanonicalMass P κ B hBm γ G) : - C.rational.terminal =ᵐ[P] D.rational.terminal := by - filter_upwards [C.process_indistinguishable D,C.paths,D.paths,C.exhaustion,D.exhaustion] - with ω he hc hd hlc hld - let := (C.rational.terminal_radon ω).1 - let := (D.rational.terminal_radon ω).1 - exact radon_terminal_eq_of_common_exhaustion (C.rational.terminal ω) (D.rational.terminal ω) - (fun t => C.process t ω) hc.dominated (fun t => by rw [he t]; exact hd.dominated t) - hlc (by simpa only [he] using hld) - -end CanonicalMass -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassConstruction +import OAI.Probability.SLE.Mass.RadonTerminalUniqueness + +namespace OAI + +/-! The constructed canonical mass does not depend on the convex extraction. +The uniqueness argument uses only integrable terminal compact tests and the +conditional remainder kernel, rather than second moments of a competitor. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +namespace CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma uniqueness_process_mean_le (t : ℝ≥0) (E : Set Ω) : + (P.restrict E).bind (C.process t) ≤ initialGreenMeasure κ := by + have hdom : ∀ᵐ ω ∂P, C.process t ω ≤ C.rational.terminal ω := + C.paths.mono (fun _ h => h.dominated t) + have h := restricted_bind_le_of_ae_le P (C.process t) C.rational.terminal + (C.progressive.adapted.measurable (i := t)) C.rational.terminal_measurable hdom E + rwa [C.rational.mean] at h + +lemma uniqueness_terminal_mean_le (E : Set Ω) : + (P.restrict E).bind C.rational.terminal ≤ initialGreenMeasure κ := by + have h := restricted_bind_le_of_ae_le P C.rational.terminal C.rational.terminal + C.rational.terminal_measurable C.rational.terminal_measurable (ae_of_all _ (fun _ => le_rfl)) E + rwa [C.rational.mean] at h + +lemma uniqueness_test_integrable (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)) : + Integrable (fun ω => ∫ z, f z ∂C.process t ω) P := by + let (ω : Ω) : (C.process t ω).Regular := (C.radon t ω).1 + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + apply (integrable_integral_bind_of_sections P (C.process t) + (C.progressive.adapted.measurable (i := t)) f f.continuous.measurable + (fun _ => f.integrable) ?_).1 + have hfI : Integrable f (initialGreenMeasure κ) := f.integrable + exact hfI.mono_measure (by + simpa only [Measure.restrict_univ] using C.uniqueness_process_mean_le t univ) + +lemma uniqueness_terminal_test_integrable (f : C_c(upperHalfPlane,ℝ)) : + Integrable (fun ω => ∫ z, f z ∂C.rational.terminal ω) P := by + let (ω : Ω) : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + apply (integrable_integral_bind_of_sections P C.rational.terminal C.rational.terminal_measurable + f f.continuous.measurable (fun _ => f.integrable) ?_).1 + rw [C.rational.mean] + exact f.integrable + +lemma uniqueness_test_adapted (f : C_c(upperHalfPlane,ℝ)) : + StronglyAdapted (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) + (fun t ω => ∫ z, f z ∂C.process t ω) := by + intro t + exact (@measurable_halfPlane_compact_test_integral Ω + ((Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) t) + (C.process t) + (C.progressive.adapted t) (fun ω => (C.radon t ω).1) f).stronglyMeasurable + +lemma uniqueness_cross_mean (D : CanonicalMass P κ B hBm γ G) + (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) t] E) : + (P.restrict E).bind C.rational.terminal+(P.restrict E).bind (D.process t) = + (P.restrict E).bind D.rational.terminal+(P.restrict E).bind (C.process t) := by + let ℱ := Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) + have hS : IsStoppingTime ℱ (fun _ : Ω => (t:WithTop ℝ≥0)) := isStoppingTime_const ℱ t + have hES : MeasurableSet[hS.measurableSpace] E := by + change MeasurableSet[(isStoppingTime_const ℱ t).measurableSpace] E + rw [IsStoppingTime.measurableSpace_const] + exact hE + have hc := C.kernel (fun _ => (t:WithTop ℝ≥0)) hS E hES + have hd := D.kernel (fun _ => (t:WithTop ℝ≥0)) hS E hES + have hfinite : (t:WithTop ℝ≥0) < ∞ := WithTop.coe_lt_top t + have heC : (fun ω => extendedCanonicalMass C.process C.rational.terminal + ((t:WithTop ℝ≥0):ℝ≥0∞) ω) = C.process t := by + funext ω + unfold extendedCanonicalMass + split_ifs with ht + · congr 1 + · exact (ht hfinite).elim + have heD : (fun ω => extendedCanonicalMass D.process D.rational.terminal + ((t:WithTop ℝ≥0):ℝ≥0∞) ω) = D.process t := by + funext ω + unfold extendedCanonicalMass + split_ifs with ht + · congr 1 + · exact (ht hfinite).elim + rw [heC] at hc + rw [heD] at hd + rw [hc,hd] + ac_rfl + +lemma uniqueness_test_terminal_mean (D : CanonicalMass P κ B hBm γ G) + (f : C_c(upperHalfPlane,ℝ)) (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) t] E) : + (∫ ω in E, (∫ z, f z ∂C.process t ω)-(∫ z, f z ∂D.process t ω) ∂P) = + ∫ ω in E, (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂D.rational.terminal ω) ∂P := by + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + let (ω : Ω) : (C.rational.terminal ω).Regular := (C.rational.terminal_radon ω).1 + let (ω : Ω) : (D.rational.terminal ω).Regular := (D.rational.terminal_radon ω).1 + let (ω : Ω) : (C.process t ω).Regular := (C.radon t ω).1 + let (ω : Ω) : (D.process t ω).Regular := (D.radon t ω).1 + have hiC := f.integrable.mono_measure (C.uniqueness_process_mean_le t E) + have hiD := f.integrable.mono_measure (D.uniqueness_process_mean_le t E) + have hiCT := f.integrable.mono_measure (C.uniqueness_terminal_mean_le E) + have hiDT := f.integrable.mono_measure (D.uniqueness_terminal_mean_le E) + have hcross := congrArg (fun μ : Measure upperHalfPlane => ∫ z, f z ∂μ) + (C.uniqueness_cross_mean D t E hE) + rw [integral_add_measure hiCT hiD,integral_add_measure hiDT hiC] at hcross + have heC := integral_bind_of_integrable_sections (P.restrict E) (C.process t) + (C.progressive.adapted.measurable (i := t)) f f.continuous.measurable (fun _ => f.integrable) hiC + have heD := integral_bind_of_integrable_sections (P.restrict E) (D.process t) + (D.progressive.adapted.measurable (i := t)) f f.continuous.measurable (fun _ => f.integrable) hiD + have heCT := integral_bind_of_integrable_sections (P.restrict E) C.rational.terminal + C.rational.terminal_measurable f f.continuous.measurable (fun _ => f.integrable) hiCT + have heDT := integral_bind_of_integrable_sections (P.restrict E) D.rational.terminal + D.rational.terminal_measurable f f.continuous.measurable (fun _ => f.integrable) hiDT + rw [integral_sub (C.uniqueness_test_integrable t f).integrableOn + (D.uniqueness_test_integrable t f).integrableOn, + integral_sub (C.uniqueness_terminal_test_integrable f).integrableOn + (D.uniqueness_terminal_test_integrable f).integrableOn,heC,heD,heCT,heDT] + linarith + +variable [IsFiniteMeasure P] + +/-- Two canonical mass constructions for the same trace have identical finite +time measures on one full-measure event. -/ +theorem process_indistinguishable (D : CanonicalMass P κ B hBm γ G) : + ∀ᵐ ω ∂P, ∀ t, C.process t ω = D.process t ω := by + apply random_radon_process_eq_of_positive_tests P (localTestFamily upperHalfPlane) + C.process D.process (fun t ω => (C.radon t ω).1) (fun t ω => (D.radon t ω).1) + intro f hf + apply increasing_mass_unique_of_terminal_mean + (fun t ω => ∫ z, f z ∂C.process t ω) (fun t ω => ∫ z, f z ∂D.process t ω) + (fun ω => (∫ z, f z ∂C.rational.terminal ω)-(∫ z, f z ∂D.rational.terminal ω)) + ((C.uniqueness_terminal_test_integrable f).sub (D.uniqueness_terminal_test_integrable f)) + (C.uniqueness_test_adapted f) (D.uniqueness_test_adapted f) + (fun t => C.uniqueness_test_integrable t f) (fun t => D.uniqueness_test_integrable t f) + (C.uniqueness_test_terminal_mean D f) + · filter_upwards [C.paths] with ω hω + intro s t hst + let : (C.process t ω).Regular := (C.radon t ω).1 + exact integral_mono_measure (hω.monotone hst) (ae_of_all _ hf) f.integrable + · filter_upwards [D.paths] with ω hω + intro s t hst + let : (D.process t ω).Regular := (D.radon t ω).1 + exact integral_mono_measure (hω.monotone hst) (ae_of_all _ hf) f.integrable + · filter_upwards [C.paths] with ω hω + simp only [hω.zero,integral_zero_measure,Pi.zero_apply] + · filter_upwards [D.paths] with ω hω + simp only [hω.zero,integral_zero_measure,Pi.zero_apply] + · filter_upwards [C.paths,D.paths] with ω hc hd + exact (hc.continuous_tests f).sub (hd.continuous_tests f) + +/-- The locally exhausted terminal measures also agree. -/ +theorem terminal_ae_eq (D : CanonicalMass P κ B hBm γ G) : + C.rational.terminal =ᵐ[P] D.rational.terminal := by + filter_upwards [C.process_indistinguishable D,C.paths,D.paths,C.exhaustion,D.exhaustion] + with ω he hc hd hlc hld + let := (C.rational.terminal_radon ω).1 + let := (D.rational.terminal_radon ω).1 + exact radon_terminal_eq_of_common_exhaustion (C.rational.terminal ω) (D.rational.terminal ω) + (fun t => C.process t ω) hc.dominated (fun t => by rw [he t]; exact hd.dominated t) + hlc (by simpa only [he] using hld) + +end CanonicalMass +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalPhysicalMass.lean b/lean/OAI/Probability/SLE/Mass/CanonicalPhysicalMass.lean index d386007956973a6ba46d014f905030731eef3b55..38254bc89ba0faea17294b6afda8e0dba645c184 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalPhysicalMass.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalPhysicalMass.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Geometry.BoundedTerminalMean - -namespace OAI - -/-! The upper cover is charged to the canonical terminal measure on the -physical plane. Its mean is the initial Green measure, also on sets meeting -the real boundary. -/ -noncomputable section -open Set Filter MeasureTheory Metric -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -def RationalCanonicalMass.physicalTerminal (C : RationalCanonicalMass P κ B hBm γ G) - (ω : Ω) : Measure ℂ := (C.terminal ω).map Subtype.val - -lemma RationalCanonicalMass.physicalTerminal_measurable - (C : RationalCanonicalMass P κ B hBm γ G) : Measurable C.physicalTerminal := - (Measure.measurable_map Subtype.val measurable_subtype_coe).comp C.terminal_measurable - -lemma RationalCanonicalMass.physicalTerminal_apply - (C : RationalCanonicalMass P κ B hBm γ G) (ω : Ω) {A : Set ℂ} (hA : MeasurableSet A) : - C.physicalTerminal ω A = C.terminal ω (Subtype.val ⁻¹' A) := - Measure.map_apply measurable_subtype_coe hA - -lemma RationalCanonicalMass.physicalTerminal_mean - (C : RationalCanonicalMass P κ B hBm γ G) : - P.bind C.physicalTerminal = (initialGreenMeasure κ).map Subtype.val := by - apply Measure.ext - intro A hA - rw [Measure.bind_apply hA C.physicalTerminal_measurable.aemeasurable, - Measure.map_apply measurable_subtype_coe hA,← C.mean, - Measure.bind_apply (hA.preimage measurable_subtype_coe) C.terminal_measurable.aemeasurable] - apply lintegral_congr - intro ω - exact C.physicalTerminal_apply ω hA - -lemma RationalCanonicalMass.physicalTerminal_bounded_mean_lt_top - (C : RationalCanonicalMass P κ B hBm γ G) (hκ : 0<κ) (hκ8 : κ<8) (R : ℝ) : - (∫⁻ ω,C.physicalTerminal ω (closedBall 0 R) ∂P) < ∞ := by - simp_rw [C.physicalTerminal_apply _ isClosed_closedBall.measurableSet] - exact C.terminal_bounded_mean_lt_top hκ hκ8 R - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.BoundedTerminalMean + +namespace OAI + +/-! The upper cover is charged to the canonical terminal measure on the +physical plane. Its mean is the initial Green measure, also on sets meeting +the real boundary. -/ +noncomputable section +open Set Filter MeasureTheory Metric +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +def RationalCanonicalMass.physicalTerminal (C : RationalCanonicalMass P κ B hBm γ G) + (ω : Ω) : Measure ℂ := (C.terminal ω).map Subtype.val + +lemma RationalCanonicalMass.physicalTerminal_measurable + (C : RationalCanonicalMass P κ B hBm γ G) : Measurable C.physicalTerminal := + (Measure.measurable_map Subtype.val measurable_subtype_coe).comp C.terminal_measurable + +lemma RationalCanonicalMass.physicalTerminal_apply + (C : RationalCanonicalMass P κ B hBm γ G) (ω : Ω) {A : Set ℂ} (hA : MeasurableSet A) : + C.physicalTerminal ω A = C.terminal ω (Subtype.val ⁻¹' A) := + Measure.map_apply measurable_subtype_coe hA + +lemma RationalCanonicalMass.physicalTerminal_mean + (C : RationalCanonicalMass P κ B hBm γ G) : + P.bind C.physicalTerminal = (initialGreenMeasure κ).map Subtype.val := by + apply Measure.ext + intro A hA + rw [Measure.bind_apply hA C.physicalTerminal_measurable.aemeasurable, + Measure.map_apply measurable_subtype_coe hA,← C.mean, + Measure.bind_apply (hA.preimage measurable_subtype_coe) C.terminal_measurable.aemeasurable] + apply lintegral_congr + intro ω + exact C.physicalTerminal_apply ω hA + +lemma RationalCanonicalMass.physicalTerminal_bounded_mean_lt_top + (C : RationalCanonicalMass P κ B hBm γ G) (hκ : 0<κ) (hκ8 : κ<8) (R : ℝ) : + (∫⁻ ω,C.physicalTerminal ω (closedBall 0 R) ∂P) < ∞ := by + simp_rw [C.physicalTerminal_apply _ isClosed_closedBall.measurableSet] + exact C.terminal_bounded_mean_lt_top hκ hκ8 R + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalRadonLimit.lean b/lean/OAI/Probability/SLE/Mass/CanonicalRadonLimit.lean index ff6c2eeaebc93bffda96fda854164b00a32cf78b..17648e0fec85316efdeceebd1365078d4c9263ee 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalRadonLimit.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalRadonLimit.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Mass.SubSigmaRadonLimit - -namespace OAI - -/-! A deterministic choice of the vague limit, with value zero when no -regular limit exists. The same choice is measurable for every sigma-algebra -making the approximating compact tests measurable. -/ - -noncomputable section -open Set Filter MeasureTheory Classical -open scoped Topology CompactlySupported - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] - -def HasRegularTestLimit (μ : ℕ → Measure X) (ν : Measure X) : Prop := - ν.Regular ∧ ∀ f : C_c(X,ℝ), - Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) - -def canonicalRadonLimit (μ : ℕ → Measure X) : Measure X := - if h : ∃ ν, HasRegularTestLimit μ ν then h.choose else 0 - -omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X] in -theorem canonicalRadonLimit_regular (μ : ℕ → Measure X) : - (canonicalRadonLimit μ).Regular := by - classical - unfold canonicalRadonLimit - split_ifs with h - · exact h.choose_spec.1 - · infer_instance - -omit [SecondCountableTopology X] in -theorem canonicalRadonLimit_identified (μ : ℕ → Measure X) (ν : Measure X) - (hν : HasRegularTestLimit μ ν) : canonicalRadonLimit μ = ν := by - classical - have hex : ∃ η, HasRegularTestLimit μ η := ⟨ν,hν⟩ - have hspec := hex.choose_spec - let := hspec.1 - let := hν.1 - rw [canonicalRadonLimit,dite_eq_left hex] - exact radon_test_limit_unique μ _ ν hspec.2 hν.2 - -theorem canonicalRadonLimit_measurable - {Ω : Type*} [MeasurableSpace Ω] (μ : ℕ → Ω → Measure X) - (hfinite : ∀ n ω, IsFiniteMeasureOnCompacts (μ n ω)) - (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω)) : - Measurable (fun ω => canonicalRadonLimit (fun n => μ n ω)) := by - classical - let D := localTestFamily X - let G : Set Ω := {ω | ∀ j, ∃ a : ℝ, - Tendsto (fun n => ∫ x, D.test j x ∂μ n ω) atTop (𝓝 a)} - have hG : MeasurableSet G := by - simpa only [G,Set.ofPred_forall] using MeasurableSet.iInter - (fun j => measurableSet_exists_tendsto (fun n => hmeas n (D.test j))) - have hex (ω : Ω) : ω ∈ G ↔ ∃ ν, HasRegularTestLimit (fun n => μ n ω) ν := by - constructor - · intro hω - let (n : ℕ) : IsFiniteMeasureOnCompacts (μ n ω) := hfinite n ω - obtain ⟨ν,hr,_,hl⟩ := D.exists_radon_limit (fun n => μ n ω) hω - exact ⟨ν,hr,hl⟩ - · rintro ⟨ν,_,hl⟩ j - exact ⟨∫ x, D.test j x ∂ν,hl (D.test j)⟩ - apply measurable_radon_of_tests _ (fun ω => canonicalRadonLimit_regular _) - intro f - apply measurable_of_tendsto_metrizable - (f := fun n ω => if ω ∈ G then ∫ x, f x ∂μ n ω else 0) - · intro n - exact (hmeas n f).ite hG measurable_const - · apply tendsto_pi_nhds.mpr - intro ω - by_cases hω : ω ∈ G - · obtain ⟨ν,hν⟩ := (hex ω).mp hω - rw [canonicalRadonLimit_identified _ ν hν] - simpa only [ite_eq_left hω] using hν.2 f - · have he : canonicalRadonLimit (fun n => μ n ω) = 0 := by - rw [canonicalRadonLimit,dite_eq_right (mt (hex ω).mpr hω)] - simpa only [ite_eq_right hω,he,integral_zero_measure] using - (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.SubSigmaRadonLimit + +namespace OAI + +/-! A deterministic choice of the vague limit, with value zero when no +regular limit exists. The same choice is measurable for every sigma-algebra +making the approximating compact tests measurable. -/ + +noncomputable section +open Set Filter MeasureTheory Classical +open scoped Topology CompactlySupported + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] + +def HasRegularTestLimit (μ : ℕ → Measure X) (ν : Measure X) : Prop := + ν.Regular ∧ ∀ f : C_c(X,ℝ), + Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) + +def canonicalRadonLimit (μ : ℕ → Measure X) : Measure X := + if h : ∃ ν, HasRegularTestLimit μ ν then h.choose else 0 + +omit [T2Space X] [LocallyCompactSpace X] [SecondCountableTopology X] [BorelSpace X] in +theorem canonicalRadonLimit_regular (μ : ℕ → Measure X) : + (canonicalRadonLimit μ).Regular := by + classical + unfold canonicalRadonLimit + split_ifs with h + · exact h.choose_spec.1 + · infer_instance + +omit [SecondCountableTopology X] in +theorem canonicalRadonLimit_identified (μ : ℕ → Measure X) (ν : Measure X) + (hν : HasRegularTestLimit μ ν) : canonicalRadonLimit μ = ν := by + classical + have hex : ∃ η, HasRegularTestLimit μ η := ⟨ν,hν⟩ + have hspec := hex.choose_spec + let := hspec.1 + let := hν.1 + rw [canonicalRadonLimit,dite_eq_left hex] + exact radon_test_limit_unique μ _ ν hspec.2 hν.2 + +theorem canonicalRadonLimit_measurable + {Ω : Type*} [MeasurableSpace Ω] (μ : ℕ → Ω → Measure X) + (hfinite : ∀ n ω, IsFiniteMeasureOnCompacts (μ n ω)) + (hmeas : ∀ n (f : C_c(X,ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω)) : + Measurable (fun ω => canonicalRadonLimit (fun n => μ n ω)) := by + classical + let D := localTestFamily X + let G : Set Ω := {ω | ∀ j, ∃ a : ℝ, + Tendsto (fun n => ∫ x, D.test j x ∂μ n ω) atTop (𝓝 a)} + have hG : MeasurableSet G := by + simpa only [G,Set.ofPred_forall] using MeasurableSet.iInter + (fun j => measurableSet_exists_tendsto (fun n => hmeas n (D.test j))) + have hex (ω : Ω) : ω ∈ G ↔ ∃ ν, HasRegularTestLimit (fun n => μ n ω) ν := by + constructor + · intro hω + let (n : ℕ) : IsFiniteMeasureOnCompacts (μ n ω) := hfinite n ω + obtain ⟨ν,hr,_,hl⟩ := D.exists_radon_limit (fun n => μ n ω) hω + exact ⟨ν,hr,hl⟩ + · rintro ⟨ν,_,hl⟩ j + exact ⟨∫ x, D.test j x ∂ν,hl (D.test j)⟩ + apply measurable_radon_of_tests _ (fun ω => canonicalRadonLimit_regular _) + intro f + apply measurable_of_tendsto_metrizable + (f := fun n ω => if ω ∈ G then ∫ x, f x ∂μ n ω else 0) + · intro n + exact (hmeas n f).ite hG measurable_const + · apply tendsto_pi_nhds.mpr + intro ω + by_cases hω : ω ∈ G + · obtain ⟨ν,hν⟩ := (hex ω).mp hω + rw [canonicalRadonLimit_identified _ ν hν] + simpa only [ite_eq_left hω] using hν.2 f + · have he : canonicalRadonLimit (fun n => μ n ω) = 0 := by + rw [canonicalRadonLimit,dite_eq_right (mt (hex ω).mpr hω)] + simpa only [ite_eq_right hω,he,integral_zero_measure] using + (tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CanonicalRawMass.lean b/lean/OAI/Probability/SLE/Mass/CanonicalRawMass.lean index f207a9a18b1db93a6e5c94a0ae6d6811ad9cea07..2b7ac2be8d816395e43aae86bcb69ba478586e5f 100644 --- a/lean/OAI/Probability/SLE/Mass/CanonicalRawMass.lean +++ b/lean/OAI/Probability/SLE/Mass/CanonicalRawMass.lean @@ -1,39 +1,39 @@ -import OAI.Probability.SLE.Mass.CanonicalMassConstruction -import OAI.Probability.SLE.Mass.RawMassFunctional - -namespace OAI - -/-! The canonical finite-time mass admits one raw nonanticipating functional -on continuous real driver paths. No future-dependent modification is used. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance canonicalRawDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance canonicalRawDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -theorem CanonicalMass.exists_raw_nonanticipating_functional - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - (hBc : ∀ ω, Continuous (fun t => B t ω)) : - ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane, - IsProgressive driverPathFiltration R ∧ - (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧ - (∀ t p q, (∀ s ≤ t, p s=q s) → R t p=R t q) ∧ - ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω) = C.process t ω := by - apply exists_raw_progressive_radon_factor P B - (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBc C.process - C.progressive.adapted (fun t ω => (C.radon t ω).1) - filter_upwards [C.paths] with ω hω - exact hω.continuous_tests - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassConstruction +import OAI.Probability.SLE.Mass.RawMassFunctional + +namespace OAI + +/-! The canonical finite-time mass admits one raw nonanticipating functional +on continuous real driver paths. No future-dependent modification is used. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance canonicalRawDriverMeasurable : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance canonicalRawDriverBorel : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +theorem CanonicalMass.exists_raw_nonanticipating_functional + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + (hBc : ∀ ω, Continuous (fun t => B t ω)) : + ∃ R : ℝ≥0 → C(ℝ≥0,ℝ) → Measure upperHalfPlane, + IsProgressive driverPathFiltration R ∧ + (∀ t p, (R t p).Regular ∧ IsFiniteMeasureOnCompacts (R t p)) ∧ + (∀ t p q, (∀ s ≤ t, p s=q s) → R t p=R t q) ∧ + ∀ᵐ ω ∂P, ∀ t, R t (sourceDriverPath B hBc ω) = C.process t ω := by + apply exists_raw_progressive_radon_factor P B + (fun t => hBm.comp (measurable_const.prodMk measurable_id)) hBc C.process + C.progressive.adapted (fun t ω => (C.radon t ω).1) + filter_upwards [C.paths] with ω hω + exact hω.continuous_tests + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CellMassComparison.lean b/lean/OAI/Probability/SLE/Mass/CellMassComparison.lean index 1d033fd80ec7964f6494d02e12817ed9c3862d94..94474c89621a23ee3192653c68f2caf09301bca6 100644 --- a/lean/OAI/Probability/SLE/Mass/CellMassComparison.lean +++ b/lean/OAI/Probability/SLE/Mass/CellMassComparison.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Mass.InverseMassDilation -import OAI.Probability.SLE.Mass.CanonicalMassContinuity - -namespace OAI - -/-! At a fixed conformal-radius level the normalized disk mass bounds the -mass of a small cell and is bounded by the mass of a larger physical ball. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Multipoint - -lemma Multipoint.ContinuousMassPath.zero_on_surviving - {γ : ℝ≥0 → ℂ} {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} - {N : ℝ≥0 → Measure upperHalfPlane} (hp : ContinuousMassPath γ μ ν N) - (t : ℝ≥0) : N t (Subtype.val ⁻¹' survivingDomain γ t)=0 := by - have h := hp.increment_support 0 t (show 0 ≤ t from zero_le) - rw [hp.zero,Measure.sub_zero] at h - apply le_zero_iff.mp - apply (measure_mono ?_).trans_eq h - intro z hz - rintro ⟨s,hs,he⟩ - apply hz.1.2 - refine ⟨s.toNNReal,⟨zero_le,?_⟩,he⟩ - simpa only [Real.toNNReal_coe] using Real.toNNReal_mono hs.2 - -lemma Multipoint.ContinuousMassPath.zero_on_surviving_subset - {γ : ℝ≥0 → ℂ} {μ : ℚ → Measure upperHalfPlane} {ν : Measure upperHalfPlane} - {N : ℝ≥0 → Measure upperHalfPlane} (hp : ContinuousMassPath γ μ ν N) - (t : ℝ≥0) (A : Set ℂ) (hA : A ⊆ survivingDomain γ t) : - N t (Subtype.val ⁻¹' A)=0 := - le_zero_iff.mp ((measure_mono (preimage_mono hA)).trans_eq (hp.zero_on_surviving t)) - -lemma measure_sub_apply_of_zero - {X : Type*} [MeasurableSpace X] (μ ν : Measure X) [SigmaFinite μ] - (hle : μ ≤ ν) (A : Set X) (hz : μ A=0) : (ν-μ) A=ν A := by - have he := congrArg (fun ρ : Measure X => ρ A) (sigmaFinite_sub_add_cancel_of_le hle) - simpa only [Measure.add_apply,hz,add_zero] using he - -lemma IsCapacityTwoWitness.cell_mass_comparison - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) - (κ : ℝ) (hκ : 0<κ) (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) - (r : ℝ) (hr : 0 F t ((w:ℂ)+(U t:ℂ))) - (fun w : upperHalfPlane => ENNReal.ofReal - (‖deriv (F t) ((w:ℂ)+(U t:ℂ))‖^dimension κ)) (dilationMass κ a ha μ) - let L := ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) - (Target.diskSourceHalfPlane θ) μ) - ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*L ≤ - ν (closedBall z ((4096*Real.pi)*r)) ∧ - ∀ A : Set ℂ, A ⊆ closedBall z (2*r) → - ν A ≤ ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*L := by - let f := physicalDiskChart id F G t z - have hRp : 0 ρ A) (sigmaFinite_sub_add_cancel_of_le hle) + simpa only [Measure.add_apply,hz,add_zero] using he + +lemma IsCapacityTwoWitness.cell_mass_comparison + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) + (κ : ℝ) (hκ : 0<κ) (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) + (r : ℝ) (hr : 0 F t ((w:ℂ)+(U t:ℂ))) + (fun w : upperHalfPlane => ENNReal.ofReal + (‖deriv (F t) ((w:ℂ)+(U t:ℂ))‖^dimension κ)) (dilationMass κ a ha μ) + let L := ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) + (Target.diskSourceHalfPlane θ) μ) + ENNReal.ofReal (((128*Real.pi)*r)^dimension κ)*L ≤ + ν (closedBall z ((4096*Real.pi)*r)) ∧ + ∀ A : Set ℂ, A ⊆ closedBall z (2*r) → + ν A ≤ ENNReal.ofReal (((32768*Real.pi)*r)^dimension κ)*L := by + let f := physicalDiskChart id F G t z + have hRp : 0 Fₙ n (γ ω t)) (v n) R} := by - obtain ⟨c, ell, v₀, hc, hell, hv₀, htransfer⟩ := eventually_compact_sausage_transfer - κ Fₙ F hFn hinj hF hne hconv K hK hKn hKH R hR himage - have hcoeff : 0 < c ^ 2 * ell ^ codimension κ := - mul_pos (sq_pos_of_pos hc) (Real.rpow_pos_of_pos hell _) - have hsmall : ∀ᶠ n in atTop, v n < v₀ := hlim.eventually (gt_mem_nhds hv₀) - have hlim' : Tendsto (fun n => ell * v n) atTop (𝓝 0) := by - simpa only [mul_zero] using tendsto_const_nhds.mul hlim - have htrans : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n → True ∧ - ∃ J ∈ I, ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) * - sourceSausageMass κ (γ ω) J.1 J.2 (ell * v n) ≤ - wholeSausageCharge κ (fun t => Fₙ n (γ ω t)) (v n) R := by - filter_upwards [htransfer, hretain, hsmall] with n hn hr hs - intro ω hω - obtain ⟨J, hJI, hJK⟩ := hr ω hω - refine ⟨trivial, J, hJI, (hn (γ ω) J.1 J.2 hJK (v n) (hv n) hs).trans ?_⟩ - apply mul_le_mul' le_rfl (measure_mono (inter_subset_inter_right R ?_)) - rintro z ⟨t, _, hdist⟩ - exact ⟨t, by simpa only [dist_comm] using hdist⟩ - obtain ⟨a, ha, hmass⟩ := ribbon_passage_mass_of_eventually_finite_intervals - P κ hκ hκ8 γ hSLE I hI (fun n => ell * v n) - (fun n => mul_pos hell (hv n)) hlim' G q hq hG - (fun n ω => wholeSausageCharge κ (fun t => Fₙ n (γ ω t)) (v n) R) - (fun _ _ => True) (c ^ 2 * ell ^ codimension κ) hcoeff htrans - exact ⟨a, ha, by simpa only [true_and] using hmass⟩ - -/-- A countable family of actual retained compact intervals is enough. -The finite extraction, common mesh, derivative constants, and unconditional -small-sausage subtraction are all performed here. -/ -theorem countable_compact_conformal_ribbon_mass - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) - (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) - (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) - (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) - (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) - (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) - (hKH : ∀ i, K i ⊆ upperHalfPlane) - (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R) - (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) - (E : Set Ω) (D : ℕ → Set Ω) - (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) - (hcover : P (E \ ⋃ i, D i) = 0) - (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) - (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) - (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) - (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) : - ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, - ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ - wholeSausageCharge κ (fun t => Fₙ n (γ ω t)) (v n) R} := by - classical - obtain ⟨N, hN⟩ := varying_events_after_finite_retention P E D G hE hD hcover q hq hG - let A := ⋃ i ∈ Finset.range N, D i - let Q := ⋃ i ∈ Finset.range N, K i - let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A - have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) - have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) - have hQR : F '' Q ⊆ R := by - rintro w ⟨z, hz, rfl⟩ - obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz - exact himage i (mem_image_of_mem F hi) - have hQn : Q.Nonempty := by - have hHpos : 0 < P (H 0) := (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le (hN 0) - obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 - refine ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, ?_⟩⟩ - exact hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩) - let I := (Finset.range N).image J - have hI : ∀ L ∈ I, L.1 < L.2 := by - intro L hL - obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL - exact hJ i - have hr : ∀ n ω, ω ∈ H n → ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q := by - intro n ω hω - obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 - exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, - (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl)⟩ - obtain ⟨a, ha, hmass⟩ := compact_conformal_ribbon_mass P κ hκ hκ8 γ hSLE - Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI - v hv hlim H (q / 2) (by positivity) (Eventually.of_forall hN) - (Eventually.of_forall hr) - refine ⟨a, ha, ?_⟩ - filter_upwards [hmass] with n hn - have hqq : q / 2 / 2 = q / 4 := by ring - rw [hqq] at hn - exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.ConformalCompactSausage +import OAI.Probability.SLE.Gauge.RibbonPassageMass +import OAI.Probability.SLE.SausageCharge +import OAI.Probability.SLE.Gauge.RetainedPassageExtraction + +namespace OAI + +/-! The compact-retention and sausage part of the uniform ribbon step, +assembled for actual locally convergent conformal maps and actual SLE. +Only the passage event and its retained intervals are supplied by geometry. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal NNReal + +namespace SLEExactGauge + +theorem compact_conformal_ribbon_mass + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (K : Set ℂ) (hK : IsCompact K) (hKn : K.Nonempty) (hKH : K ⊆ upperHalfPlane) + (R : Set ℂ) (hR : IsOpen R) (himage : F '' K ⊆ R) + (I : Finset (ℝ≥0 × ℝ≥0)) (hI : ∀ J ∈ I, J.1 < J.2) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ᶠ n in atTop, ENNReal.ofReal q ≤ P (G n)) + (hretain : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n → + ∃ J ∈ I, γ ω '' Icc J.1 J.2 ⊆ K) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 2) ≤ P {ω | ω ∈ G n ∧ ENNReal.ofReal a ≤ + wholeSausageCharge κ (fun t => Fₙ n (γ ω t)) (v n) R} := by + obtain ⟨c, ell, v₀, hc, hell, hv₀, htransfer⟩ := eventually_compact_sausage_transfer + κ Fₙ F hFn hinj hF hne hconv K hK hKn hKH R hR himage + have hcoeff : 0 < c ^ 2 * ell ^ codimension κ := + mul_pos (sq_pos_of_pos hc) (Real.rpow_pos_of_pos hell _) + have hsmall : ∀ᶠ n in atTop, v n < v₀ := hlim.eventually (gt_mem_nhds hv₀) + have hlim' : Tendsto (fun n => ell * v n) atTop (𝓝 0) := by + simpa only [mul_zero] using tendsto_const_nhds.mul hlim + have htrans : ∀ᶠ n in atTop, ∀ ω, ω ∈ G n → True ∧ + ∃ J ∈ I, ENNReal.ofReal (c ^ 2 * ell ^ codimension κ) * + sourceSausageMass κ (γ ω) J.1 J.2 (ell * v n) ≤ + wholeSausageCharge κ (fun t => Fₙ n (γ ω t)) (v n) R := by + filter_upwards [htransfer, hretain, hsmall] with n hn hr hs + intro ω hω + obtain ⟨J, hJI, hJK⟩ := hr ω hω + refine ⟨trivial, J, hJI, (hn (γ ω) J.1 J.2 hJK (v n) (hv n) hs).trans ?_⟩ + apply mul_le_mul' le_rfl (measure_mono (inter_subset_inter_right R ?_)) + rintro z ⟨t, _, hdist⟩ + exact ⟨t, by simpa only [dist_comm] using hdist⟩ + obtain ⟨a, ha, hmass⟩ := ribbon_passage_mass_of_eventually_finite_intervals + P κ hκ hκ8 γ hSLE I hI (fun n => ell * v n) + (fun n => mul_pos hell (hv n)) hlim' G q hq hG + (fun n ω => wholeSausageCharge κ (fun t => Fₙ n (γ ω t)) (v n) R) + (fun _ _ => True) (c ^ 2 * ell ^ codimension κ) hcoeff htrans + exact ⟨a, ha, by simpa only [true_and] using hmass⟩ + +/-- A countable family of actual retained compact intervals is enough. +The finite extraction, common mesh, derivative constants, and unconditional +small-sausage subtraction are all performed here. -/ +theorem countable_compact_conformal_ribbon_mass + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + (κ : ℝ) (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (hSLE : IsOrdinaryChordalSLE κ γ P) + (Fₙ : ℕ → ℂ → ℂ) (F : ℂ → ℂ) + (hFn : ∀ n, DifferentiableOn ℂ (Fₙ n) upperHalfPlane) + (hinj : ∀ n, InjOn (Fₙ n) upperHalfPlane) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hne : ∀ z ∈ upperHalfPlane, deriv F z ≠ 0) + (hconv : TendstoLocallyUniformlyOn Fₙ F atTop upperHalfPlane) + (K : ℕ → Set ℂ) (hK : ∀ i, IsCompact (K i)) + (hKH : ∀ i, K i ⊆ upperHalfPlane) + (R : Set ℂ) (hR : IsOpen R) (himage : ∀ i, F '' K i ⊆ R) + (J : ℕ → ℝ≥0 × ℝ≥0) (hJ : ∀ i, (J i).1 < (J i).2) + (E : Set Ω) (D : ℕ → Set Ω) + (hE : MeasurableSet E) (hD : ∀ i, MeasurableSet (D i)) + (hcover : P (E \ ⋃ i, D i) = 0) + (hretain : ∀ i ω, ω ∈ D i → γ ω '' Icc (J i).1 (J i).2 ⊆ K i) + (v : ℕ → ℝ) (hv : ∀ n, 0 < v n) (hlim : Tendsto v atTop (𝓝 0)) + (G : ℕ → Set Ω) (q : ℝ) (hq : 0 < q) + (hG : ∀ n, ENNReal.ofReal q ≤ P (G n ∩ E)) : + ∃ a : ℝ, 0 < a ∧ ∀ᶠ n in atTop, + ENNReal.ofReal (q / 4) ≤ P {ω | ω ∈ G n ∩ E ∧ ENNReal.ofReal a ≤ + wholeSausageCharge κ (fun t => Fₙ n (γ ω t)) (v n) R} := by + classical + obtain ⟨N, hN⟩ := varying_events_after_finite_retention P E D G hE hD hcover q hq hG + let A := ⋃ i ∈ Finset.range N, D i + let Q := ⋃ i ∈ Finset.range N, K i + let H : ℕ → Set Ω := fun n => G n ∩ E ∩ A + have hQ : IsCompact Q := (Finset.range N).isCompact_biUnion (fun i _ => hK i) + have hQH : Q ⊆ upperHalfPlane := iUnion₂_subset (fun i _ => hKH i) + have hQR : F '' Q ⊆ R := by + rintro w ⟨z, hz, rfl⟩ + obtain ⟨i, _, hi⟩ := mem_iUnion₂.mp hz + exact himage i (mem_image_of_mem F hi) + have hQn : Q.Nonempty := by + have hHpos : 0 < P (H 0) := (ENNReal.ofReal_pos.mpr (by positivity : 0 < q / 2)).trans_le (hN 0) + obtain ⟨ω, hω⟩ := nonempty_of_measure_ne_zero hHpos.ne' + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 + refine ⟨γ ω (J i).1, mem_iUnion₂.mpr ⟨i, hi, ?_⟩⟩ + exact hretain i ω hDi (mem_image_of_mem (γ ω) ⟨le_rfl, (hJ i).le⟩) + let I := (Finset.range N).image J + have hI : ∀ L ∈ I, L.1 < L.2 := by + intro L hL + obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hL + exact hJ i + have hr : ∀ n ω, ω ∈ H n → ∃ L ∈ I, γ ω '' Icc L.1 L.2 ⊆ Q := by + intro n ω hω + obtain ⟨i, hi, hDi⟩ := mem_iUnion₂.mp hω.2 + exact ⟨J i, Finset.mem_image.mpr ⟨i, hi, rfl⟩, + (hretain i ω hDi).trans (subset_iUnion₂_of_subset i hi Subset.rfl)⟩ + obtain ⟨a, ha, hmass⟩ := compact_conformal_ribbon_mass P κ hκ hκ8 γ hSLE + Fₙ F hFn hinj hF hne hconv Q hQ hQn hQH R hR hQR I hI + v hv hlim H (q / 2) (by positivity) (Eventually.of_forall hN) + (Eventually.of_forall hr) + refine ⟨a, ha, ?_⟩ + filter_upwards [hmass] with n hn + have hqq : q / 2 / 2 = q / 4 := by ring + rw [hqq] at hn + exact hn.trans (measure_mono (fun _ hω => ⟨hω.1.1, hω.2⟩)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CompactEntryLevels.lean b/lean/OAI/Probability/SLE/Mass/CompactEntryLevels.lean index 86809e29c12ec848584dea812c92b6134bf74c2f..1fcbc9246787e5d32ed41c904f9b567d3f267910 100644 --- a/lean/OAI/Probability/SLE/Mass/CompactEntryLevels.lean +++ b/lean/OAI/Probability/SLE/Mass/CompactEntryLevels.lean @@ -1,52 +1,52 @@ -import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels - -namespace OAI - -/-! Uniform control of the prescribed logarithmic levels on a compact -subset of the open half-plane. -/ -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge.Target - -lemma continuous_entryPhysicalLevel (κ : ℝ) (W : WienerLaw) (e : Entry) - {A : ℝ} (hA : 0 entryPhysicalLevel κ W e A z) := by - have hf : Continuous (fun z : upperHalfPlane => (2*(z:ℂ).im)/A) := by fun_prop - have hn (z : upperHalfPlane) : (2*(z:ℂ).im)/A ≠ 0 := - ne_of_gt (div_pos (mul_pos (by norm_num) z.property) hA) - exact ((hf.log hn).add continuous_const).add continuous_const - -lemma entryPhysicalLevel_bounded_on_compact (κ : ℝ) (W : WienerLaw) (e : Entry) - {A : ℝ} (hA : 0 hM ⟨z,hz,rfl⟩⟩ - -lemma eventually_entryPhysicalLevel_large_on_compact {κ : ℝ} (hκ : 0<κ) - (W : WienerLaw) {A : ℝ} (hA : 0 Real.exp (-(n:ℝ))) atTop (nhds 0) := - Real.tendsto_exp_neg_atTop_nhds_zero.comp tendsto_natCast_atTop_atTop - have hh : ∀ᶠ n : ℕ in atTop, Real.exp (-(n:ℝ)) < (z₀:ℂ).im := - he.eventually (gt_mem_nhds z₀.property) - obtain ⟨N,hN⟩ := eventually_atTop.mp - (hh.and (eventually_entryPhysicalLevel_large hκ W hA L)) - refine ⟨N,?_⟩ - intro e he z hz - exact (hN e.val.1 he).2 e.val.2 z - ((hN e.val.1 he).1.le.trans (hmin hz)) - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Gauge.PrescribedRadiusLevels + +namespace OAI + +/-! Uniform control of the prescribed logarithmic levels on a compact +subset of the open half-plane. -/ +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge.Target + +lemma continuous_entryPhysicalLevel (κ : ℝ) (W : WienerLaw) (e : Entry) + {A : ℝ} (hA : 0 entryPhysicalLevel κ W e A z) := by + have hf : Continuous (fun z : upperHalfPlane => (2*(z:ℂ).im)/A) := by fun_prop + have hn (z : upperHalfPlane) : (2*(z:ℂ).im)/A ≠ 0 := + ne_of_gt (div_pos (mul_pos (by norm_num) z.property) hA) + exact ((hf.log hn).add continuous_const).add continuous_const + +lemma entryPhysicalLevel_bounded_on_compact (κ : ℝ) (W : WienerLaw) (e : Entry) + {A : ℝ} (hA : 0 hM ⟨z,hz,rfl⟩⟩ + +lemma eventually_entryPhysicalLevel_large_on_compact {κ : ℝ} (hκ : 0<κ) + (W : WienerLaw) {A : ℝ} (hA : 0 Real.exp (-(n:ℝ))) atTop (nhds 0) := + Real.tendsto_exp_neg_atTop_nhds_zero.comp tendsto_natCast_atTop_atTop + have hh : ∀ᶠ n : ℕ in atTop, Real.exp (-(n:ℝ)) < (z₀:ℂ).im := + he.eventually (gt_mem_nhds z₀.property) + obtain ⟨N,hN⟩ := eventually_atTop.mp + (hh.and (eventually_entryPhysicalLevel_large hκ W hA L)) + refine ⟨N,?_⟩ + intro e he z hz + exact (hN e.val.1 he).2 e.val.2 z + ((hN e.val.1 he).1.le.trans (hmin hz)) + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CompactImageMassBounds.lean b/lean/OAI/Probability/SLE/Mass/CompactImageMassBounds.lean index 1a5513271993e04b4c6714621c132e0dbefa1577..aa8533e60fbf6875d984d18aa01b43ff45175ebf 100644 --- a/lean/OAI/Probability/SLE/Mass/CompactImageMassBounds.lean +++ b/lean/OAI/Probability/SLE/Mass/CompactImageMassBounds.lean @@ -1,31 +1,31 @@ -import OAI.Probability.SLE.Mass.WeightedImageRandomMass - -namespace OAI - -/-! Pointwise conformal weight bounds imply actual image-mass bounds. -/ - -noncomputable section -open Set MeasureTheory -open scoped ENNReal - -namespace SLEExactGauge - -lemma weightedImageMass_image_bounds - {X : Type*} [MeasurableSpace X] (J : upperHalfPlane → X) - (hJ : MeasurableEmbedding J) (μ : Measure upperHalfPlane) - (w v : upperHalfPlane → ℝ≥0∞) (K : Set upperHalfPlane) (hK : MeasurableSet K) - (c C : ℝ≥0∞) (hc : c ≠ ∞) (hC : C ≠ ∞) - (hlower : ∀ z∈K, c*v z ≤ w z) (hupper : ∀ z∈K, w z ≤ C*v z) : - c*(∫⁻ z in K, v z ∂μ) ≤ weightedImageMass J w μ (J '' K) ∧ - weightedImageMass J w μ (J '' K) ≤ C*(∫⁻ z in K, v z ∂μ) := by - have he : weightedImageMass J w μ (J '' K) = ∫⁻ z in K, w z ∂μ := by - rw [weightedImageMass,Measure.map_apply hJ.measurable (hJ.measurableSet_image.mpr hK), - Set.preimage_image_eq _ hJ.injective,withDensity_apply _ hK] - rw [he,← lintegral_const_mul' _ _ hc,← lintegral_const_mul' _ _ hC] - exact ⟨setLIntegral_mono' hK hlower,setLIntegral_mono' hK hupper⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.WeightedImageRandomMass + +namespace OAI + +/-! Pointwise conformal weight bounds imply actual image-mass bounds. -/ + +noncomputable section +open Set MeasureTheory +open scoped ENNReal + +namespace SLEExactGauge + +lemma weightedImageMass_image_bounds + {X : Type*} [MeasurableSpace X] (J : upperHalfPlane → X) + (hJ : MeasurableEmbedding J) (μ : Measure upperHalfPlane) + (w v : upperHalfPlane → ℝ≥0∞) (K : Set upperHalfPlane) (hK : MeasurableSet K) + (c C : ℝ≥0∞) (hc : c ≠ ∞) (hC : C ≠ ∞) + (hlower : ∀ z∈K, c*v z ≤ w z) (hupper : ∀ z∈K, w z ≤ C*v z) : + c*(∫⁻ z in K, v z ∂μ) ≤ weightedImageMass J w μ (J '' K) ∧ + weightedImageMass J w μ (J '' K) ≤ C*(∫⁻ z in K, v z ∂μ) := by + have he : weightedImageMass J w μ (J '' K) = ∫⁻ z in K, w z ∂μ := by + rw [weightedImageMass,Measure.map_apply hJ.measurable (hJ.measurableSet_image.mpr hK), + Set.preimage_image_eq _ hJ.injective,withDensity_apply _ hK] + rw [he,← lintegral_const_mul' _ _ hc,← lintegral_const_mul' _ _ hC] + exact ⟨setLIntegral_mono' hK hlower,setLIntegral_mono' hK hupper⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CompactLevelMoments.lean b/lean/OAI/Probability/SLE/Mass/CompactLevelMoments.lean index ec67dc7aa16046edf8cff491d9328611cc12150d..578f843a62a9e0e21b25ca6fb13219d3f8aa0e2b 100644 --- a/lean/OAI/Probability/SLE/Mass/CompactLevelMoments.lean +++ b/lean/OAI/Probability/SLE/Mass/CompactLevelMoments.lean @@ -1,201 +1,201 @@ -import OAI.Probability.SLE.Multipoint.MultipointCenteredWeights -import OAI.Probability.SLE.Multipoint.MultipointFixedCenterKernel - -namespace OAI - -/-! Integrating the actual multipoint bound on an interior compact set. -The bound is independent of the common radius level. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -lemma ae_tuple_mem {ι : Type*} [Fintype ι] (K : Set ℂ) (hK : MeasurableSet K) : - ∀ᵐ z : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), ∀ i, z i ∈ K := by - apply ae_all_iff.mpr - intro i - exact MeasureTheory.Measure.tendsto_eval_ae_ae.eventually (ae_restrict_mem hK) - -lemma compact_positive_radius_cap {K : Set ℂ} (hK : IsCompact K) - (hKH : K ⊆ upperHalfPlane) : ∃ b : ℝ, 0 < b ∧ ∀ z ∈ K, b ≤ 2*z.im := by - exact hK.exists_forall_le' (by fun_prop) (fun z hz => mul_pos (by norm_num) (hKH hz)) - -def compactLevelMomentBound (κ b : ℝ) (K : Set ℂ) (m : ℕ) : ℝ≥0∞ := - ENNReal.ofReal (multipointConstant κ m) * - ∫⁻ z : Fin m → ℂ, cappedDistanceKernel (codimension κ) b z - ∂Measure.pi (fun _ : Fin m => volume.restrict K) - -theorem levelMass_moment_le_cappedKernel - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ) - (u : ℝ) (hu : 0 ≤ u) : - (∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) ≤ compactLevelMomentBound κ b K m := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - rw [levelMass_moment_identity P hm hBc κ u hu γ G F hw K m] - unfold compactLevelMomentBound - rw [← lintegral_const_mul _ (measurable_cappedDistanceKernel _ _)] - apply lintegral_mono_ae - filter_upwards [ae_tuple_mem (ι := Fin m) K hK,ae_tuple_injective (ι := Fin m) K] with z hz hzi - have h := stoppedGreen_common_level_product_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw - hgrowth z (fun i => hKH (hz i)) hzi (cappedDistance b z) - (fun i => cap_pos hb hzi i) - (fun i => (cap_le_fixed b z i).trans (hbR (z i) (hz i))) - (fun i j hij => cap_le_dist b z i j hij) u - apply h.trans_eq - rw [ENNReal.ofReal_mul (by positivity)] - congr 1 - · simp only [multipointConstant,Fintype.card_fin] - · exact ENNReal.ofReal_prod_of_nonneg - (fun i _ => Real.rpow_nonneg (cap_pos hb hzi i).le _) - -lemma compactLevelMomentBound_lt_top {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (b : ℝ) (K : Set ℂ) (hK : Bornology.IsBounded K) (m : ℕ) : - compactLevelMomentBound κ b K m < ⊤ := by - apply ENNReal.mul_lt_top ENNReal.ofReal_lt_top - exact cappedDistanceKernel_integral_lt_top (codimension_pos κ hκ8) - (by unfold codimension dimension; linarith) b K hK - -theorem uniform_levelMass_compact_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) (m : ℕ) : - ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → - (∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) ≤ M := by - obtain ⟨b,hb,hbR⟩ := compact_positive_radius_cap hK hKH - exact ⟨compactLevelMomentBound κ b K m, - compactLevelMomentBound_lt_top hκ hκ8 b K hK.isBounded m, - fun u hu => levelMass_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw - hgrowth K hK.measurableSet hKH b hb hbR m u hu⟩ - -theorem uniform_levelApproximation_compact_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) (m : ℕ) : - ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, - (∫⁻ ω, (levelApproximation κ B γ G u t ω K) ^ m ∂P) ≤ M := by - obtain ⟨M,hM,hu⟩ := uniform_levelMass_compact_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth K hK hKH m - refine ⟨M,hM,fun u hu₀ t => ?_⟩ - apply le_trans (lintegral_mono (fun ω => ?_)) (hu u hu₀) - exact pow_le_pow_left' (levelApproximation_apply_le κ B γ G u t ω hK.measurableSet) m - -def centeredLevelMomentBound (κ b : ℝ) (z₀ : ℂ) (K : Set ℂ) (m : ℕ) : ℝ≥0∞ := - ENNReal.ofReal (multipointConstant κ (m+1)) * - ∫⁻ z : Fin m → ℂ, cappedDistanceKernel (codimension κ) b (Fin.cons z₀ z) - ∂Measure.pi (fun _ : Fin m => volume.restrict K) - -theorem levelMass_centered_moment_le_cappedKernel - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (z₀ : ℂ) (hz₀ : z₀ ∈ upperHalfPlane) (e₀ : ℝ) (he₀ : 0 < e₀) (heR : e₀ ≤ 2*z₀.im) - (b : ℝ) (hb : 0 < b) (hb₀ : b ≤ 2*z₀.im) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ) - (u : ℝ) (hu : 0 ≤ u) : - (∫⁻ ω in {ω | ∃ t : ℝ≥0, z₀ ∈ survivingDomain (γ ω) t ∧ - physicalConformalRadius (fun z : ℂ => z) (G ω) t z₀ ≤ e₀}, - levelMass κ B γ G u K ω ^ m ∂P) ≤ - ENNReal.ofReal (e₀^(codimension κ)) * centeredLevelMomentBound κ b z₀ K m := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let E := {ω | ∃ t : ℝ≥0, z₀ ∈ survivingDomain (γ ω) t ∧ - physicalConformalRadius (fun z : ℂ => z) (G ω) t z₀ ≤ e₀} - have hE : MeasurableSet E := measurableSet_physical_radius_hit hm hBc κ γ G F hw z₀ hz₀ e₀ he₀ heR - rw [levelMass_moment_identity (P.restrict E) hm hBc κ u hu γ G F hw K m] - have hbound : (∫⁻ z : Fin m → ℂ, ∫⁻ ω in E, - ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ*B t ω) - (γ ω) (G ω) (z i) u) ∂P ∂Measure.pi (fun _ : Fin m => volume.restrict K)) ≤ - (ENNReal.ofReal (multipointConstant κ (m+1)) * ENNReal.ofReal (e₀^codimension κ)) * - ∫⁻ z : Fin m → ℂ, cappedDistanceKernel (codimension κ) b (Fin.cons z₀ z) - ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by - have hc : Measurable (fun z : Fin m → ℂ => (Fin.cons z₀ z : Fin (m+1) → ℂ)) := by - apply Measurable.of_eval - intro i - exact Fin.cases measurable_const (fun j => measurable_pi_apply j) i - have hk : Measurable (fun z : Fin m → ℂ => cappedDistanceKernel (codimension κ) b - (Fin.cons z₀ z)) := by - simpa only [Function.comp_def] using - (measurable_cappedDistanceKernel (codimension κ) b).comp hc - rw [← lintegral_const_mul _ hk] - apply lintegral_mono_ae - filter_upwards [ae_tuple_mem (ι := Fin m) K hK,ae_tuple_injective (ι := Fin m) K, - ae_tuple_avoids (ι := Fin m) K z₀] with z hz hzi havoid - let z' : Fin (m+1) → ℂ := Fin.cons z₀ z - have hz' : ∀ i, z' i ∈ upperHalfPlane := Fin.cases hz₀ (fun i => hKH (hz i)) - have hzi' : Function.Injective z' := Fin.cons_injective_of_injective - (by rintro ⟨i,hi⟩; exact havoid i hi) hzi - have hbR' : ∀ i, b ≤ 2*(z' i).im := Fin.cases hb₀ (fun i => hbR (z i) (hz i)) - have h := stoppedGreen_centered_common_level_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw - hgrowth m z' hz' hzi' (cappedDistance b z') (fun i => cap_pos hb hzi' i) - (fun i => (cap_le_fixed b z' i).trans (hbR' i)) - (fun i j hij => cap_le_dist b z' i j hij) e₀ he₀ u - rw [← lintegral_indicator hE] - apply h.trans_eq - rw [ENNReal.ofReal_mul (mul_nonneg (multipointConstant_pos κ _).le - (Real.rpow_nonneg he₀.le _)), - ENNReal.ofReal_mul (multipointConstant_pos κ _).le] - congr 1 - exact ENNReal.ofReal_prod_of_nonneg - (fun i _ => Real.rpow_nonneg (cap_pos hb hzi' i).le _) - exact hbound.trans_eq (by unfold centeredLevelMomentBound; ring) - -theorem uniform_levelMass_centered_compact_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) - (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ) : - ∃ C : ℝ≥0∞, C < ⊤ ∧ ∀ z₀ : ℂ, z₀ ∈ upperHalfPlane → b ≤ 2*z₀.im → - ∀ e₀ : ℝ, 0 < e₀ → e₀ ≤ 2*z₀.im → ∀ u : ℝ, 0 ≤ u → - (∫⁻ ω in {ω | ∃ t : ℝ≥0, z₀ ∈ survivingDomain (γ ω) t ∧ - physicalConformalRadius (fun z : ℂ => z) (G ω) t z₀ ≤ e₀}, - levelMass κ B γ G u K ω ^ m ∂P) ≤ C * ENNReal.ofReal (e₀^codimension κ) := by - obtain ⟨D,hD,hbound⟩ := cappedDistanceKernel_fixed_center_bound m - (codimension_pos κ hκ8) (by unfold codimension dimension; linarith) b K hK.isBounded - refine ⟨ENNReal.ofReal (multipointConstant κ (m+1)) * D, - ENNReal.mul_lt_top ENNReal.ofReal_lt_top hD,?_⟩ - intro z₀ hz₀ hb₀ e₀ he₀ heR u hu - apply (levelMass_centered_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw - hgrowth K hK.measurableSet hKH z₀ hz₀ e₀ he₀ heR b hb hb₀ hbR m u hu).trans - unfold centeredLevelMomentBound - calc - _ ≤ ENNReal.ofReal (e₀^codimension κ) * - (ENNReal.ofReal (multipointConstant κ (m+1)) * D) := by gcongr; exact hbound z₀ - _ = _ := mul_comm _ _ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Multipoint.MultipointCenteredWeights +import OAI.Probability.SLE.Multipoint.MultipointFixedCenterKernel + +namespace OAI + +/-! Integrating the actual multipoint bound on an interior compact set. +The bound is independent of the common radius level. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +lemma ae_tuple_mem {ι : Type*} [Fintype ι] (K : Set ℂ) (hK : MeasurableSet K) : + ∀ᵐ z : ι → ℂ ∂Measure.pi (fun _ : ι => volume.restrict K), ∀ i, z i ∈ K := by + apply ae_all_iff.mpr + intro i + exact MeasureTheory.Measure.tendsto_eval_ae_ae.eventually (ae_restrict_mem hK) + +lemma compact_positive_radius_cap {K : Set ℂ} (hK : IsCompact K) + (hKH : K ⊆ upperHalfPlane) : ∃ b : ℝ, 0 < b ∧ ∀ z ∈ K, b ≤ 2*z.im := by + exact hK.exists_forall_le' (by fun_prop) (fun z hz => mul_pos (by norm_num) (hKH hz)) + +def compactLevelMomentBound (κ b : ℝ) (K : Set ℂ) (m : ℕ) : ℝ≥0∞ := + ENNReal.ofReal (multipointConstant κ m) * + ∫⁻ z : Fin m → ℂ, cappedDistanceKernel (codimension κ) b z + ∂Measure.pi (fun _ : Fin m => volume.restrict K) + +theorem levelMass_moment_le_cappedKernel + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ) + (u : ℝ) (hu : 0 ≤ u) : + (∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) ≤ compactLevelMomentBound κ b K m := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + rw [levelMass_moment_identity P hm hBc κ u hu γ G F hw K m] + unfold compactLevelMomentBound + rw [← lintegral_const_mul _ (measurable_cappedDistanceKernel _ _)] + apply lintegral_mono_ae + filter_upwards [ae_tuple_mem (ι := Fin m) K hK,ae_tuple_injective (ι := Fin m) K] with z hz hzi + have h := stoppedGreen_common_level_product_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw + hgrowth z (fun i => hKH (hz i)) hzi (cappedDistance b z) + (fun i => cap_pos hb hzi i) + (fun i => (cap_le_fixed b z i).trans (hbR (z i) (hz i))) + (fun i j hij => cap_le_dist b z i j hij) u + apply h.trans_eq + rw [ENNReal.ofReal_mul (by positivity)] + congr 1 + · simp only [multipointConstant,Fintype.card_fin] + · exact ENNReal.ofReal_prod_of_nonneg + (fun i _ => Real.rpow_nonneg (cap_pos hb hzi i).le _) + +lemma compactLevelMomentBound_lt_top {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (b : ℝ) (K : Set ℂ) (hK : Bornology.IsBounded K) (m : ℕ) : + compactLevelMomentBound κ b K m < ⊤ := by + apply ENNReal.mul_lt_top ENNReal.ofReal_lt_top + exact cappedDistanceKernel_integral_lt_top (codimension_pos κ hκ8) + (by unfold codimension dimension; linarith) b K hK + +theorem uniform_levelMass_compact_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) (m : ℕ) : + ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → + (∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) ≤ M := by + obtain ⟨b,hb,hbR⟩ := compact_positive_radius_cap hK hKH + exact ⟨compactLevelMomentBound κ b K m, + compactLevelMomentBound_lt_top hκ hκ8 b K hK.isBounded m, + fun u hu => levelMass_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw + hgrowth K hK.measurableSet hKH b hb hbR m u hu⟩ + +theorem uniform_levelApproximation_compact_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) (m : ℕ) : + ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, + (∫⁻ ω, (levelApproximation κ B γ G u t ω K) ^ m ∂P) ≤ M := by + obtain ⟨M,hM,hu⟩ := uniform_levelMass_compact_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth K hK hKH m + refine ⟨M,hM,fun u hu₀ t => ?_⟩ + apply le_trans (lintegral_mono (fun ω => ?_)) (hu u hu₀) + exact pow_le_pow_left' (levelApproximation_apply_le κ B γ G u t ω hK.measurableSet) m + +def centeredLevelMomentBound (κ b : ℝ) (z₀ : ℂ) (K : Set ℂ) (m : ℕ) : ℝ≥0∞ := + ENNReal.ofReal (multipointConstant κ (m+1)) * + ∫⁻ z : Fin m → ℂ, cappedDistanceKernel (codimension κ) b (Fin.cons z₀ z) + ∂Measure.pi (fun _ : Fin m => volume.restrict K) + +theorem levelMass_centered_moment_le_cappedKernel + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (z₀ : ℂ) (hz₀ : z₀ ∈ upperHalfPlane) (e₀ : ℝ) (he₀ : 0 < e₀) (heR : e₀ ≤ 2*z₀.im) + (b : ℝ) (hb : 0 < b) (hb₀ : b ≤ 2*z₀.im) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ) + (u : ℝ) (hu : 0 ≤ u) : + (∫⁻ ω in {ω | ∃ t : ℝ≥0, z₀ ∈ survivingDomain (γ ω) t ∧ + physicalConformalRadius (fun z : ℂ => z) (G ω) t z₀ ≤ e₀}, + levelMass κ B γ G u K ω ^ m ∂P) ≤ + ENNReal.ofReal (e₀^(codimension κ)) * centeredLevelMomentBound κ b z₀ K m := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let E := {ω | ∃ t : ℝ≥0, z₀ ∈ survivingDomain (γ ω) t ∧ + physicalConformalRadius (fun z : ℂ => z) (G ω) t z₀ ≤ e₀} + have hE : MeasurableSet E := measurableSet_physical_radius_hit hm hBc κ γ G F hw z₀ hz₀ e₀ he₀ heR + rw [levelMass_moment_identity (P.restrict E) hm hBc κ u hu γ G F hw K m] + have hbound : (∫⁻ z : Fin m → ℂ, ∫⁻ ω in E, + ENNReal.ofReal (∏ i, stoppedGreen κ (fun t => Real.sqrt κ*B t ω) + (γ ω) (G ω) (z i) u) ∂P ∂Measure.pi (fun _ : Fin m => volume.restrict K)) ≤ + (ENNReal.ofReal (multipointConstant κ (m+1)) * ENNReal.ofReal (e₀^codimension κ)) * + ∫⁻ z : Fin m → ℂ, cappedDistanceKernel (codimension κ) b (Fin.cons z₀ z) + ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by + have hc : Measurable (fun z : Fin m → ℂ => (Fin.cons z₀ z : Fin (m+1) → ℂ)) := by + apply Measurable.of_eval + intro i + exact Fin.cases measurable_const (fun j => measurable_pi_apply j) i + have hk : Measurable (fun z : Fin m → ℂ => cappedDistanceKernel (codimension κ) b + (Fin.cons z₀ z)) := by + simpa only [Function.comp_def] using + (measurable_cappedDistanceKernel (codimension κ) b).comp hc + rw [← lintegral_const_mul _ hk] + apply lintegral_mono_ae + filter_upwards [ae_tuple_mem (ι := Fin m) K hK,ae_tuple_injective (ι := Fin m) K, + ae_tuple_avoids (ι := Fin m) K z₀] with z hz hzi havoid + let z' : Fin (m+1) → ℂ := Fin.cons z₀ z + have hz' : ∀ i, z' i ∈ upperHalfPlane := Fin.cases hz₀ (fun i => hKH (hz i)) + have hzi' : Function.Injective z' := Fin.cons_injective_of_injective + (by rintro ⟨i,hi⟩; exact havoid i hi) hzi + have hbR' : ∀ i, b ≤ 2*(z' i).im := Fin.cases hb₀ (fun i => hbR (z i) (hz i)) + have h := stoppedGreen_centered_common_level_bound hB hm hBm hBc hB0 hκ hκ8 γ G F hw + hgrowth m z' hz' hzi' (cappedDistance b z') (fun i => cap_pos hb hzi' i) + (fun i => (cap_le_fixed b z' i).trans (hbR' i)) + (fun i j hij => cap_le_dist b z' i j hij) e₀ he₀ u + rw [← lintegral_indicator hE] + apply h.trans_eq + rw [ENNReal.ofReal_mul (mul_nonneg (multipointConstant_pos κ _).le + (Real.rpow_nonneg he₀.le _)), + ENNReal.ofReal_mul (multipointConstant_pos κ _).le] + congr 1 + exact ENNReal.ofReal_prod_of_nonneg + (fun i _ => Real.rpow_nonneg (cap_pos hb hzi' i).le _) + exact hbound.trans_eq (by unfold centeredLevelMomentBound; ring) + +theorem uniform_levelMass_centered_compact_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) + (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) (m : ℕ) : + ∃ C : ℝ≥0∞, C < ⊤ ∧ ∀ z₀ : ℂ, z₀ ∈ upperHalfPlane → b ≤ 2*z₀.im → + ∀ e₀ : ℝ, 0 < e₀ → e₀ ≤ 2*z₀.im → ∀ u : ℝ, 0 ≤ u → + (∫⁻ ω in {ω | ∃ t : ℝ≥0, z₀ ∈ survivingDomain (γ ω) t ∧ + physicalConformalRadius (fun z : ℂ => z) (G ω) t z₀ ≤ e₀}, + levelMass κ B γ G u K ω ^ m ∂P) ≤ C * ENNReal.ofReal (e₀^codimension κ) := by + obtain ⟨D,hD,hbound⟩ := cappedDistanceKernel_fixed_center_bound m + (codimension_pos κ hκ8) (by unfold codimension dimension; linarith) b K hK.isBounded + refine ⟨ENNReal.ofReal (multipointConstant κ (m+1)) * D, + ENNReal.mul_lt_top ENNReal.ofReal_lt_top hD,?_⟩ + intro z₀ hz₀ hb₀ e₀ he₀ heR u hu + apply (levelMass_centered_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw + hgrowth K hK.measurableSet hKH z₀ hz₀ e₀ he₀ heR b hb hb₀ hbR m u hu).trans + unfold centeredLevelMomentBound + calc + _ ≤ ENNReal.ofReal (e₀^codimension κ) * + (ENNReal.ofReal (multipointConstant κ (m+1)) * D) := by gcongr; exact hbound z₀ + _ = _ := mul_comm _ _ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CompactSpatialMass.lean b/lean/OAI/Probability/SLE/Mass/CompactSpatialMass.lean index 056d3fe35f8278a295017e4be62f975705118a28..beafcd9b4f14d5501acd049f2ba10f83dd2d67b7 100644 --- a/lean/OAI/Probability/SLE/Mass/CompactSpatialMass.lean +++ b/lean/OAI/Probability/SLE/Mass/CompactSpatialMass.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Mass.CanonicalMassConstruction - -namespace OAI - -/-! Compact restrictions of half-plane measures, viewed as physical measures. -These preserve measurable dependence and compact finiteness. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -/-- The physical measure carried by a fixed compact part of the half-plane. -/ -def compactSpatialMass (μ : Measure upperHalfPlane) (K : Set upperHalfPlane) : Measure ℂ := - (μ.restrict K).map (Subtype.val : upperHalfPlane → ℂ) - -lemma compactSpatialMass_apply (μ : Measure upperHalfPlane) (K : Set upperHalfPlane) - (A : Set ℂ) (hA : MeasurableSet A) : - compactSpatialMass μ K A = μ (K ∩ Subtype.val ⁻¹' A) := by - rw [compactSpatialMass,Measure.map_apply measurable_subtype_coe hA, - Measure.restrict_apply (hA.preimage measurable_subtype_coe),inter_comm] - -lemma compactSpatialMass_measurable {Ω : Type*} [MeasurableSpace Ω] - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) - (K : Set upperHalfPlane) (hK : MeasurableSet K) : - Measurable (fun ω => compactSpatialMass (μ ω) K) := by - apply Measure.measurable_of_measurable_coe - intro A hA - simp_rw [compactSpatialMass_apply _ _ _ hA] - exact (Measure.measurable_coe (hK.inter (hA.preimage measurable_subtype_coe))).comp hμ - -lemma compactSpatialMass_finite (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] - (K : Set upperHalfPlane) (hK : IsCompact K) : IsFiniteMeasure (compactSpatialMass μ K) := by - constructor - rw [compactSpatialMass_apply _ _ _ MeasurableSet.univ] - simpa only [preimage_univ,inter_univ] using hK.measure_lt_top - -lemma compactSpatialMass_compact_cell (K : Set upperHalfPlane) (hK : IsCompact K) - (z : ℂ) (r : ℝ) : IsCompact (K ∩ Subtype.val ⁻¹' closedBall z r) := - hK.inter_right (isClosed_closedBall.preimage continuous_subtype_val) - -namespace Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma compact_spatial_terminal_measurable (K : Set upperHalfPlane) (hK : IsCompact K) : - Measurable (fun ω => compactSpatialMass (C.rational.terminal ω) K) := - compactSpatialMass_measurable _ C.rational.terminal_measurable K hK.measurableSet - -lemma compact_spatial_terminal_finite (K : Set upperHalfPlane) (hK : IsCompact K) (ω : Ω) : - IsFiniteMeasure (compactSpatialMass (C.rational.terminal ω) K) := by - let := (C.rational.terminal_radon ω).2 - exact compactSpatialMass_finite _ K hK - -lemma compact_spatial_initial_finite (K : Set upperHalfPlane) (hK : IsCompact K) : - IsFiniteMeasure (compactSpatialMass (initialGreenMeasure κ) K) := by - let := initialGreenMeasure_finite_on_compacts κ - exact compactSpatialMass_finite _ K hK - -lemma compact_spatial_terminal_mean (K : Set upperHalfPlane) (hK : IsCompact K) - (A : Set ℂ) (hA : MeasurableSet A) : - (∫⁻ ω, compactSpatialMass (C.rational.terminal ω) K A ∂P) = - compactSpatialMass (initialGreenMeasure κ) K A := by - simp_rw [compactSpatialMass_apply _ _ _ hA] - rw [← Measure.bind_apply (hK.measurableSet.inter (hA.preimage measurable_subtype_coe)) - C.rational.terminal_measurable.aemeasurable,C.rational.mean] - -end Multipoint.CanonicalMass -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassConstruction + +namespace OAI + +/-! Compact restrictions of half-plane measures, viewed as physical measures. +These preserve measurable dependence and compact finiteness. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +/-- The physical measure carried by a fixed compact part of the half-plane. -/ +def compactSpatialMass (μ : Measure upperHalfPlane) (K : Set upperHalfPlane) : Measure ℂ := + (μ.restrict K).map (Subtype.val : upperHalfPlane → ℂ) + +lemma compactSpatialMass_apply (μ : Measure upperHalfPlane) (K : Set upperHalfPlane) + (A : Set ℂ) (hA : MeasurableSet A) : + compactSpatialMass μ K A = μ (K ∩ Subtype.val ⁻¹' A) := by + rw [compactSpatialMass,Measure.map_apply measurable_subtype_coe hA, + Measure.restrict_apply (hA.preimage measurable_subtype_coe),inter_comm] + +lemma compactSpatialMass_measurable {Ω : Type*} [MeasurableSpace Ω] + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) + (K : Set upperHalfPlane) (hK : MeasurableSet K) : + Measurable (fun ω => compactSpatialMass (μ ω) K) := by + apply Measure.measurable_of_measurable_coe + intro A hA + simp_rw [compactSpatialMass_apply _ _ _ hA] + exact (Measure.measurable_coe (hK.inter (hA.preimage measurable_subtype_coe))).comp hμ + +lemma compactSpatialMass_finite (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] + (K : Set upperHalfPlane) (hK : IsCompact K) : IsFiniteMeasure (compactSpatialMass μ K) := by + constructor + rw [compactSpatialMass_apply _ _ _ MeasurableSet.univ] + simpa only [preimage_univ,inter_univ] using hK.measure_lt_top + +lemma compactSpatialMass_compact_cell (K : Set upperHalfPlane) (hK : IsCompact K) + (z : ℂ) (r : ℝ) : IsCompact (K ∩ Subtype.val ⁻¹' closedBall z r) := + hK.inter_right (isClosed_closedBall.preimage continuous_subtype_val) + +namespace Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma compact_spatial_terminal_measurable (K : Set upperHalfPlane) (hK : IsCompact K) : + Measurable (fun ω => compactSpatialMass (C.rational.terminal ω) K) := + compactSpatialMass_measurable _ C.rational.terminal_measurable K hK.measurableSet + +lemma compact_spatial_terminal_finite (K : Set upperHalfPlane) (hK : IsCompact K) (ω : Ω) : + IsFiniteMeasure (compactSpatialMass (C.rational.terminal ω) K) := by + let := (C.rational.terminal_radon ω).2 + exact compactSpatialMass_finite _ K hK + +lemma compact_spatial_initial_finite (K : Set upperHalfPlane) (hK : IsCompact K) : + IsFiniteMeasure (compactSpatialMass (initialGreenMeasure κ) K) := by + let := initialGreenMeasure_finite_on_compacts κ + exact compactSpatialMass_finite _ K hK + +lemma compact_spatial_terminal_mean (K : Set upperHalfPlane) (hK : IsCompact K) + (A : Set ℂ) (hA : MeasurableSet A) : + (∫⁻ ω, compactSpatialMass (C.rational.terminal ω) K A ∂P) = + compactSpatialMass (initialGreenMeasure κ) K A := by + simp_rw [compactSpatialMass_apply _ _ _ hA] + rw [← Measure.bind_apply (hK.measurableSet.inter (hA.preimage measurable_subtype_coe)) + C.rational.terminal_measurable.aemeasurable,C.rational.mean] + +end Multipoint.CanonicalMass +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/ConditionalRadonRandomTests.lean b/lean/OAI/Probability/SLE/Mass/ConditionalRadonRandomTests.lean index 76c328c520760f0456d516d15aa1f3b323da5343..905c860c73f36050ffff5f221eb4f5a10271e5b7 100644 --- a/lean/OAI/Probability/SLE/Mass/ConditionalRadonRandomTests.lean +++ b/lean/OAI/Probability/SLE/Mass/ConditionalRadonRandomTests.lean @@ -1,123 +1,123 @@ -import OAI.Probability.SLE.Mass.RadonRandomTesting -import OAI.Probability.SLE.Mass.RadonTestFamily - -namespace OAI - -/-! Random spatial testing for conditional kernels of locally finite -half-plane measures, allowing exceptional non-locally-finite values. -/ - -noncomputable section -open Set Filter MeasureTheory Classical -open scoped ENNReal - -namespace SLEExactGauge - -local instance conditionalTestLocallyCompact : LocallyCompactSpace upperHalfPlane := - isOpen_upperHalfPlane.locallyCompactSpace - -private lemma locallyFiniteKernelVersion - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) - (hfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (μ ω)) : - ∃ ρ : Ω → Measure upperHalfPlane, Measurable ρ ∧ - (∀ ω, IsFiniteMeasureOnCompacts (ρ ω)) ∧ ρ =ᵐ[P] μ := by - let D := localTestFamily upperHalfPlane - let E := {ω | ∀ n, μ ω (D.compactSet n) < ∞} - have hE : MeasurableSet E := by - simp only [E,Set.ofPred_forall] - apply MeasurableSet.iInter - intro n - simpa only [Function.comp_def] using measurableSet_lt - ((Measure.measurable_coe (D.isCompact n).measurableSet).comp hμ) - (measurable_const : Measurable (fun _ : Ω => (∞:ℝ≥0∞))) - let ρ (ω : Ω) := if ω ∈ E then μ ω else 0 - refine ⟨ρ,hμ.ite hE measurable_const,?_,?_⟩ - · intro ω - by_cases hω : ω ∈ E - · have he : ρ ω=μ ω := ite_eq_left hω - rw [he] - constructor - intro K hK - obtain ⟨n,hn⟩ := D.exhaustive K hK - exact lt_of_le_of_lt (measure_mono hn) (hω n) - · simp only [ρ,ite_eq_right hω] - infer_instance - · filter_upwards [hfin] with ω hω - have : ω ∈ E := fun n => by let := hω; exact (D.isCompact n).measure_lt_top - exact ite_eq_left this - -lemma ae_finite_on_compacts_of_compact_mean - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) - (hmean : ∀ K : Set upperHalfPlane, IsCompact K → P.bind μ K < ∞) : - ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (μ ω) := by - let D := localTestFamily upperHalfPlane - have hfin (n : ℕ) : ∀ᵐ ω ∂P, μ ω (D.compactSet n) < ∞ := by - apply ae_lt_top ((Measure.measurable_coe (D.isCompact n).measurableSet).comp hμ) - change (∫⁻ ω, μ ω (D.compactSet n) ∂P) ≠ ∞ - rw [← Measure.bind_apply (D.isCompact n).measurableSet hμ.aemeasurable] - exact (hmean _ (D.isCompact n)).ne - filter_upwards [ae_all_iff.mpr hfin] with ω hω - constructor - intro K hK - obtain ⟨n,hn⟩ := D.exhaustive K hK - exact lt_of_le_of_lt (measure_mono hn) (hω n) - -/-- Equality of conditional mean measures gives arbitrary jointly -past-measurable tests. Only compact mean masses need be finite. -/ -theorem conditional_halfPlane_random_test - {Ω : Type*} [mΩ : MeasurableSpace Ω] - (m : MeasurableSpace Ω) (hm : m ≤ mΩ) (P : @Measure Ω mΩ) - (μ ν : Ω → Measure upperHalfPlane) (hμ : Measurable[mΩ] μ) (hν : Measurable[mΩ] ν) - (hμfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (μ ω)) - (hνfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (ν ω)) - (hmean : ∀ K : Set upperHalfPlane, IsCompact K → P.bind μ K < ∞) - (hevent : ∀ A : Set Ω, MeasurableSet[m] A → - (P.restrict A).bind μ=(P.restrict A).bind ν) - (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable[m.prod inferInstance] f) : - (∫⁻ ω, ∫⁻ z, f (ω,z) ∂μ ω ∂P) = ∫⁻ ω, ∫⁻ z, f (ω,z) ∂ν ω ∂P := by - let : MeasurableSpace Ω := mΩ - obtain ⟨ρ,hρ,hρfin,hρeq⟩ := locallyFiniteKernelVersion P μ hμ hμfin - obtain ⟨σ,hσ,hσfin,hσeq⟩ := locallyFiniteKernelVersion P ν hν hνfin - let D := localTestFamily upperHalfPlane - have hE (A : Set Ω) (hA : MeasurableSet[m] A) : - (P.restrict A).bind ρ=(P.restrict A).bind σ := by - rw [Measure.bind_congr_right (ae_restrict_of_ae hρeq), - Measure.bind_congr_right (ae_restrict_of_ae hσeq),hevent A hA] - have hcov : ⋃ n, disjointed D.compactSet n=univ := by - rw [iUnion_disjointed] - apply eq_univ_iff_forall.mpr - intro z - obtain ⟨n,hn⟩ := D.exhaustive {z} isCompact_singleton - exact mem_iUnion.mpr ⟨n,hn (mem_singleton z)⟩ - have hρK (n : ℕ) (ω : Ω) : ρ ω (disjointed D.compactSet n) < ∞ := by - let := hρfin ω - exact lt_of_le_of_lt (measure_mono (disjointed_subset D.compactSet n)) - (D.isCompact n).measure_lt_top - have hσK (n : ℕ) (ω : Ω) : σ ω (disjointed D.compactSet n) < ∞ := by - let := hσfin ω - exact lt_of_le_of_lt (measure_mono (disjointed_subset D.compactSet n)) - (D.isCompact n).measure_lt_top - have hmeanρ (n : ℕ) : P.bind ρ (disjointed D.compactSet n) < ∞ := by - rw [Measure.bind_congr_right hρeq] - exact lt_of_le_of_lt (measure_mono (disjointed_subset D.compactSet n)) - (hmean _ (D.isCompact n)) - have h := conditional_random_spatial_test m hm P ρ σ hρ hσ - (disjointed D.compactSet) (MeasurableSet.disjointed fun n => (D.isCompact n).measurableSet) - (disjoint_disjointed D.compactSet) hcov hρK hσK hmeanρ hE f hf - calc - _ = ∫⁻ ω, ∫⁻ z, f (ω,z) ∂ρ ω ∂P := by - apply lintegral_congr_ae - filter_upwards [hρeq] with ω hω - rw [hω] - _ = ∫⁻ ω, ∫⁻ z, f (ω,z) ∂σ ω ∂P := h - _ = _ := by - apply lintegral_congr_ae - filter_upwards [hσeq] with ω hω - rw [hω] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonRandomTesting +import OAI.Probability.SLE.Mass.RadonTestFamily + +namespace OAI + +/-! Random spatial testing for conditional kernels of locally finite +half-plane measures, allowing exceptional non-locally-finite values. -/ + +noncomputable section +open Set Filter MeasureTheory Classical +open scoped ENNReal + +namespace SLEExactGauge + +local instance conditionalTestLocallyCompact : LocallyCompactSpace upperHalfPlane := + isOpen_upperHalfPlane.locallyCompactSpace + +private lemma locallyFiniteKernelVersion + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) + (hfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (μ ω)) : + ∃ ρ : Ω → Measure upperHalfPlane, Measurable ρ ∧ + (∀ ω, IsFiniteMeasureOnCompacts (ρ ω)) ∧ ρ =ᵐ[P] μ := by + let D := localTestFamily upperHalfPlane + let E := {ω | ∀ n, μ ω (D.compactSet n) < ∞} + have hE : MeasurableSet E := by + simp only [E,Set.ofPred_forall] + apply MeasurableSet.iInter + intro n + simpa only [Function.comp_def] using measurableSet_lt + ((Measure.measurable_coe (D.isCompact n).measurableSet).comp hμ) + (measurable_const : Measurable (fun _ : Ω => (∞:ℝ≥0∞))) + let ρ (ω : Ω) := if ω ∈ E then μ ω else 0 + refine ⟨ρ,hμ.ite hE measurable_const,?_,?_⟩ + · intro ω + by_cases hω : ω ∈ E + · have he : ρ ω=μ ω := ite_eq_left hω + rw [he] + constructor + intro K hK + obtain ⟨n,hn⟩ := D.exhaustive K hK + exact lt_of_le_of_lt (measure_mono hn) (hω n) + · simp only [ρ,ite_eq_right hω] + infer_instance + · filter_upwards [hfin] with ω hω + have : ω ∈ E := fun n => by let := hω; exact (D.isCompact n).measure_lt_top + exact ite_eq_left this + +lemma ae_finite_on_compacts_of_compact_mean + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) + (hmean : ∀ K : Set upperHalfPlane, IsCompact K → P.bind μ K < ∞) : + ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (μ ω) := by + let D := localTestFamily upperHalfPlane + have hfin (n : ℕ) : ∀ᵐ ω ∂P, μ ω (D.compactSet n) < ∞ := by + apply ae_lt_top ((Measure.measurable_coe (D.isCompact n).measurableSet).comp hμ) + change (∫⁻ ω, μ ω (D.compactSet n) ∂P) ≠ ∞ + rw [← Measure.bind_apply (D.isCompact n).measurableSet hμ.aemeasurable] + exact (hmean _ (D.isCompact n)).ne + filter_upwards [ae_all_iff.mpr hfin] with ω hω + constructor + intro K hK + obtain ⟨n,hn⟩ := D.exhaustive K hK + exact lt_of_le_of_lt (measure_mono hn) (hω n) + +/-- Equality of conditional mean measures gives arbitrary jointly +past-measurable tests. Only compact mean masses need be finite. -/ +theorem conditional_halfPlane_random_test + {Ω : Type*} [mΩ : MeasurableSpace Ω] + (m : MeasurableSpace Ω) (hm : m ≤ mΩ) (P : @Measure Ω mΩ) + (μ ν : Ω → Measure upperHalfPlane) (hμ : Measurable[mΩ] μ) (hν : Measurable[mΩ] ν) + (hμfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (μ ω)) + (hνfin : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (ν ω)) + (hmean : ∀ K : Set upperHalfPlane, IsCompact K → P.bind μ K < ∞) + (hevent : ∀ A : Set Ω, MeasurableSet[m] A → + (P.restrict A).bind μ=(P.restrict A).bind ν) + (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable[m.prod inferInstance] f) : + (∫⁻ ω, ∫⁻ z, f (ω,z) ∂μ ω ∂P) = ∫⁻ ω, ∫⁻ z, f (ω,z) ∂ν ω ∂P := by + let : MeasurableSpace Ω := mΩ + obtain ⟨ρ,hρ,hρfin,hρeq⟩ := locallyFiniteKernelVersion P μ hμ hμfin + obtain ⟨σ,hσ,hσfin,hσeq⟩ := locallyFiniteKernelVersion P ν hν hνfin + let D := localTestFamily upperHalfPlane + have hE (A : Set Ω) (hA : MeasurableSet[m] A) : + (P.restrict A).bind ρ=(P.restrict A).bind σ := by + rw [Measure.bind_congr_right (ae_restrict_of_ae hρeq), + Measure.bind_congr_right (ae_restrict_of_ae hσeq),hevent A hA] + have hcov : ⋃ n, disjointed D.compactSet n=univ := by + rw [iUnion_disjointed] + apply eq_univ_iff_forall.mpr + intro z + obtain ⟨n,hn⟩ := D.exhaustive {z} isCompact_singleton + exact mem_iUnion.mpr ⟨n,hn (mem_singleton z)⟩ + have hρK (n : ℕ) (ω : Ω) : ρ ω (disjointed D.compactSet n) < ∞ := by + let := hρfin ω + exact lt_of_le_of_lt (measure_mono (disjointed_subset D.compactSet n)) + (D.isCompact n).measure_lt_top + have hσK (n : ℕ) (ω : Ω) : σ ω (disjointed D.compactSet n) < ∞ := by + let := hσfin ω + exact lt_of_le_of_lt (measure_mono (disjointed_subset D.compactSet n)) + (D.isCompact n).measure_lt_top + have hmeanρ (n : ℕ) : P.bind ρ (disjointed D.compactSet n) < ∞ := by + rw [Measure.bind_congr_right hρeq] + exact lt_of_le_of_lt (measure_mono (disjointed_subset D.compactSet n)) + (hmean _ (D.isCompact n)) + have h := conditional_random_spatial_test m hm P ρ σ hρ hσ + (disjointed D.compactSet) (MeasurableSet.disjointed fun n => (D.isCompact n).measurableSet) + (disjoint_disjointed D.compactSet) hcov hρK hσK hmeanρ hE f hf + calc + _ = ∫⁻ ω, ∫⁻ z, f (ω,z) ∂ρ ω ∂P := by + apply lintegral_congr_ae + filter_upwards [hρeq] with ω hω + rw [hω] + _ = ∫⁻ ω, ∫⁻ z, f (ω,z) ∂σ ω ∂P := h + _ = _ := by + apply lintegral_congr_ae + filter_upwards [hσeq] with ω hω + rw [hω] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/ConformalMassTransport.lean b/lean/OAI/Probability/SLE/Mass/ConformalMassTransport.lean index 11f7059c86b0dd19aaff61b54a039a525642745a..eb1bc878c32d44327a308317d809839164c650e8 100644 --- a/lean/OAI/Probability/SLE/Mass/ConformalMassTransport.lean +++ b/lean/OAI/Probability/SLE/Mass/ConformalMassTransport.lean @@ -1,129 +1,129 @@ -import OAI.Probability.SLE.Mass.CanonicalMassAllVisits -import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap - -namespace OAI - -/-! The weighted pushforward used for canonical mass in a conformal -image. Compact tests pull back with the same deterministic convex sequence; -the construction does not extract a new subsequence for the image domain. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -variable {D : Set ℂ} - -/-- For a conformal identification, `w` is the continuous weight `|J'|^d`. -/ -def conformalMass (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane, ℝ≥0)) - (μ : Measure upperHalfPlane) : Measure D := - (μ.withDensity (fun z => (w z : ℝ≥0∞))).map J - -def conformalMassTest (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane, ℝ≥0)) - (f : C_c(D, ℝ)) : C_c(upperHalfPlane, ℝ) := - (⟨fun z => (w z : ℝ), continuous_subtype_val.comp w.continuous⟩ : - C(upperHalfPlane, ℝ)) • f.comp J.toCocompactMap - -@[simp] lemma conformalMassTest_apply (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (f : C_c(D, ℝ)) (z : upperHalfPlane) : - conformalMassTest J w f z = (w z : ℝ) * f (J z) := rfl - -lemma conformalMass_integral (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) (f : C_c(D, ℝ)) : - (∫ z, f z ∂conformalMass J w μ) = ∫ z, conformalMassTest J w f z ∂μ := by - rw [conformalMass,integral_map_of_stronglyMeasurable J.continuous.measurable - (show StronglyMeasurable (fun z => f z) from f.continuous.stronglyMeasurable), - integral_withDensity_eq_integral_smul w.continuous.measurable] - rfl - -lemma conformalMass_radon (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) - [IsFiniteMeasureOnCompacts μ] : - (conformalMass J w μ).Regular ∧ IsFiniteMeasureOnCompacts (conformalMass J w μ) := by - let : IsLocallyFiniteMeasure (μ.withDensity (fun z => (w z : ℝ≥0∞))) := - IsLocallyFiniteMeasure.withDensity_coe w.continuous - have : (μ.withDensity (fun z => (w z : ℝ≥0∞))).Regular := inferInstance - exact ⟨Measure.Regular.map J,Measure.IsFiniteMeasureOnCompacts.map _ J⟩ - -lemma conformalMass_measurable (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) : Measurable (conformalMass J w) := by - apply (Measure.measurable_map J J.continuous.measurable).comp - apply Measure.measurable_of_measurable_coe - intro A hA - simp only [withDensity_apply _ hA] - simp_rw [← lintegral_indicator hA] - exact Measure.measurable_lintegral - ((w.continuous.measurable.coe_nnreal_ennreal).indicator hA) - -@[simp] lemma conformalMass_zero (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) : conformalMass J w 0 = 0 := by - simp [conformalMass] - -lemma conformalMass_add (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (μ ν : Measure upperHalfPlane) : - conformalMass J w (μ+ν) = conformalMass J w μ + conformalMass J w ν := by - simp only [conformalMass,withDensity_add_measure] - exact Measure.map_add _ _ J.continuous.measurable - -lemma conformalMass_mono (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) {μ ν : Measure upperHalfPlane} (h : μ ≤ ν) : - conformalMass J w μ ≤ conformalMass J w ν := by - apply Measure.map_mono ?_ J.continuous.measurable - apply Measure.le_iff.mpr - intro A hA - simp only [withDensity_apply _ hA] - exact lintegral_mono' (Measure.restrict_mono le_rfl h) le_rfl - -lemma conformalMass_nullSingleton (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) [NullSingletonClass μ] : - NullSingletonClass (conformalMass J w μ) := by - constructor - intro z - rw [conformalMass,Measure.map_apply J.continuous.measurable (measurableSet_singleton z)] - rw [show J ⁻¹' {z} = {J.symm z} by - ext x - simp only [mem_preimage,mem_singleton_iff] - constructor - · intro h - exact J.injective (h.trans (J.apply_symm_apply z).symm) - · intro h - rw [h,J.apply_symm_apply]] - exact measure_singleton _ - -lemma conformalMass_carried (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) - (S : Set upperHalfPlane) (hS : MeasurableSet S) (hμ : μ Sᶜ = 0) : - conformalMass J w μ (J '' S)ᶜ = 0 := by - rw [conformalMass,Measure.map_apply J.continuous.measurable - (J.measurableEmbedding.measurableSet_image.mpr hS).compl] - apply withDensity_absolutelyContinuous μ _ - simpa only [preimage_compl,Set.preimage_image_eq _ J.injective] using hμ - -/-- Vague convergence transfers for every image-domain test on the same -event and with the same sequence. -/ -lemma conformalMass_test_tendsto (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (μ : ℕ → Measure upperHalfPlane) - (ν : Measure upperHalfPlane) - (h : ∀ f : C_c(upperHalfPlane, ℝ), - Tendsto (fun n => ∫ z, f z ∂μ n) atTop (𝓝 (∫ z, f z ∂ν))) - (f : C_c(D, ℝ)) : - Tendsto (fun n => ∫ z, f z ∂conformalMass J w (μ n)) atTop - (𝓝 (∫ z, f z ∂conformalMass J w ν)) := by - simp only [conformalMass_integral] - exact h (conformalMassTest J w f) - -lemma conformalMass_test_continuous (J : upperHalfPlane ≃ₜ D) - (w : C(upperHalfPlane, ℝ≥0)) (μ : ℝ≥0 → Measure upperHalfPlane) - (h : ∀ f : C_c(upperHalfPlane, ℝ), Continuous (fun t => ∫ z, f z ∂μ t)) - (f : C_c(D, ℝ)) : Continuous (fun t => ∫ z, f z ∂conformalMass J w (μ t)) := by - simp only [conformalMass_integral] - exact h (conformalMassTest J w f) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassAllVisits +import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap + +namespace OAI + +/-! The weighted pushforward used for canonical mass in a conformal +image. Compact tests pull back with the same deterministic convex sequence; +the construction does not extract a new subsequence for the image domain. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +variable {D : Set ℂ} + +/-- For a conformal identification, `w` is the continuous weight `|J'|^d`. -/ +def conformalMass (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane, ℝ≥0)) + (μ : Measure upperHalfPlane) : Measure D := + (μ.withDensity (fun z => (w z : ℝ≥0∞))).map J + +def conformalMassTest (J : upperHalfPlane ≃ₜ D) (w : C(upperHalfPlane, ℝ≥0)) + (f : C_c(D, ℝ)) : C_c(upperHalfPlane, ℝ) := + (⟨fun z => (w z : ℝ), continuous_subtype_val.comp w.continuous⟩ : + C(upperHalfPlane, ℝ)) • f.comp J.toCocompactMap + +@[simp] lemma conformalMassTest_apply (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (f : C_c(D, ℝ)) (z : upperHalfPlane) : + conformalMassTest J w f z = (w z : ℝ) * f (J z) := rfl + +lemma conformalMass_integral (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) (f : C_c(D, ℝ)) : + (∫ z, f z ∂conformalMass J w μ) = ∫ z, conformalMassTest J w f z ∂μ := by + rw [conformalMass,integral_map_of_stronglyMeasurable J.continuous.measurable + (show StronglyMeasurable (fun z => f z) from f.continuous.stronglyMeasurable), + integral_withDensity_eq_integral_smul w.continuous.measurable] + rfl + +lemma conformalMass_radon (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) + [IsFiniteMeasureOnCompacts μ] : + (conformalMass J w μ).Regular ∧ IsFiniteMeasureOnCompacts (conformalMass J w μ) := by + let : IsLocallyFiniteMeasure (μ.withDensity (fun z => (w z : ℝ≥0∞))) := + IsLocallyFiniteMeasure.withDensity_coe w.continuous + have : (μ.withDensity (fun z => (w z : ℝ≥0∞))).Regular := inferInstance + exact ⟨Measure.Regular.map J,Measure.IsFiniteMeasureOnCompacts.map _ J⟩ + +lemma conformalMass_measurable (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) : Measurable (conformalMass J w) := by + apply (Measure.measurable_map J J.continuous.measurable).comp + apply Measure.measurable_of_measurable_coe + intro A hA + simp only [withDensity_apply _ hA] + simp_rw [← lintegral_indicator hA] + exact Measure.measurable_lintegral + ((w.continuous.measurable.coe_nnreal_ennreal).indicator hA) + +@[simp] lemma conformalMass_zero (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) : conformalMass J w 0 = 0 := by + simp [conformalMass] + +lemma conformalMass_add (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (μ ν : Measure upperHalfPlane) : + conformalMass J w (μ+ν) = conformalMass J w μ + conformalMass J w ν := by + simp only [conformalMass,withDensity_add_measure] + exact Measure.map_add _ _ J.continuous.measurable + +lemma conformalMass_mono (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) {μ ν : Measure upperHalfPlane} (h : μ ≤ ν) : + conformalMass J w μ ≤ conformalMass J w ν := by + apply Measure.map_mono ?_ J.continuous.measurable + apply Measure.le_iff.mpr + intro A hA + simp only [withDensity_apply _ hA] + exact lintegral_mono' (Measure.restrict_mono le_rfl h) le_rfl + +lemma conformalMass_nullSingleton (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) [NullSingletonClass μ] : + NullSingletonClass (conformalMass J w μ) := by + constructor + intro z + rw [conformalMass,Measure.map_apply J.continuous.measurable (measurableSet_singleton z)] + rw [show J ⁻¹' {z} = {J.symm z} by + ext x + simp only [mem_preimage,mem_singleton_iff] + constructor + · intro h + exact J.injective (h.trans (J.apply_symm_apply z).symm) + · intro h + rw [h,J.apply_symm_apply]] + exact measure_singleton _ + +lemma conformalMass_carried (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (μ : Measure upperHalfPlane) + (S : Set upperHalfPlane) (hS : MeasurableSet S) (hμ : μ Sᶜ = 0) : + conformalMass J w μ (J '' S)ᶜ = 0 := by + rw [conformalMass,Measure.map_apply J.continuous.measurable + (J.measurableEmbedding.measurableSet_image.mpr hS).compl] + apply withDensity_absolutelyContinuous μ _ + simpa only [preimage_compl,Set.preimage_image_eq _ J.injective] using hμ + +/-- Vague convergence transfers for every image-domain test on the same +event and with the same sequence. -/ +lemma conformalMass_test_tendsto (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (μ : ℕ → Measure upperHalfPlane) + (ν : Measure upperHalfPlane) + (h : ∀ f : C_c(upperHalfPlane, ℝ), + Tendsto (fun n => ∫ z, f z ∂μ n) atTop (𝓝 (∫ z, f z ∂ν))) + (f : C_c(D, ℝ)) : + Tendsto (fun n => ∫ z, f z ∂conformalMass J w (μ n)) atTop + (𝓝 (∫ z, f z ∂conformalMass J w ν)) := by + simp only [conformalMass_integral] + exact h (conformalMassTest J w f) + +lemma conformalMass_test_continuous (J : upperHalfPlane ≃ₜ D) + (w : C(upperHalfPlane, ℝ≥0)) (μ : ℝ≥0 → Measure upperHalfPlane) + (h : ∀ f : C_c(upperHalfPlane, ℝ), Continuous (fun t => ∫ z, f z ∂μ t)) + (f : C_c(D, ℝ)) : Continuous (fun t => ∫ z, f z ∂conformalMass J w (μ t)) := by + simp only [conformalMass_integral] + exact h (conformalMassTest J w f) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/ConvexCurrentMean.lean b/lean/OAI/Probability/SLE/Mass/ConvexCurrentMean.lean index e56c75b9e11530526fed3c5eb70e6f39ca661208..4c780203d470aa5f93cfe62ef6adc97a3cdc375b 100644 --- a/lean/OAI/Probability/SLE/Mass/ConvexCurrentMean.lean +++ b/lean/OAI/Probability/SLE/Mass/ConvexCurrentMean.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Mass.LevelConditionalMean - -namespace OAI - -/-! Convex tail averages of current conditional mean measures have the -same signed compact-test limit as the original radius levels. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem convexCurrentMean_test_tendsto - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0) (E : Set Ω) (f : C_c(upperHalfPlane,ℝ)) : - Tendsto (fun n => ∫ z, f z ∂convexCurrentMean (P.restrict E) κ B γ G u s w n t) - atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) := by - let ρ := initialGreenMeasure κ - let : IsFiniteMeasureOnCompacts ρ := initialGreenMeasure_finite_on_compacts κ - let μ (j : ℕ) := (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u j) t) - have hμ (j : ℕ) : μ j ≤ ρ := - (bind_mono_of_measurable (P.restrict E) - (halfPlaneCurrentMeasure_measurable hBm hBc κ (u j) (hu j) γ G F hw t) - (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t) - (fun ω => halfPlaneCurrentMeasure_le_green ω (hw ω) (u j) t)).trans - (halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E) - have hfi (j : ℕ) : Integrable f (μ j) := f.integrable.mono_measure (hμ j) - have he (n : ℕ) : (∫ z, f z ∂convexCurrentMean (P.restrict E) κ B γ G u s w n t) = - ∑ i ∈ s n, w n i * (∫ z, f z ∂μ (n+i)) := by - have hi (i : ℕ) : Integrable f (ENNReal.ofReal (w n i) • μ (n+i)) := - (hfi (n+i)).smul_measure ENNReal.ofReal_ne_top - rw [convexCurrentMean,integral_finsetSum_measure (fun i _ => hi i)] - apply Finset.sum_congr rfl - intro i his - rw [integral_smul_measure,ENNReal.toReal_ofReal (hw₀ n i his),smul_eq_mul] - simp only [he] - exact real_convex_tail_tendsto (fun j => ∫ z, f z ∂μ j) - (halfPlaneCurrentMeasure_mean_test_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw - u hu hmono hlim t E f) s w hw₀ hw₁ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelConditionalMean + +namespace OAI + +/-! Convex tail averages of current conditional mean measures have the +same signed compact-test limit as the original radius levels. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem convexCurrentMean_test_tendsto + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0) (E : Set Ω) (f : C_c(upperHalfPlane,ℝ)) : + Tendsto (fun n => ∫ z, f z ∂convexCurrentMean (P.restrict E) κ B γ G u s w n t) + atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) := by + let ρ := initialGreenMeasure κ + let : IsFiniteMeasureOnCompacts ρ := initialGreenMeasure_finite_on_compacts κ + let μ (j : ℕ) := (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u j) t) + have hμ (j : ℕ) : μ j ≤ ρ := + (bind_mono_of_measurable (P.restrict E) + (halfPlaneCurrentMeasure_measurable hBm hBc κ (u j) (hu j) γ G F hw t) + (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t) + (fun ω => halfPlaneCurrentMeasure_le_green ω (hw ω) (u j) t)).trans + (halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E) + have hfi (j : ℕ) : Integrable f (μ j) := f.integrable.mono_measure (hμ j) + have he (n : ℕ) : (∫ z, f z ∂convexCurrentMean (P.restrict E) κ B γ G u s w n t) = + ∑ i ∈ s n, w n i * (∫ z, f z ∂μ (n+i)) := by + have hi (i : ℕ) : Integrable f (ENNReal.ofReal (w n i) • μ (n+i)) := + (hfi (n+i)).smul_measure ENNReal.ofReal_ne_top + rw [convexCurrentMean,integral_finsetSum_measure (fun i _ => hi i)] + apply Finset.sum_congr rfl + intro i his + rw [integral_smul_measure,ENNReal.toReal_ofReal (hw₀ n i his),smul_eq_mul] + simp only [he] + exact real_convex_tail_tendsto (fun j => ∫ z, f z ∂μ j) + (halfPlaneCurrentMeasure_mean_test_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw + u hu hmono hlim t E f) s w hw₀ hw₁ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/ConvexMeanMeasure.lean b/lean/OAI/Probability/SLE/Mass/ConvexMeanMeasure.lean index ffeeba54d362c4cd9c74bc6114d9f812091bd0b7..ab93301283121d39ddf3feddeb43cc8fffb6b644 100644 --- a/lean/OAI/Probability/SLE/Mass/ConvexMeanMeasure.lean +++ b/lean/OAI/Probability/SLE/Mass/ConvexMeanMeasure.lean @@ -1,86 +1,86 @@ -import OAI.Probability.SLE.Mass.LevelMeanMeasure -import OAI.Probability.SLE.Mass.LevelRadonExtraction - -namespace OAI - -/-! The common convex coefficients preserve the exact terminal mean -measure, including signed compact tests. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -lemma convexLevelMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (n : ℕ) (t : ℝ≥0∞) : Measurable (convexLevelMeasure κ B γ G u s w n t) := by - apply Measure.measurable_of_measurable_coe - intro A hA - simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - apply Finset.measurable_sum - intro i hi - exact measurable_const.mul ((Measure.measurable_coe hA).comp - (halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t)) - -theorem convexLevelMeasure_terminal_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (n : ℕ) : P.bind (convexLevelMeasure κ B γ G u s w n ∞) = initialGreenMeasure κ := by - apply Measure.ext - intro A hA - rw [Measure.bind_apply hA (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞).aemeasurable] - simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - rw [lintegral_finsetSum (s n) (f := fun i ω => ENNReal.ofReal (w n i) * - halfPlaneLevelMeasure κ B γ G (u (n+i)) ∞ ω A) - (fun i _ => measurable_const.mul ((Measure.measurable_coe hA).comp - (halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw ∞)))] - have he (i : ℕ) : (∫⁻ ω, halfPlaneLevelMeasure κ B γ G (u (n+i)) ∞ ω A ∂P) = - initialGreenMeasure κ A := by - rw [← Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw ∞).aemeasurable, - halfPlaneLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw (u (n+i)) (hu _)] - simp_rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,he] - rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg (hw₀ n),hw₁ n,ENNReal.ofReal_one,one_mul] - -theorem convexLevelMeasure_terminal_test_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (n : ℕ) (f : C_c(upperHalfPlane,ℝ)) : - Integrable (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) P ∧ - (∫ ω, (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) ∂P) = - ∫ z, f z ∂initialGreenMeasure κ := by - have he := convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw - u hu s w hw₀ hw₁ n - have hi (ω : Ω) : Integrable f (convexLevelMeasure κ B γ G u s w n ∞ ω) := by - let := convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n ∞ - exact f.integrable - have hiP : Integrable f (P.bind (convexLevelMeasure κ B γ G u s w n ∞)) := by - rw [he] - let := initialGreenMeasure_finite_on_compacts κ - exact f.integrable - simpa only [he] using integrable_integral_bind_of_sections P _ - (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞) f f.continuous.measurable hi hiP - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelMeanMeasure +import OAI.Probability.SLE.Mass.LevelRadonExtraction + +namespace OAI + +/-! The common convex coefficients preserve the exact terminal mean +measure, including signed compact tests. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +lemma convexLevelMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (n : ℕ) (t : ℝ≥0∞) : Measurable (convexLevelMeasure κ B γ G u s w n t) := by + apply Measure.measurable_of_measurable_coe + intro A hA + simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + apply Finset.measurable_sum + intro i hi + exact measurable_const.mul ((Measure.measurable_coe hA).comp + (halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t)) + +theorem convexLevelMeasure_terminal_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (n : ℕ) : P.bind (convexLevelMeasure κ B γ G u s w n ∞) = initialGreenMeasure κ := by + apply Measure.ext + intro A hA + rw [Measure.bind_apply hA (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞).aemeasurable] + simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + rw [lintegral_finsetSum (s n) (f := fun i ω => ENNReal.ofReal (w n i) * + halfPlaneLevelMeasure κ B γ G (u (n+i)) ∞ ω A) + (fun i _ => measurable_const.mul ((Measure.measurable_coe hA).comp + (halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw ∞)))] + have he (i : ℕ) : (∫⁻ ω, halfPlaneLevelMeasure κ B γ G (u (n+i)) ∞ ω A ∂P) = + initialGreenMeasure κ A := by + rw [← Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw ∞).aemeasurable, + halfPlaneLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw (u (n+i)) (hu _)] + simp_rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,he] + rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg (hw₀ n),hw₁ n,ENNReal.ofReal_one,one_mul] + +theorem convexLevelMeasure_terminal_test_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (n : ℕ) (f : C_c(upperHalfPlane,ℝ)) : + Integrable (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) P ∧ + (∫ ω, (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) ∂P) = + ∫ z, f z ∂initialGreenMeasure κ := by + have he := convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw + u hu s w hw₀ hw₁ n + have hi (ω : Ω) : Integrable f (convexLevelMeasure κ B γ G u s w n ∞ ω) := by + let := convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n ∞ + exact f.integrable + have hiP : Integrable f (P.bind (convexLevelMeasure κ B γ G u s w n ∞)) := by + rw [he] + let := initialGreenMeasure_finite_on_compacts κ + exact f.integrable + simpa only [he] using integrable_integral_bind_of_sections P _ + (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞) f f.continuous.measurable hi hiP + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/ConvexMomentLimits.lean b/lean/OAI/Probability/SLE/Mass/ConvexMomentLimits.lean index 806da04ee9413a41a634194550a1ea5b93d86188..f1fefc65ce301791a9af297d261cc8317fe240dc 100644 --- a/lean/OAI/Probability/SLE/Mass/ConvexMomentLimits.lean +++ b/lean/OAI/Probability/SLE/Mass/ConvexMomentLimits.lean @@ -1,71 +1,71 @@ -import Mathlib.Analysis.MeanInequalitiesPow -import Mathlib.MeasureTheory.Integral.Lebesgue.Add - -namespace OAI - -/-! Moment bounds survive finite positive convex combinations and -almost-sure limits. These two steps preserve the actual level bounds. -/ - -noncomputable section -open Set Filter MeasureTheory Classical -open scoped ENNReal Topology - -namespace SLEExactGauge - -lemma ofReal_pow_convex_sum_le {ι : Type*} (s : Finset ι) - (w a : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw₁ : ∑ i ∈ s, w i = 1) - (ha : ∀ i ∈ s, 0 ≤ a i) (m : ℕ) : - ENNReal.ofReal (∑ i ∈ s, w i*a i)^m ≤ - ∑ i ∈ s, ENNReal.ofReal (w i) * ENNReal.ofReal (a i)^m := by - have hs : 0 ≤ ∑ i ∈ s, w i*a i := Finset.sum_nonneg (fun i hi => mul_nonneg (hw i hi) (ha i hi)) - rw [← ENNReal.ofReal_pow hs] - apply (ENNReal.ofReal_le_ofReal (Real.pow_arith_mean_le_arith_mean_pow s w a hw hw₁ ha m)).trans_eq - rw [ENNReal.ofReal_sum_of_nonneg (fun i hi => mul_nonneg (hw i hi) (pow_nonneg (ha i hi) m))] - apply Finset.sum_congr rfl - intro i hi - rw [ENNReal.ofReal_mul (hw i hi),ENNReal.ofReal_pow (ha i hi)] - -lemma lintegral_pow_convex_sum_le - {Ω ι : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (s : Finset ι) (w : ι → ℝ) (a : ι → Ω → ℝ) - (hw : ∀ i ∈ s, 0 ≤ w i) (hw₁ : ∑ i ∈ s, w i = 1) - (ha : ∀ i ∈ s, ∀ ω, 0 ≤ a i ω) (ham : ∀ i ∈ s, Measurable (a i)) - (m : ℕ) (M : ℝ≥0∞) (hM : ∀ i ∈ s, (∫⁻ ω, ENNReal.ofReal (a i ω)^m ∂P) ≤ M) : - (∫⁻ ω, ENNReal.ofReal (∑ i ∈ s, w i*a i ω)^m ∂P) ≤ M := by - calc - _ ≤ ∫⁻ ω, ∑ i ∈ s, ENNReal.ofReal (w i) * ENNReal.ofReal (a i ω)^m ∂P := - lintegral_mono (fun ω => ofReal_pow_convex_sum_le s w (fun i => a i ω) hw hw₁ - (fun i hi => ha i hi ω) m) - _ = ∑ i ∈ s, ENNReal.ofReal (w i) * ∫⁻ ω, ENNReal.ofReal (a i ω)^m ∂P := by - rw [lintegral_finsetSum s (f := fun i ω => ENNReal.ofReal (w i) * ENNReal.ofReal (a i ω)^m) - (fun i hi => measurable_const.mul ((ham i hi).ennreal_ofReal.pow_const m))] - apply Finset.sum_congr rfl - intro i hi - exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top - _ ≤ ∑ i ∈ s, ENNReal.ofReal (w i) * M := - Finset.sum_le_sum (fun i hi => mul_le_mul' le_rfl (hM i hi)) - _ = M := by - rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg hw,hw₁,ENNReal.ofReal_one,one_mul] - -lemma lintegral_le_of_ae_tendsto_bound - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (f : ℕ → Ω → ℝ≥0∞) (F : Ω → ℝ≥0∞) - (hf : ∀ n, AEMeasurable (f n) P) - (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => f n ω) atTop (𝓝 (F ω))) - (M : ℝ≥0∞) (hM : ∀ n, (∫⁻ ω, f n ω ∂P) ≤ M) : - (∫⁻ ω, F ω ∂P) ≤ M := by - calc - _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := by - apply lintegral_congr_ae - filter_upwards [hlim] with ω hω - exact hω.liminf_eq.symm - _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' hf - _ ≤ M := by - have h : ∀ᶠ n : ℕ in atTop, (∫⁻ ω, f n ω ∂P) ≤ M := Eventually.of_forall hM - simpa only [liminf_const] using liminf_le_liminf h - -end SLEExactGauge - -end - -end OAI +import Mathlib.Analysis.MeanInequalitiesPow +import Mathlib.MeasureTheory.Integral.Lebesgue.Add + +namespace OAI + +/-! Moment bounds survive finite positive convex combinations and +almost-sure limits. These two steps preserve the actual level bounds. -/ + +noncomputable section +open Set Filter MeasureTheory Classical +open scoped ENNReal Topology + +namespace SLEExactGauge + +lemma ofReal_pow_convex_sum_le {ι : Type*} (s : Finset ι) + (w a : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw₁ : ∑ i ∈ s, w i = 1) + (ha : ∀ i ∈ s, 0 ≤ a i) (m : ℕ) : + ENNReal.ofReal (∑ i ∈ s, w i*a i)^m ≤ + ∑ i ∈ s, ENNReal.ofReal (w i) * ENNReal.ofReal (a i)^m := by + have hs : 0 ≤ ∑ i ∈ s, w i*a i := Finset.sum_nonneg (fun i hi => mul_nonneg (hw i hi) (ha i hi)) + rw [← ENNReal.ofReal_pow hs] + apply (ENNReal.ofReal_le_ofReal (Real.pow_arith_mean_le_arith_mean_pow s w a hw hw₁ ha m)).trans_eq + rw [ENNReal.ofReal_sum_of_nonneg (fun i hi => mul_nonneg (hw i hi) (pow_nonneg (ha i hi) m))] + apply Finset.sum_congr rfl + intro i hi + rw [ENNReal.ofReal_mul (hw i hi),ENNReal.ofReal_pow (ha i hi)] + +lemma lintegral_pow_convex_sum_le + {Ω ι : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (s : Finset ι) (w : ι → ℝ) (a : ι → Ω → ℝ) + (hw : ∀ i ∈ s, 0 ≤ w i) (hw₁ : ∑ i ∈ s, w i = 1) + (ha : ∀ i ∈ s, ∀ ω, 0 ≤ a i ω) (ham : ∀ i ∈ s, Measurable (a i)) + (m : ℕ) (M : ℝ≥0∞) (hM : ∀ i ∈ s, (∫⁻ ω, ENNReal.ofReal (a i ω)^m ∂P) ≤ M) : + (∫⁻ ω, ENNReal.ofReal (∑ i ∈ s, w i*a i ω)^m ∂P) ≤ M := by + calc + _ ≤ ∫⁻ ω, ∑ i ∈ s, ENNReal.ofReal (w i) * ENNReal.ofReal (a i ω)^m ∂P := + lintegral_mono (fun ω => ofReal_pow_convex_sum_le s w (fun i => a i ω) hw hw₁ + (fun i hi => ha i hi ω) m) + _ = ∑ i ∈ s, ENNReal.ofReal (w i) * ∫⁻ ω, ENNReal.ofReal (a i ω)^m ∂P := by + rw [lintegral_finsetSum s (f := fun i ω => ENNReal.ofReal (w i) * ENNReal.ofReal (a i ω)^m) + (fun i hi => measurable_const.mul ((ham i hi).ennreal_ofReal.pow_const m))] + apply Finset.sum_congr rfl + intro i hi + exact lintegral_const_mul' _ _ ENNReal.ofReal_ne_top + _ ≤ ∑ i ∈ s, ENNReal.ofReal (w i) * M := + Finset.sum_le_sum (fun i hi => mul_le_mul' le_rfl (hM i hi)) + _ = M := by + rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg hw,hw₁,ENNReal.ofReal_one,one_mul] + +lemma lintegral_le_of_ae_tendsto_bound + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (f : ℕ → Ω → ℝ≥0∞) (F : Ω → ℝ≥0∞) + (hf : ∀ n, AEMeasurable (f n) P) + (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => f n ω) atTop (𝓝 (F ω))) + (M : ℝ≥0∞) (hM : ∀ n, (∫⁻ ω, f n ω ∂P) ≤ M) : + (∫⁻ ω, F ω ∂P) ≤ M := by + calc + _ = ∫⁻ ω, liminf (fun n => f n ω) atTop ∂P := by + apply lintegral_congr_ae + filter_upwards [hlim] with ω hω + exact hω.liminf_eq.symm + _ ≤ liminf (fun n => ∫⁻ ω, f n ω ∂P) atTop := lintegral_liminf_le' hf + _ ≤ M := by + have h : ∀ᶠ n : ℕ in atTop, (∫⁻ ω, f n ω ∂P) ≤ M := Eventually.of_forall hM + simpa only [liminf_const] using liminf_le_liminf h + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CountableLevelAdapted.lean b/lean/OAI/Probability/SLE/Mass/CountableLevelAdapted.lean index 920571751bc3227b138170dc96ecdbaf1ae8800f..a3385eb3f57b2dc672fdeaac3fb82d13f65cc732 100644 --- a/lean/OAI/Probability/SLE/Mass/CountableLevelAdapted.lean +++ b/lean/OAI/Probability/SLE/Mass/CountableLevelAdapted.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Mass.LevelApproximationAdapted - -namespace OAI - -/-! A common family of actual convex limits admits simultaneous versions -measurable in the original natural past at each prescribed finite time. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge.Multipoint - -theorem countable_level_limits_have_past_versions - {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (t : ℕ → ℝ≥0) - (ν : ℕ → Ω → Measure upperHalfPlane) - (hν : ∀ᵐ ω ∂P, ∀ j, (ν j ω).Regular) - (hlim : ∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν j ω))) : - ∃ η : ℕ → Ω → Measure upperHalfPlane, - (∀ j, Measurable[Filtration.natural B (fun r => (hm r).stronglyMeasurable) (t j)] (η j)) ∧ - (∀ j ω, (η j ω).Regular ∧ IsFiniteMeasureOnCompacts (η j ω)) ∧ - ∀ᵐ ω ∂P, ∀ j, η j ω = ν j ω := by - have hex (j : ℕ) : ∃ η : Ω → Measure upperHalfPlane, - Measurable[Filtration.natural B (fun r => (hm r).stronglyMeasurable) (t j)] η ∧ - (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ η =ᵐ[P] ν j := by - apply level_radon_limit_has_past_measurable_version P hm hc hκ hκ8 γ G F hw - u hu s w hw₀ (t j) (ν j) - · filter_upwards [hν] with ω hω - exact hω j - · filter_upwards [hlim] with ω hω - exact hω j - choose η hηm hηr hηeq using hex - exact ⟨η,hηm,hηr,ae_all_iff.mpr hηeq⟩ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximationAdapted + +namespace OAI + +/-! A common family of actual convex limits admits simultaneous versions +measurable in the original natural past at each prescribed finite time. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge.Multipoint + +theorem countable_level_limits_have_past_versions + {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (t : ℕ → ℝ≥0) + (ν : ℕ → Ω → Measure upperHalfPlane) + (hν : ∀ᵐ ω ∂P, ∀ j, (ν j ω).Regular) + (hlim : ∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν j ω))) : + ∃ η : ℕ → Ω → Measure upperHalfPlane, + (∀ j, Measurable[Filtration.natural B (fun r => (hm r).stronglyMeasurable) (t j)] (η j)) ∧ + (∀ j ω, (η j ω).Regular ∧ IsFiniteMeasureOnCompacts (η j ω)) ∧ + ∀ᵐ ω ∂P, ∀ j, η j ω = ν j ω := by + have hex (j : ℕ) : ∃ η : Ω → Measure upperHalfPlane, + Measurable[Filtration.natural B (fun r => (hm r).stronglyMeasurable) (t j)] η ∧ + (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ η =ᵐ[P] ν j := by + apply level_radon_limit_has_past_measurable_version P hm hc hκ hκ8 γ G F hw + u hu s w hw₀ (t j) (ν j) + · filter_upwards [hν] with ω hω + exact hω j + · filter_upwards [hlim] with ω hω + exact hω j + choose η hηm hηr hηeq using hex + exact ⟨η,hηm,hηr,ae_all_iff.mpr hηeq⟩ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/CountableLevelKernel.lean b/lean/OAI/Probability/SLE/Mass/CountableLevelKernel.lean index d56874f416eb16fa99ea61d742b304bb32ef6572..e110d420881f3f9cc06cf88c97cb5ebc870b8834 100644 --- a/lean/OAI/Probability/SLE/Mass/CountableLevelKernel.lean +++ b/lean/OAI/Probability/SLE/Mass/CountableLevelKernel.lean @@ -1,89 +1,89 @@ -import OAI.Probability.SLE.Mass.LevelRadonKernel -import OAI.Probability.SLE.Mass.LevelRadonPairings -import OAI.Probability.SLE.Mass.LevelRadonMean - -namespace OAI - -/-! Actual common Radon limits with their exact conditional kernels at -any prescribed countable set of deterministic times. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -def levelMassTimes (t : ℕ → ℝ≥0) : ℕ → ℝ≥0∞ - | 0 => ∞ - | n+1 => t n - -@[simp] lemma levelMassTimes_zero (t : ℕ → ℝ≥0) : levelMassTimes t 0 = ∞ := rfl -@[simp] lemma levelMassTimes_succ (t : ℕ → ℝ≥0) (n : ℕ) : levelMassTimes t (n+1) = t n := rfl - -theorem exists_countable_level_radon_kernels - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (t : ℕ → ℝ≥0) : - ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, ∃ ν : ℕ → Ω → Measure upperHalfPlane, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Measurable (ν j)) ∧ - (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ - (∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n - (levelMassTimes t j) ω) atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ - P.bind (ν 0) = initialGreenMeasure κ ∧ - ∀ j (E : Set Ω), - MeasurableSet[Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) (t j)] E → - (P.restrict E).bind (ν 0) = (P.restrict E).bind (ν (j+1)) + - (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G (t j)) := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let u (n : ℕ) : ℝ := n - have hu (n : ℕ) : 0 ≤ u n := Nat.cast_nonneg n - have hmono : Monotone u := fun _ _ h => Nat.cast_le.mpr h - have huLim : Tendsto u atTop atTop := tendsto_natCast_atTop_atTop - obtain ⟨s,w,ν,hw₀,hw₁,hνm,hν,hlim,hpair⟩ := exists_measurable_level_radon_limits_with_pairings - hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth u hu (levelMassTimes t) - refine ⟨s,w,ν,hw₀,hw₁,hνm,hν,hlim,?_,?_⟩ - · apply level_radon_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - u hu s w hw₀ hw₁ (ν 0) (hνm 0) (fun ω => (hν 0 ω).1) - · filter_upwards [hlim] with ω hω - simpa only [levelMassTimes_zero] using hω 0 - · intro k - simpa only [one_mul,levelMassTimes_zero] using hpair 0 k (fun _ => 1) (memLp_const 1) - · intro j E hE - let v (i : Fin 2) := if i = 0 then ν (j+1) else ν 0 - have hvm (i : Fin 2) : Measurable (v i) := by - dsimp only [v]; split_ifs <;> exact hνm _ - have hvr (i : Fin 2) (ω : Ω) : (v i ω).Regular ∧ IsFiniteMeasureOnCompacts (v i ω) := by - dsimp only [v]; split_ifs <;> exact hν _ ω - have hvl (i : Fin 2) : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n - (if i = 0 then (t j : ℝ≥0∞) else ∞) ω) atTop (𝓝 (∫ z, f z ∂v i ω)) := by - filter_upwards [hlim] with ω hω - by_cases hi : i = 0 - · simpa only [v,ite_eq_left hi,levelMassTimes_succ] using hω (j+1) - · simpa only [v,ite_eq_right hi,levelMassTimes_zero] using hω 0 - have hvp (i : Fin 2) (k : ℕ) (a : Ω → ℝ) (ha : MemLp a 2 P) : - Tendsto (fun n => ∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z - ∂convexLevelMeasure κ B γ G u s w n (if i = 0 then (t j : ℝ≥0∞) else ∞) ω) ∂P) - atTop (𝓝 (∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z ∂v i ω) ∂P)) := by - by_cases hi : i = 0 - · simpa only [v,ite_eq_left hi,levelMassTimes_succ] using hpair (j+1) k a ha - · simpa only [v,ite_eq_right hi,levelMassTimes_zero] using hpair 0 k a ha - have hk := level_radon_conditional_kernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - u hu hmono huLim s w hw₀ hw₁ (t j) v hvm hvr hvl hvp E hE - simpa [v] using hk - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelRadonKernel +import OAI.Probability.SLE.Mass.LevelRadonPairings +import OAI.Probability.SLE.Mass.LevelRadonMean + +namespace OAI + +/-! Actual common Radon limits with their exact conditional kernels at +any prescribed countable set of deterministic times. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +def levelMassTimes (t : ℕ → ℝ≥0) : ℕ → ℝ≥0∞ + | 0 => ∞ + | n+1 => t n + +@[simp] lemma levelMassTimes_zero (t : ℕ → ℝ≥0) : levelMassTimes t 0 = ∞ := rfl +@[simp] lemma levelMassTimes_succ (t : ℕ → ℝ≥0) (n : ℕ) : levelMassTimes t (n+1) = t n := rfl + +theorem exists_countable_level_radon_kernels + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (t : ℕ → ℝ≥0) : + ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, ∃ ν : ℕ → Ω → Measure upperHalfPlane, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Measurable (ν j)) ∧ + (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ + (∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G (fun n => (n:ℝ)) s w n + (levelMassTimes t j) ω) atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ + P.bind (ν 0) = initialGreenMeasure κ ∧ + ∀ j (E : Set Ω), + MeasurableSet[Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) (t j)] E → + (P.restrict E).bind (ν 0) = (P.restrict E).bind (ν (j+1)) + + (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G (t j)) := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let u (n : ℕ) : ℝ := n + have hu (n : ℕ) : 0 ≤ u n := Nat.cast_nonneg n + have hmono : Monotone u := fun _ _ h => Nat.cast_le.mpr h + have huLim : Tendsto u atTop atTop := tendsto_natCast_atTop_atTop + obtain ⟨s,w,ν,hw₀,hw₁,hνm,hν,hlim,hpair⟩ := exists_measurable_level_radon_limits_with_pairings + hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth u hu (levelMassTimes t) + refine ⟨s,w,ν,hw₀,hw₁,hνm,hν,hlim,?_,?_⟩ + · apply level_radon_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + u hu s w hw₀ hw₁ (ν 0) (hνm 0) (fun ω => (hν 0 ω).1) + · filter_upwards [hlim] with ω hω + simpa only [levelMassTimes_zero] using hω 0 + · intro k + simpa only [one_mul,levelMassTimes_zero] using hpair 0 k (fun _ => 1) (memLp_const 1) + · intro j E hE + let v (i : Fin 2) := if i = 0 then ν (j+1) else ν 0 + have hvm (i : Fin 2) : Measurable (v i) := by + dsimp only [v]; split_ifs <;> exact hνm _ + have hvr (i : Fin 2) (ω : Ω) : (v i ω).Regular ∧ IsFiniteMeasureOnCompacts (v i ω) := by + dsimp only [v]; split_ifs <;> exact hν _ ω + have hvl (i : Fin 2) : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n + (if i = 0 then (t j : ℝ≥0∞) else ∞) ω) atTop (𝓝 (∫ z, f z ∂v i ω)) := by + filter_upwards [hlim] with ω hω + by_cases hi : i = 0 + · simpa only [v,ite_eq_left hi,levelMassTimes_succ] using hω (j+1) + · simpa only [v,ite_eq_right hi,levelMassTimes_zero] using hω 0 + have hvp (i : Fin 2) (k : ℕ) (a : Ω → ℝ) (ha : MemLp a 2 P) : + Tendsto (fun n => ∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z + ∂convexLevelMeasure κ B γ G u s w n (if i = 0 then (t j : ℝ≥0∞) else ∞) ω) ∂P) + atTop (𝓝 (∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z ∂v i ω) ∂P)) := by + by_cases hi : i = 0 + · simpa only [v,ite_eq_left hi,levelMassTimes_succ] using hpair (j+1) k a ha + · simpa only [v,ite_eq_right hi,levelMassTimes_zero] using hpair 0 k a ha + have hk := level_radon_conditional_kernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + u hu hmono huLim s w hw₀ hw₁ (t j) v hvm hvr hvl hvp E hE + simpa [v] using hk + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DenseCoverMassTests.lean b/lean/OAI/Probability/SLE/Mass/DenseCoverMassTests.lean index dfbb8b4e6119263b555f0b10c7cf40a9077c25e9..4b96b2046d613627d805a80ee833847f963255a9 100644 --- a/lean/OAI/Probability/SLE/Mass/DenseCoverMassTests.lean +++ b/lean/OAI/Probability/SLE/Mass/DenseCoverMassTests.lean @@ -1,116 +1,116 @@ -import OAI.Probability.SLE.Gauge.DenseCoverAdaptedTests -import OAI.Probability.SLE.Geometry.FrontSuccessStopped -import OAI.Probability.SLE.Gauge.CoverEvents - -namespace OAI - -/-! Actual sausage tests at successive stopping times. Their terminal -candidate implication is deterministic, whereas their failure calculation -uses only information observable at each test endpoint. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open FrontSuccessMeasurability - -local instance denseMassPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ -local instance denseMassPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ - -def stoppedWholeSausageCharge (κ : ℝ) (f : C(ℝ≥0, ℂ)) - (τ : WithTop ℝ≥0) (v : ℝ) (D : Set ℂ) : ℝ≥0∞ := - wholeSausageCharge κ (stoppedAtPath f τ) v D - -theorem stoppedWholeSausageCharge_le_whole - (κ : ℝ) (f : C(ℝ≥0, ℂ)) (τ : WithTop ℝ≥0) (v : ℝ) (D : Set ℂ) : - stoppedWholeSausageCharge κ f τ v D ≤ wholeSausageCharge κ f v D := by - apply mul_le_mul' le_rfl - apply measure_mono - rintro z ⟨hz, t, ht⟩ - refine ⟨hz, (min τ (t : WithTop ℝ≥0)).untopA, ?_⟩ - simpa only [stoppedAtPath_apply] using ht - -theorem stoppedWholeSausageCharge_finite - (κ : ℝ) (f : C(ℝ≥0, ℂ)) (τ : ℝ≥0) (v : ℝ) (D : Set ℂ) : - stoppedWholeSausageCharge κ f (τ : WithTop ℝ≥0) v D = - segmentSausageCharge κ f 0 τ v D := by - simp only [stoppedWholeSausageCharge, stoppedAtPath, WithTop.coe_ne_top, ite_false, - WithTop.untopD_coe, segmentSausageCharge, wholeSausageCharge, - sourceSausageSet_eq_clamped f 0 τ bot_le] - -theorem stoppedWholeSausageCharge_measurable_at_stop - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) - (hc : ∀ ω, Continuous (γ ω)) - (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (κ v : ℝ) (D : Set ℂ) : - Measurable[hτ.measurableSpace] (fun ω => - stoppedWholeSausageCharge κ (sourceTracePath γ hc ω) (τ ω) v D) := - (wholeSausageCharge_measurable κ v D).comp - (stoppedAtPath_measurable ℱ γ ha hc τ hτ) - -def clockStopFiltration - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (τ : ℕ → Ω → WithTop ℝ≥0) (hτ : ∀ j, IsStoppingTime ℱ (τ j)) - (hmono : ∀ ω, Monotone (fun j => τ j ω)) : Filtration ℕ mΩ := - ⟨fun j => (hτ j).measurableSpace, - fun _ _ hij => IsStoppingTime.measurableSpace_mono _ _ (fun ω => hmono ω hij), - fun j => (hτ j).measurableSpace_le⟩ - -def stoppedMassTest - {Ω : Type*} (κ v : ℝ) (f : Ω → C(ℝ≥0, ℂ)) - (τ : Ω → WithTop ℝ≥0) (D : Set ℂ) (b : ℝ) : Set Ω := - {ω | ENNReal.ofReal b ≤ stoppedWholeSausageCharge κ (f ω) (τ ω) v D} - -theorem stoppedMassTest_measurable_at_stop - {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) - (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) - (hc : ∀ ω, Continuous (γ ω)) - (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (κ v b : ℝ) (D : Set ℂ) : - MeasurableSet[hτ.measurableSpace] - (stoppedMassTest κ v (sourceTracePath γ hc) τ D b) := - measurableSet_le measurable_const - (stoppedWholeSausageCharge_measurable_at_stop ℱ γ ha hc τ hτ κ v D) - -/-- No terminal candidate forces failure of every earlier adapted test. -The terminal candidate is never put inside a finite change of measure. -/ -theorem no_terminal_candidate_subset_stopped_failures - {Ω : Type*} (κ v : ℝ) (f : Ω → C(ℝ≥0, ℂ)) - (τ : ℕ → Ω → WithTop ℝ≥0) (D : ℕ → Set ℂ) (b : ℕ → ℝ) (I : Finset ℕ) : - {ω | ∀ j ∈ I, wholeSausageCharge κ (f ω) v (D j) < ENNReal.ofReal (b j)} ⊆ - ⋂ j ∈ I, (stoppedMassTest κ v f (τ (j + 1)) (D j) (b j))ᶜ := by - intro ω hω - apply mem_iInter₂.mpr - intro j hj hsuccess - exact (not_le_of_gt (hω j hj)) - (hsuccess.trans (stoppedWholeSausageCharge_le_whole κ (f ω) _ v (D j))) - -/-- The actual successive stopped-sausage tests give the stretched -exponential missed-candidate bound under their finite stopped law. -/ -theorem stoppedMassTests_cover_failure - {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] - (ℱ : Filtration ℝ≥0 mΩ) - (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) - (hc : ∀ ω, Continuous (γ ω)) - (τ : ℕ → Ω → WithTop ℝ≥0) (hτ : ∀ j, IsStoppingTime ℱ (τ j)) - (hmono : ∀ ω, Monotone (fun j => τ j ω)) - (κ v : ℝ) (D : ℕ → Set ℂ) (b : ℕ → ℝ) (n : ℕ) (hn : 1 ≤ n) - (hstep : ∀ j (B : Set Ω), MeasurableSet[(hτ j).measurableSpace] B → - ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ - μ (B ∩ stoppedMassTest κ v (sourceTracePath γ hc) (τ (j + 1)) (D j) (b j))) : - μ {ω | ∀ j ∈ coverTestWindow n, - wholeSausageCharge κ (γ ω) v (D j) < ENNReal.ofReal (b j)} ≤ - ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by - let f := sourceTracePath γ hc - let success := fun j => stoppedMassTest κ v f (τ (j + 1)) (D j) (b j) - refine (measure_mono (no_terminal_candidate_subset_stopped_failures - κ v f τ D b (coverTestWindow n))).trans ?_ - exact cover_window_failure_bound μ (clockStopFiltration ℱ τ hτ hmono) success - (fun j => stoppedMassTest_measurable_at_stop ℱ γ ha hc (τ (j + 1)) (hτ (j + 1)) - κ v (b j) (D j)) n hn hstep - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.DenseCoverAdaptedTests +import OAI.Probability.SLE.Geometry.FrontSuccessStopped +import OAI.Probability.SLE.Gauge.CoverEvents + +namespace OAI + +/-! Actual sausage tests at successive stopping times. Their terminal +candidate implication is deterministic, whereas their failure calculation +uses only information observable at each test endpoint. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open FrontSuccessMeasurability + +local instance denseMassPathMeasurable : MeasurableSpace C(ℝ≥0, ℂ) := borel _ +local instance denseMassPathBorel : BorelSpace C(ℝ≥0, ℂ) := ⟨rfl⟩ + +def stoppedWholeSausageCharge (κ : ℝ) (f : C(ℝ≥0, ℂ)) + (τ : WithTop ℝ≥0) (v : ℝ) (D : Set ℂ) : ℝ≥0∞ := + wholeSausageCharge κ (stoppedAtPath f τ) v D + +theorem stoppedWholeSausageCharge_le_whole + (κ : ℝ) (f : C(ℝ≥0, ℂ)) (τ : WithTop ℝ≥0) (v : ℝ) (D : Set ℂ) : + stoppedWholeSausageCharge κ f τ v D ≤ wholeSausageCharge κ f v D := by + apply mul_le_mul' le_rfl + apply measure_mono + rintro z ⟨hz, t, ht⟩ + refine ⟨hz, (min τ (t : WithTop ℝ≥0)).untopA, ?_⟩ + simpa only [stoppedAtPath_apply] using ht + +theorem stoppedWholeSausageCharge_finite + (κ : ℝ) (f : C(ℝ≥0, ℂ)) (τ : ℝ≥0) (v : ℝ) (D : Set ℂ) : + stoppedWholeSausageCharge κ f (τ : WithTop ℝ≥0) v D = + segmentSausageCharge κ f 0 τ v D := by + simp only [stoppedWholeSausageCharge, stoppedAtPath, WithTop.coe_ne_top, ite_false, + WithTop.untopD_coe, segmentSausageCharge, wholeSausageCharge, + sourceSausageSet_eq_clamped f 0 τ bot_le] + +theorem stoppedWholeSausageCharge_measurable_at_stop + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) + (hc : ∀ ω, Continuous (γ ω)) + (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (κ v : ℝ) (D : Set ℂ) : + Measurable[hτ.measurableSpace] (fun ω => + stoppedWholeSausageCharge κ (sourceTracePath γ hc ω) (τ ω) v D) := + (wholeSausageCharge_measurable κ v D).comp + (stoppedAtPath_measurable ℱ γ ha hc τ hτ) + +def clockStopFiltration + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (τ : ℕ → Ω → WithTop ℝ≥0) (hτ : ∀ j, IsStoppingTime ℱ (τ j)) + (hmono : ∀ ω, Monotone (fun j => τ j ω)) : Filtration ℕ mΩ := + ⟨fun j => (hτ j).measurableSpace, + fun _ _ hij => IsStoppingTime.measurableSpace_mono _ _ (fun ω => hmono ω hij), + fun j => (hτ j).measurableSpace_le⟩ + +def stoppedMassTest + {Ω : Type*} (κ v : ℝ) (f : Ω → C(ℝ≥0, ℂ)) + (τ : Ω → WithTop ℝ≥0) (D : Set ℂ) (b : ℝ) : Set Ω := + {ω | ENNReal.ofReal b ≤ stoppedWholeSausageCharge κ (f ω) (τ ω) v D} + +theorem stoppedMassTest_measurable_at_stop + {Ω : Type*} [mΩ : MeasurableSpace Ω] (ℱ : Filtration ℝ≥0 mΩ) + (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) + (hc : ∀ ω, Continuous (γ ω)) + (τ : Ω → WithTop ℝ≥0) (hτ : IsStoppingTime ℱ τ) (κ v b : ℝ) (D : Set ℂ) : + MeasurableSet[hτ.measurableSpace] + (stoppedMassTest κ v (sourceTracePath γ hc) τ D b) := + measurableSet_le measurable_const + (stoppedWholeSausageCharge_measurable_at_stop ℱ γ ha hc τ hτ κ v D) + +/-- No terminal candidate forces failure of every earlier adapted test. +The terminal candidate is never put inside a finite change of measure. -/ +theorem no_terminal_candidate_subset_stopped_failures + {Ω : Type*} (κ v : ℝ) (f : Ω → C(ℝ≥0, ℂ)) + (τ : ℕ → Ω → WithTop ℝ≥0) (D : ℕ → Set ℂ) (b : ℕ → ℝ) (I : Finset ℕ) : + {ω | ∀ j ∈ I, wholeSausageCharge κ (f ω) v (D j) < ENNReal.ofReal (b j)} ⊆ + ⋂ j ∈ I, (stoppedMassTest κ v f (τ (j + 1)) (D j) (b j))ᶜ := by + intro ω hω + apply mem_iInter₂.mpr + intro j hj hsuccess + exact (not_le_of_gt (hω j hj)) + (hsuccess.trans (stoppedWholeSausageCharge_le_whole κ (f ω) _ v (D j))) + +/-- The actual successive stopped-sausage tests give the stretched +exponential missed-candidate bound under their finite stopped law. -/ +theorem stoppedMassTests_cover_failure + {Ω : Type*} [mΩ : MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] + (ℱ : Filtration ℝ≥0 mΩ) + (γ : Ω → ℝ≥0 → ℂ) (ha : Adapted ℱ (fun t ω => γ ω t)) + (hc : ∀ ω, Continuous (γ ω)) + (τ : ℕ → Ω → WithTop ℝ≥0) (hτ : ∀ j, IsStoppingTime ℱ (τ j)) + (hmono : ∀ ω, Monotone (fun j => τ j ω)) + (κ v : ℝ) (D : ℕ → Set ℂ) (b : ℕ → ℝ) (n : ℕ) (hn : 1 ≤ n) + (hstep : ∀ j (B : Set Ω), MeasurableSet[(hτ j).measurableSpace] B → + ENNReal.ofReal (1 / Real.sqrt (n : ℝ)) * μ B ≤ + μ (B ∩ stoppedMassTest κ v (sourceTracePath γ hc) (τ (j + 1)) (D j) (b j))) : + μ {ω | ∀ j ∈ coverTestWindow n, + wholeSausageCharge κ (γ ω) v (D j) < ENNReal.ofReal (b j)} ≤ + ENNReal.ofReal (Real.exp (-Real.sqrt (n : ℝ) / 4)) := by + let f := sourceTracePath γ hc + let success := fun j => stoppedMassTest κ v f (τ (j + 1)) (D j) (b j) + refine (measure_mono (no_terminal_candidate_subset_stopped_failures + κ v f τ D b (coverTestWindow n))).trans ?_ + exact cover_window_failure_bound μ (clockStopFiltration ℱ τ hτ hmono) success + (fun j => stoppedMassTest_measurable_at_stop ℱ γ ha hc (τ (j + 1)) (hτ (j + 1)) + κ v (b j) (D j)) n hn hstep + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DilationMass.lean b/lean/OAI/Probability/SLE/Mass/DilationMass.lean index 7cd51a3e434258192486d5e6764c7c09e527141a..0f484c958015422af02c19e5a906875acd59e13c 100644 --- a/lean/OAI/Probability/SLE/Mass/DilationMass.lean +++ b/lean/OAI/Probability/SLE/Mass/DilationMass.lean @@ -1,165 +1,165 @@ -import OAI.Probability.SLE.Mass.ConformalMassTransport -import OAI.Probability.SLE.RetainedSeedMoments - -namespace OAI - -/-! Exact area and measure transport for positive real dilations of the -half-plane. These are the spatial factors in capacity scaling. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge - -lemma source_scale_area_lintegral (c : ℝ≥0) (hc : 0 < c) - (f : ℂ → ℝ≥0∞) (K : Set ℂ) : - (∫⁻ z in (fun z : ℂ => (c:ℂ)*z) '' K, f z) = - ENNReal.ofReal ((c:ℝ)^2) * ∫⁻ z in K, f ((c:ℂ)*z) := by - have hcR : 0 < (c:ℝ) := hc - have hcC : (c:ℂ) ≠ 0 := by exact_mod_cast hc.ne' - let e := Homeomorph.mulLeft₀ (c:ℂ) hcC - have hvol (s : Set ℂ) : volume (e ⁻¹' s)= - ENNReal.ofReal (((c:ℝ)^2)⁻¹)*volume s := by - change volume ((fun z : ℂ => (c:ℂ)*z) ⁻¹' s)=_ - simpa only [Complex.real_smul,smul_eq_mul,Complex.finrank_real_complex, - abs_of_nonneg (inv_nonneg.mpr (sq_nonneg (c:ℝ)))] using - (volume : Measure ℂ).addHaar_preimage_smul hcR.ne' s - have hp : MeasurePreserving e (ENNReal.ofReal ((c:ℝ)^2) • (volume : Measure ℂ)) volume := by - refine ⟨e.continuous.measurable,?_⟩ - apply Measure.ext - intro s hs - rw [e.measurableEmbedding.map_apply,Measure.smul_apply,smul_eq_mul,hvol,← mul_assoc, - ← ENNReal.ofReal_mul (sq_nonneg (c:ℝ)),mul_inv_cancel₀ (pow_ne_zero 2 hcR.ne'), - ENNReal.ofReal_one,one_mul] - have h := hp.setLIntegral_comp_emb e.measurableEmbedding f K - rw [Measure.restrict_smul,lintegral_smul_measure] at h - exact h.symm - -def halfPlaneDilation (c : ℝ≥0) (hc : 0 < c) : upperHalfPlane ≃ₜ upperHalfPlane where - toFun z := ⟨(c:ℂ)*(z:ℂ),by - change 0 < ((c:ℂ)*(z:ℂ)).im - simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero] - using mul_pos (show 0 < (c:ℝ) from hc) z.property⟩ - invFun z := ⟨(c:ℂ)⁻¹*(z:ℂ),by - change 0 < ((c:ℂ)⁻¹*(z:ℂ)).im - rw [← Complex.ofReal_inv] - simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero] - using mul_pos (inv_pos.mpr (show 0 < (c:ℝ) from hc)) z.property⟩ - left_inv z := by - apply Subtype.ext - change (c:ℂ)⁻¹*((c:ℂ)*(z:ℂ))=(z:ℂ) - exact inv_mul_cancel_left₀ (by exact_mod_cast hc.ne') _ - right_inv z := by - apply Subtype.ext - change (c:ℂ)*((c:ℂ)⁻¹*(z:ℂ))=(z:ℂ) - exact mul_inv_cancel_left₀ (by exact_mod_cast hc.ne') _ - continuous_toFun := by fun_prop - continuous_invFun := by fun_prop - -@[simp] lemma halfPlaneDilation_apply (c : ℝ≥0) (hc : 0 < c) (z : upperHalfPlane) : - ((halfPlaneDilation c hc z : upperHalfPlane):ℂ)=(c:ℂ)*(z:ℂ) := rfl - -namespace Multipoint - -def dilationMass (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) (μ : Measure upperHalfPlane) : - Measure upperHalfPlane := - conformalMass (halfPlaneDilation c hc) (ContinuousMap.const _ (Real.toNNReal ((c:ℝ)^dimension κ))) μ - -lemma dilationMass_apply (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) - (μ : Measure upperHalfPlane) (A : Set upperHalfPlane) (hA : MeasurableSet A) : - dilationMass κ c hc μ A=ENNReal.ofReal ((c:ℝ)^dimension κ)* - μ ((halfPlaneDilation c hc) ⁻¹' A) := by - rw [dilationMass,conformalMass,Measure.map_apply (halfPlaneDilation c hc).continuous.measurable hA, - withDensity_apply _ ((halfPlaneDilation c hc).continuous.measurable hA)] - simp only [ContinuousMap.const_apply] - change (∫⁻ _ in (halfPlaneDilation c hc) ⁻¹' A, ENNReal.ofReal ((c:ℝ)^dimension κ) ∂μ)=_ - exact setLIntegral_const _ _ - -lemma dilationMass_realDensity (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) - (ρ ρ' : ℂ → ℝ) - (hρ : ∀ z ∈ upperHalfPlane, ρ' ((c:ℂ)*z)=(c:ℝ)^(-codimension κ)*ρ z) : - (volume.withDensity (fun z => ENNReal.ofReal (ρ' z))).comap Subtype.val = - dilationMass κ c hc ((volume.withDensity (fun z => ENNReal.ofReal (ρ z))).comap Subtype.val) := by - let J := halfPlaneDilation c hc - have hpos : 0 < (c:ℝ) := hc - have hpower : ENNReal.ofReal ((c:ℝ)^2)*ENNReal.ofReal ((c:ℝ)^(-codimension κ))= - ENNReal.ofReal ((c:ℝ)^dimension κ) := by - rw [← ENNReal.ofReal_mul (sq_nonneg _),← Real.rpow_natCast (c:ℝ) 2, - ← Real.rpow_add hpos] - have hexp : (2:ℝ)+ -codimension κ=dimension κ := by unfold codimension; ring - norm_num only [Nat.cast_ofNat] - rw [hexp] - apply Measure.ext - intro A hA - have hpre : MeasurableSet (J ⁻¹' A) := J.continuous.measurable hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - have hKI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hpre - have himage : (fun z : ℂ => (c:ℂ)*z) '' ((Subtype.val : upperHalfPlane → ℂ) '' (J ⁻¹' A))= - (Subtype.val : upperHalfPlane → ℂ) '' A := by - ext z - constructor - · rintro ⟨_,⟨x,hx,rfl⟩,rfl⟩ - exact ⟨J x,hx,rfl⟩ - · rintro ⟨x,hx,rfl⟩ - refine ⟨(J.symm x:ℂ),⟨J.symm x,?_,rfl⟩,?_⟩ - · change J (J.symm x) ∈ A - simpa only [J.apply_symm_apply] using hx - · exact congrArg Subtype.val (J.apply_symm_apply x) - rw [dilationMass_apply κ c hc _ A hA, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI,withDensity_apply _ hKI,← himage, - source_scale_area_lintegral c hc] - have hi : (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' (J ⁻¹' A), - ENNReal.ofReal (ρ' ((c:ℂ)*z))) = - ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' (J ⁻¹' A), - ENNReal.ofReal ((c:ℝ)^(-codimension κ))*ENNReal.ofReal (ρ z) := by - apply setLIntegral_congr_fun hKI - rintro z ⟨x,_hx,rfl⟩ - dsimp only - rw [hρ x x.property,ENNReal.ofReal_mul (Real.rpow_nonneg c.coe_nonneg _)] - rw [hi,lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,← mul_assoc,hpower] - -lemma dilationMass_initialGreen (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) : - dilationMass κ c hc (initialGreenMeasure κ)=initialGreenMeasure κ := by - exact (dilationMass_realDensity κ c hc (initialGreen κ) (initialGreen κ) - (fun z hz => source_scale_initialGreen κ c hc z hz.le)).symm - -lemma dilationMass_bind {Ω : Type*} [MeasurableSpace Ω] - (P : Measure Ω) (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) : - P.bind (fun ω => dilationMass κ c hc (μ ω))=dilationMass κ c hc (P.bind μ) := by - have hM : Measurable (fun ω => dilationMass κ c hc (μ ω)) := - (conformalMass_measurable _ _).comp hμ - apply Measure.ext - intro A hA - have hpre : MeasurableSet ((halfPlaneDilation c hc) ⁻¹' A) := - (halfPlaneDilation c hc).continuous.measurable hA - rw [Measure.bind_apply hA hM.aemeasurable,dilationMass_apply κ c hc _ A hA] - simp_rw [dilationMass_apply κ c hc _ A hA] - rw [Measure.bind_apply hpre hμ.aemeasurable,lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] - -lemma dilationMass_mono (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) - {μ ν : Measure upperHalfPlane} (h : μ ≤ ν) : - dilationMass κ c hc μ ≤ dilationMass κ c hc ν := conformalMass_mono _ _ h - -lemma dilationMass_ne_zero_iff (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) - (μ : Measure upperHalfPlane) : dilationMass κ c hc μ ≠ 0 ↔ μ ≠ 0 := by - rw [← Measure.measure_univ_pos,← Measure.measure_univ_pos,dilationMass_apply κ c hc _ univ MeasurableSet.univ] - simp only [preimage_univ] - constructor - · intro h - by_contra hn - have hzero : μ univ=0 := le_antisymm (le_of_not_gt hn) bot_le - simp only [hzero,mul_zero,lt_self_iff_false] at h - · intro h - exact pos_iff_ne_zero.mpr (mul_ne_zero - (ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos hc _)).ne' h.ne') - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.ConformalMassTransport +import OAI.Probability.SLE.RetainedSeedMoments + +namespace OAI + +/-! Exact area and measure transport for positive real dilations of the +half-plane. These are the spatial factors in capacity scaling. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge + +lemma source_scale_area_lintegral (c : ℝ≥0) (hc : 0 < c) + (f : ℂ → ℝ≥0∞) (K : Set ℂ) : + (∫⁻ z in (fun z : ℂ => (c:ℂ)*z) '' K, f z) = + ENNReal.ofReal ((c:ℝ)^2) * ∫⁻ z in K, f ((c:ℂ)*z) := by + have hcR : 0 < (c:ℝ) := hc + have hcC : (c:ℂ) ≠ 0 := by exact_mod_cast hc.ne' + let e := Homeomorph.mulLeft₀ (c:ℂ) hcC + have hvol (s : Set ℂ) : volume (e ⁻¹' s)= + ENNReal.ofReal (((c:ℝ)^2)⁻¹)*volume s := by + change volume ((fun z : ℂ => (c:ℂ)*z) ⁻¹' s)=_ + simpa only [Complex.real_smul,smul_eq_mul,Complex.finrank_real_complex, + abs_of_nonneg (inv_nonneg.mpr (sq_nonneg (c:ℝ)))] using + (volume : Measure ℂ).addHaar_preimage_smul hcR.ne' s + have hp : MeasurePreserving e (ENNReal.ofReal ((c:ℝ)^2) • (volume : Measure ℂ)) volume := by + refine ⟨e.continuous.measurable,?_⟩ + apply Measure.ext + intro s hs + rw [e.measurableEmbedding.map_apply,Measure.smul_apply,smul_eq_mul,hvol,← mul_assoc, + ← ENNReal.ofReal_mul (sq_nonneg (c:ℝ)),mul_inv_cancel₀ (pow_ne_zero 2 hcR.ne'), + ENNReal.ofReal_one,one_mul] + have h := hp.setLIntegral_comp_emb e.measurableEmbedding f K + rw [Measure.restrict_smul,lintegral_smul_measure] at h + exact h.symm + +def halfPlaneDilation (c : ℝ≥0) (hc : 0 < c) : upperHalfPlane ≃ₜ upperHalfPlane where + toFun z := ⟨(c:ℂ)*(z:ℂ),by + change 0 < ((c:ℂ)*(z:ℂ)).im + simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero] + using mul_pos (show 0 < (c:ℝ) from hc) z.property⟩ + invFun z := ⟨(c:ℂ)⁻¹*(z:ℂ),by + change 0 < ((c:ℂ)⁻¹*(z:ℂ)).im + rw [← Complex.ofReal_inv] + simpa only [Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,add_zero] + using mul_pos (inv_pos.mpr (show 0 < (c:ℝ) from hc)) z.property⟩ + left_inv z := by + apply Subtype.ext + change (c:ℂ)⁻¹*((c:ℂ)*(z:ℂ))=(z:ℂ) + exact inv_mul_cancel_left₀ (by exact_mod_cast hc.ne') _ + right_inv z := by + apply Subtype.ext + change (c:ℂ)*((c:ℂ)⁻¹*(z:ℂ))=(z:ℂ) + exact mul_inv_cancel_left₀ (by exact_mod_cast hc.ne') _ + continuous_toFun := by fun_prop + continuous_invFun := by fun_prop + +@[simp] lemma halfPlaneDilation_apply (c : ℝ≥0) (hc : 0 < c) (z : upperHalfPlane) : + ((halfPlaneDilation c hc z : upperHalfPlane):ℂ)=(c:ℂ)*(z:ℂ) := rfl + +namespace Multipoint + +def dilationMass (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) (μ : Measure upperHalfPlane) : + Measure upperHalfPlane := + conformalMass (halfPlaneDilation c hc) (ContinuousMap.const _ (Real.toNNReal ((c:ℝ)^dimension κ))) μ + +lemma dilationMass_apply (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) + (μ : Measure upperHalfPlane) (A : Set upperHalfPlane) (hA : MeasurableSet A) : + dilationMass κ c hc μ A=ENNReal.ofReal ((c:ℝ)^dimension κ)* + μ ((halfPlaneDilation c hc) ⁻¹' A) := by + rw [dilationMass,conformalMass,Measure.map_apply (halfPlaneDilation c hc).continuous.measurable hA, + withDensity_apply _ ((halfPlaneDilation c hc).continuous.measurable hA)] + simp only [ContinuousMap.const_apply] + change (∫⁻ _ in (halfPlaneDilation c hc) ⁻¹' A, ENNReal.ofReal ((c:ℝ)^dimension κ) ∂μ)=_ + exact setLIntegral_const _ _ + +lemma dilationMass_realDensity (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) + (ρ ρ' : ℂ → ℝ) + (hρ : ∀ z ∈ upperHalfPlane, ρ' ((c:ℂ)*z)=(c:ℝ)^(-codimension κ)*ρ z) : + (volume.withDensity (fun z => ENNReal.ofReal (ρ' z))).comap Subtype.val = + dilationMass κ c hc ((volume.withDensity (fun z => ENNReal.ofReal (ρ z))).comap Subtype.val) := by + let J := halfPlaneDilation c hc + have hpos : 0 < (c:ℝ) := hc + have hpower : ENNReal.ofReal ((c:ℝ)^2)*ENNReal.ofReal ((c:ℝ)^(-codimension κ))= + ENNReal.ofReal ((c:ℝ)^dimension κ) := by + rw [← ENNReal.ofReal_mul (sq_nonneg _),← Real.rpow_natCast (c:ℝ) 2, + ← Real.rpow_add hpos] + have hexp : (2:ℝ)+ -codimension κ=dimension κ := by unfold codimension; ring + norm_num only [Nat.cast_ofNat] + rw [hexp] + apply Measure.ext + intro A hA + have hpre : MeasurableSet (J ⁻¹' A) := J.continuous.measurable hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + have hKI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hpre + have himage : (fun z : ℂ => (c:ℂ)*z) '' ((Subtype.val : upperHalfPlane → ℂ) '' (J ⁻¹' A))= + (Subtype.val : upperHalfPlane → ℂ) '' A := by + ext z + constructor + · rintro ⟨_,⟨x,hx,rfl⟩,rfl⟩ + exact ⟨J x,hx,rfl⟩ + · rintro ⟨x,hx,rfl⟩ + refine ⟨(J.symm x:ℂ),⟨J.symm x,?_,rfl⟩,?_⟩ + · change J (J.symm x) ∈ A + simpa only [J.apply_symm_apply] using hx + · exact congrArg Subtype.val (J.apply_symm_apply x) + rw [dilationMass_apply κ c hc _ A hA, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI,withDensity_apply _ hKI,← himage, + source_scale_area_lintegral c hc] + have hi : (∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' (J ⁻¹' A), + ENNReal.ofReal (ρ' ((c:ℂ)*z))) = + ∫⁻ z in (Subtype.val : upperHalfPlane → ℂ) '' (J ⁻¹' A), + ENNReal.ofReal ((c:ℝ)^(-codimension κ))*ENNReal.ofReal (ρ z) := by + apply setLIntegral_congr_fun hKI + rintro z ⟨x,_hx,rfl⟩ + dsimp only + rw [hρ x x.property,ENNReal.ofReal_mul (Real.rpow_nonneg c.coe_nonneg _)] + rw [hi,lintegral_const_mul' _ _ ENNReal.ofReal_ne_top,← mul_assoc,hpower] + +lemma dilationMass_initialGreen (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) : + dilationMass κ c hc (initialGreenMeasure κ)=initialGreenMeasure κ := by + exact (dilationMass_realDensity κ c hc (initialGreen κ) (initialGreen κ) + (fun z hz => source_scale_initialGreen κ c hc z hz.le)).symm + +lemma dilationMass_bind {Ω : Type*} [MeasurableSpace Ω] + (P : Measure Ω) (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) : + P.bind (fun ω => dilationMass κ c hc (μ ω))=dilationMass κ c hc (P.bind μ) := by + have hM : Measurable (fun ω => dilationMass κ c hc (μ ω)) := + (conformalMass_measurable _ _).comp hμ + apply Measure.ext + intro A hA + have hpre : MeasurableSet ((halfPlaneDilation c hc) ⁻¹' A) := + (halfPlaneDilation c hc).continuous.measurable hA + rw [Measure.bind_apply hA hM.aemeasurable,dilationMass_apply κ c hc _ A hA] + simp_rw [dilationMass_apply κ c hc _ A hA] + rw [Measure.bind_apply hpre hμ.aemeasurable,lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] + +lemma dilationMass_mono (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) + {μ ν : Measure upperHalfPlane} (h : μ ≤ ν) : + dilationMass κ c hc μ ≤ dilationMass κ c hc ν := conformalMass_mono _ _ h + +lemma dilationMass_ne_zero_iff (κ : ℝ) (c : ℝ≥0) (hc : 0 < c) + (μ : Measure upperHalfPlane) : dilationMass κ c hc μ ≠ 0 ↔ μ ≠ 0 := by + rw [← Measure.measure_univ_pos,← Measure.measure_univ_pos,dilationMass_apply κ c hc _ univ MeasurableSet.univ] + simp only [preimage_univ] + constructor + · intro h + by_contra hn + have hzero : μ univ=0 := le_antisymm (le_of_not_gt hn) bot_le + simp only [hzero,mul_zero,lt_self_iff_false] at h + · intro h + exact pos_iff_ne_zero.mpr (mul_ne_zero + (ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos hc _)).ne' h.ne') + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DiskChartMass.lean b/lean/OAI/Probability/SLE/Mass/DiskChartMass.lean index afef2f3fb87fdebc12d4b68290e04b515b9d9e49..b66a01c80a75166b9db5955463b2758892bb2464 100644 --- a/lean/OAI/Probability/SLE/Mass/DiskChartMass.lean +++ b/lean/OAI/Probability/SLE/Mass/DiskChartMass.lean @@ -1,113 +1,113 @@ -import OAI.Probability.SLE.Geometry.DiskQuarterDistortion -import OAI.Probability.SLE.Angle.TargetDiskMass -import OAI.Probability.SLE.Mass.CompactImageMassBounds - -namespace OAI - -/-! Exact disk-coordinate pushforward mass and its two-sided comparison. -The disk scalar is the same weighted canonical scalar used in the prescribed -moment coefficients. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Multipoint - -lemma Target.mem_diskSourceHalfPlane_iff {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (z : upperHalfPlane) : - z ∈ Target.diskSourceHalfPlane θ ↔ Target.diskMap θ z ∈ Target.diskK := by - constructor - · rintro ⟨w,hw,he⟩ - have hh := congrArg (Target.diskMap θ) he - rw [Target.diskMap_diskMapInv hθ (Target.diskK_subset_unit hw)] at hh - rwa [← hh] - · intro hz - exact ⟨Target.diskMap θ z,hz,Target.diskMapInv_diskMap hθ z.property⟩ - -lemma Multipoint.ofReal_compactWeightedMass - (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) - (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : - ENNReal.ofReal (compactWeightedMass w K μ) = ∫⁻ z in K, (w z:ℝ≥0∞) ∂μ := by - have hh := ofReal_integral_eq_lintegral_ofReal (compact_weight_integrable w K hK μ) - (ae_of_all _ (fun z => (w z).coe_nonneg)) - simpa only [compactWeightedMass,ENNReal.ofReal_coe_nnreal] using hh - -def diskChartMass (κ θ : ℝ) (f : ℂ → ℂ) (μ : Measure upperHalfPlane) : Measure ℂ := - weightedImageMass (fun z : upperHalfPlane => f (Target.diskMap θ z)) - (fun z : upperHalfPlane => ENNReal.ofReal - (‖deriv (f ∘ Target.diskMap θ) z‖^dimension κ)) μ - -lemma diskChartMass_quarter_bounds (hgrowth : PublishedDiskUnivalentGrowth) - (κ : ℝ) (hκ : 0 < κ) (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (f : ℂ → ℂ) (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) - (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : - ENNReal.ofReal ((‖deriv f 0‖/16)^dimension κ) * - ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) - (Target.diskSourceHalfPlane θ) μ) ≤ diskChartMass κ θ f μ (f '' Target.diskK) ∧ - diskChartMass κ θ f μ (f '' Target.diskK) ≤ - ENNReal.ofReal ((16*‖deriv f 0‖)^dimension κ) * - ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) - (Target.diskSourceHalfPlane θ) μ) := by - let K := Target.diskSourceHalfPlane θ - let J := fun z : upperHalfPlane => f (Target.diskMap θ z) - let v := fun z : upperHalfPlane => (Target.diskMassWeight κ θ hθ z:ℝ≥0∞) - let w := fun z : upperHalfPlane => ENNReal.ofReal (‖deriv (f ∘ Target.diskMap θ) z‖^dimension κ) - have hK : IsCompact K := Target.diskSourceHalfPlane_compact hθ - have hmap := Target.diskMap_maps hθ - have hJ : Measurable J := - (hd.continuousOn.comp_continuous - ((Target.diskHomeomorph θ hθ).continuous_toFun.subtype_val) - (fun z => hmap z.property)).measurable - have himage : IsCompact (f '' Target.diskK) := - (isCompact_closedBall (0:ℂ) (1/4)).image_of_continuousOn - (hd.continuousOn.mono Target.diskK_subset_unit) - have hpre : J ⁻¹' (f '' Target.diskK)=K := by - ext z - rw [Target.mem_diskSourceHalfPlane_iff hθ] - constructor - · rintro ⟨x,hx,hxe⟩ - have he : x=Target.diskMap θ z := hi (Target.diskK_subset_unit hx) (hmap z.property) hxe - rwa [← he] - · intro hz - exact ⟨Target.diskMap θ z,hz,rfl⟩ - have he : diskChartMass κ θ f μ (f '' Target.diskK) = ∫⁻ z in K, w z ∂μ := by - change weightedImageMass J w μ _ = _ - rw [weightedImageMass,Measure.map_apply hJ himage.measurableSet,hpre,withDensity_apply _ hK.measurableSet] - have hw (z : upperHalfPlane) : w z = - ENNReal.ofReal (‖deriv f (Target.diskMap θ z)‖^dimension κ)*v z := by - dsimp only [w,v] - rw [Target.diskMassWeight_ennreal] - symm - rw [mul_comm] - exact derivative_dimension_weight_comp κ f (Target.diskMap θ) z - (hd.differentiableAt (isOpen_ball.mem_nhds (hmap z.property))) - (Target.diskMap_hasDerivAt hθ z.property).differentiableAt - have hd0 : 0 ≤ dimension κ := by unfold dimension; linarith - have hbounds (z : upperHalfPlane) (hz : z∈K) : - ENNReal.ofReal ((‖deriv f 0‖/16)^dimension κ)*v z ≤ w z ∧ - w z ≤ ENNReal.ofReal ((16*‖deriv f 0‖)^dimension κ)*v z := by - have hz' : Target.diskMap θ z ∈ Target.diskK := (Target.mem_diskSourceHalfPlane_iff hθ z).mp hz - have hn : ‖Target.diskMap θ z‖ ≤ 1/4 := by - simpa only [Target.diskK,mem_closedBall,dist_zero_right] using hz' - have hb := univalent_quarter_derivative_bounds hgrowth hd hi hn - rw [hw] - constructor - · apply mul_le_mul' _ le_rfl - exact ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (by positivity) hb.1 hd0) - · apply mul_le_mul' _ le_rfl - exact ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (norm_nonneg _) hb.2 hd0) - rw [ofReal_compactWeightedMass _ _ hK μ,he] - change ENNReal.ofReal ((‖deriv f 0‖/16)^dimension κ)*(∫⁻ z in K, v z ∂μ) ≤ _ ∧ - _ ≤ ENNReal.ofReal ((16*‖deriv f 0‖)^dimension κ)*(∫⁻ z in K, v z ∂μ) - rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top, - ← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] - exact ⟨setLIntegral_mono' hK.measurableSet (fun z hz => (hbounds z hz).1), - setLIntegral_mono' hK.measurableSet (fun z hz => (hbounds z hz).2)⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.DiskQuarterDistortion +import OAI.Probability.SLE.Angle.TargetDiskMass +import OAI.Probability.SLE.Mass.CompactImageMassBounds + +namespace OAI + +/-! Exact disk-coordinate pushforward mass and its two-sided comparison. +The disk scalar is the same weighted canonical scalar used in the prescribed +moment coefficients. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Multipoint + +lemma Target.mem_diskSourceHalfPlane_iff {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (z : upperHalfPlane) : + z ∈ Target.diskSourceHalfPlane θ ↔ Target.diskMap θ z ∈ Target.diskK := by + constructor + · rintro ⟨w,hw,he⟩ + have hh := congrArg (Target.diskMap θ) he + rw [Target.diskMap_diskMapInv hθ (Target.diskK_subset_unit hw)] at hh + rwa [← hh] + · intro hz + exact ⟨Target.diskMap θ z,hz,Target.diskMapInv_diskMap hθ z.property⟩ + +lemma Multipoint.ofReal_compactWeightedMass + (w : C(upperHalfPlane,ℝ≥0)) (K : Set upperHalfPlane) (hK : IsCompact K) + (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : + ENNReal.ofReal (compactWeightedMass w K μ) = ∫⁻ z in K, (w z:ℝ≥0∞) ∂μ := by + have hh := ofReal_integral_eq_lintegral_ofReal (compact_weight_integrable w K hK μ) + (ae_of_all _ (fun z => (w z).coe_nonneg)) + simpa only [compactWeightedMass,ENNReal.ofReal_coe_nnreal] using hh + +def diskChartMass (κ θ : ℝ) (f : ℂ → ℂ) (μ : Measure upperHalfPlane) : Measure ℂ := + weightedImageMass (fun z : upperHalfPlane => f (Target.diskMap θ z)) + (fun z : upperHalfPlane => ENNReal.ofReal + (‖deriv (f ∘ Target.diskMap θ) z‖^dimension κ)) μ + +lemma diskChartMass_quarter_bounds (hgrowth : PublishedDiskUnivalentGrowth) + (κ : ℝ) (hκ : 0 < κ) (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (f : ℂ → ℂ) (hd : DifferentiableOn ℂ f (ball 0 1)) (hi : InjOn f (ball 0 1)) + (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : + ENNReal.ofReal ((‖deriv f 0‖/16)^dimension κ) * + ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) + (Target.diskSourceHalfPlane θ) μ) ≤ diskChartMass κ θ f μ (f '' Target.diskK) ∧ + diskChartMass κ θ f μ (f '' Target.diskK) ≤ + ENNReal.ofReal ((16*‖deriv f 0‖)^dimension κ) * + ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) + (Target.diskSourceHalfPlane θ) μ) := by + let K := Target.diskSourceHalfPlane θ + let J := fun z : upperHalfPlane => f (Target.diskMap θ z) + let v := fun z : upperHalfPlane => (Target.diskMassWeight κ θ hθ z:ℝ≥0∞) + let w := fun z : upperHalfPlane => ENNReal.ofReal (‖deriv (f ∘ Target.diskMap θ) z‖^dimension κ) + have hK : IsCompact K := Target.diskSourceHalfPlane_compact hθ + have hmap := Target.diskMap_maps hθ + have hJ : Measurable J := + (hd.continuousOn.comp_continuous + ((Target.diskHomeomorph θ hθ).continuous_toFun.subtype_val) + (fun z => hmap z.property)).measurable + have himage : IsCompact (f '' Target.diskK) := + (isCompact_closedBall (0:ℂ) (1/4)).image_of_continuousOn + (hd.continuousOn.mono Target.diskK_subset_unit) + have hpre : J ⁻¹' (f '' Target.diskK)=K := by + ext z + rw [Target.mem_diskSourceHalfPlane_iff hθ] + constructor + · rintro ⟨x,hx,hxe⟩ + have he : x=Target.diskMap θ z := hi (Target.diskK_subset_unit hx) (hmap z.property) hxe + rwa [← he] + · intro hz + exact ⟨Target.diskMap θ z,hz,rfl⟩ + have he : diskChartMass κ θ f μ (f '' Target.diskK) = ∫⁻ z in K, w z ∂μ := by + change weightedImageMass J w μ _ = _ + rw [weightedImageMass,Measure.map_apply hJ himage.measurableSet,hpre,withDensity_apply _ hK.measurableSet] + have hw (z : upperHalfPlane) : w z = + ENNReal.ofReal (‖deriv f (Target.diskMap θ z)‖^dimension κ)*v z := by + dsimp only [w,v] + rw [Target.diskMassWeight_ennreal] + symm + rw [mul_comm] + exact derivative_dimension_weight_comp κ f (Target.diskMap θ) z + (hd.differentiableAt (isOpen_ball.mem_nhds (hmap z.property))) + (Target.diskMap_hasDerivAt hθ z.property).differentiableAt + have hd0 : 0 ≤ dimension κ := by unfold dimension; linarith + have hbounds (z : upperHalfPlane) (hz : z∈K) : + ENNReal.ofReal ((‖deriv f 0‖/16)^dimension κ)*v z ≤ w z ∧ + w z ≤ ENNReal.ofReal ((16*‖deriv f 0‖)^dimension κ)*v z := by + have hz' : Target.diskMap θ z ∈ Target.diskK := (Target.mem_diskSourceHalfPlane_iff hθ z).mp hz + have hn : ‖Target.diskMap θ z‖ ≤ 1/4 := by + simpa only [Target.diskK,mem_closedBall,dist_zero_right] using hz' + have hb := univalent_quarter_derivative_bounds hgrowth hd hi hn + rw [hw] + constructor + · apply mul_le_mul' _ le_rfl + exact ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (by positivity) hb.1 hd0) + · apply mul_le_mul' _ le_rfl + exact ENNReal.ofReal_le_ofReal (Real.rpow_le_rpow (norm_nonneg _) hb.2 hd0) + rw [ofReal_compactWeightedMass _ _ hK μ,he] + change ENNReal.ofReal ((‖deriv f 0‖/16)^dimension κ)*(∫⁻ z in K, v z ∂μ) ≤ _ ∧ + _ ≤ ENNReal.ofReal ((16*‖deriv f 0‖)^dimension κ)*(∫⁻ z in K, v z ∂μ) + rw [← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top, + ← lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] + exact ⟨setLIntegral_mono' hK.measurableSet (fun z hz => (hbounds z hz).1), + setLIntegral_mono' hK.measurableSet (fun z hz => (hbounds z hz).2)⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DiskLevelMoments.lean b/lean/OAI/Probability/SLE/Mass/DiskLevelMoments.lean index c0e277cd29ede2dc90086a4a07910f6a51451956..e667b37dd6e2db2d13fa3ca695ce88ca539bbc55 100644 --- a/lean/OAI/Probability/SLE/Mass/DiskLevelMoments.lean +++ b/lean/OAI/Probability/SLE/Mass/DiskLevelMoments.lean @@ -1,126 +1,126 @@ -import OAI.Probability.SLE.Geometry.PhysicalCenteredWeights -import OAI.Probability.SLE.Mass.CompactLevelMoments -import OAI.Probability.SLE.Angle.TargetPotentialMeasurability - -namespace OAI - -/-! Centered level moments in the prescribed physical disk. The radius -cap and spatial kernel are independent of the chart angle. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -def diskStoppedGreen {Ω : Type*} (κ θ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (ω : Ω) (x : ℂ) : ℝ := - ‖deriv (Target.diskMap θ) (Target.diskMapInv θ x)‖^(-codimension κ) * - stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (Target.diskMapInv θ x) u - -def diskLevelMass {Ω : Type*} (κ θ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (K : Set ℂ) (ω : Ω) : ℝ≥0∞ := - ∫⁻ x in K, ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω x) - -def diskCenterHit {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) - (G : Ω → ℝ≥0 → ℂ → ℂ) (θ e : ℝ) : Set Ω := - {ω | ∃ t : ℝ≥0, Target.diskMapInv θ 0 ∈ survivingDomain (γ ω) t ∧ - physicalConformalRadius (Target.diskMap θ) (G ω) t (Target.diskMapInv θ 0) ≤ e} - -lemma diskStoppedGreen_nonneg {Ω : Type*} {κ θ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (ω : Ω) (x : ℂ) : 0 ≤ diskStoppedGreen κ θ B γ G u ω x := - mul_nonneg (Real.rpow_nonneg (norm_nonneg _) _) (stoppedGreen_nonneg (hw ω) κ _ _) - -lemma measurable_diskStoppedGreen - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ θ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) : - Measurable (Function.uncurry (diskStoppedGreen κ θ B γ G u)) := by - have hi : Measurable (fun p : Ω × ℂ => Target.diskMapInv θ p.2) := - (Target.measurable_diskMapInv θ).comp measurable_snd - have hd : Measurable (fun p : Ω × ℂ => - ‖deriv (Target.diskMap θ) (Target.diskMapInv θ p.2)‖^(-codimension κ)) := - ((measurable_deriv (Target.diskMap θ)).comp hi).norm.pow_const _ - exact hd.mul ((actual_stoppedGreen_joint_measurable hm hc κ u hu γ G F hw).comp - (measurable_fst.prodMk hi)) - -lemma measurableSet_diskCenterHit - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (e : ℝ) (he : 0 Target.diskMapInv θ 0))) - have heq : diskCenterHit γ G θ e = - {ω | radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (Real.log (1/e)) < ∞} := by - ext ω - exact physical_radius_hit_iff_finite_level ((hc ω).const_mul _) (hw ω) - (Target.diskMap θ) (Target.diskMapInv θ 0) 1 e zero_lt_one he he1 - (Target.initialRadius_zero hθ).symm - rw [heq] - exact measurableSet_lt hτ measurable_const - -theorem diskStoppedGreen_centered_product_bound - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (m : ℕ) - (x : Fin m → ℂ) (hx : ∀ i, x i ∈ ball 0 1) - (hinj : Function.Injective (Fin.cons (0:ℂ) x)) - (b : ℝ) (hb : 0 ENNReal.ofReal (∏ i, diskStoppedGreen κ θ B γ G u ω (x i))) ω ∂P) ≤ - ENNReal.ofReal (multipointConstant κ (m+1)*e^codimension κ) * - cappedDistanceKernel (codimension κ) b (Fin.cons 0 x) := by - let y : Fin (m+1) → ℂ := Fin.cons (0:ℂ) x - let z := fun i => Target.diskMapInv θ (y i) - have hy (i) : y i ∈ ball 0 1 := Fin.cases (by change (0:ℂ)∈ball 0 1; simp) hx i - have hz (i) : z i ∈ upperHalfPlane := Target.diskMapInv_maps hθ (hy i) - have heq (i) : Target.diskMap θ (z i)=y i := Target.diskMap_diskMapInv hθ (hy i) - have hzi : Function.Injective z := fun i j h => hinj (by - simpa only [heq] using congrArg (Target.diskMap θ) h) - have hbR (i) : b≤2*(z i).im*‖deriv (Target.diskMap θ) (z i)‖ := by - rw [Target.diskMap_initial_radius hθ (hz i),heq] - exact Fin.cases (by simpa only [y,Fin.cons_zero,norm_zero,pow_two,mul_zero,sub_zero] using hb1) hbx i - let es : Fin (m+1) → ℝ := Fin.cons e - (fun i => (2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖)*Real.exp (-u)) - have hes (i) : 0 - mul_pos (hb.trans_le (hbR i.succ)) (Real.exp_pos _)) i - have hu (i : Fin m) : Real.log - (2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖/es i.succ)=u := by - change Real.log ((2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖)/ - ((2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖)*Real.exp (-u)))=u - rw [div_mul_eq_div_div,div_self (hb.trans_le (hbR i.succ)).ne',one_div,← Real.exp_neg, - neg_neg,Real.log_exp] - have hbound := physical_stoppedGreen_centered_product_bound hB hm hBm hc h0 hκ hκ8 - γ G F hw hgrowth (Target.diskMap θ) (Target.diskMap_differentiableOn hθ) - (Target.diskMap_injOn hθ) m z hz hzi (cappedDistance b y) es hes - (fun i => cap_pos hb hinj i) (fun i => (cap_le_fixed b y i).trans (hbR i)) - (fun i j hij => by simpa only [heq] using cap_le_dist b y i j hij) - simp only [hu,es,Fin.cons_zero] at hbound - apply hbound.trans_eq - rw [ENNReal.ofReal_mul (mul_nonneg (multipointConstant_pos _ _).le - (Real.rpow_nonneg he.le _))] - congr 1 - exact ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (cap_pos hb hinj i).le _) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.PhysicalCenteredWeights +import OAI.Probability.SLE.Mass.CompactLevelMoments +import OAI.Probability.SLE.Angle.TargetPotentialMeasurability + +namespace OAI + +/-! Centered level moments in the prescribed physical disk. The radius +cap and spatial kernel are independent of the chart angle. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +def diskStoppedGreen {Ω : Type*} (κ θ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (ω : Ω) (x : ℂ) : ℝ := + ‖deriv (Target.diskMap θ) (Target.diskMapInv θ x)‖^(-codimension κ) * + stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (Target.diskMapInv θ x) u + +def diskLevelMass {Ω : Type*} (κ θ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (K : Set ℂ) (ω : Ω) : ℝ≥0∞ := + ∫⁻ x in K, ENNReal.ofReal (diskStoppedGreen κ θ B γ G u ω x) + +def diskCenterHit {Ω : Type*} (γ : Ω → ℝ≥0 → ℂ) + (G : Ω → ℝ≥0 → ℂ → ℂ) (θ e : ℝ) : Set Ω := + {ω | ∃ t : ℝ≥0, Target.diskMapInv θ 0 ∈ survivingDomain (γ ω) t ∧ + physicalConformalRadius (Target.diskMap θ) (G ω) t (Target.diskMapInv θ 0) ≤ e} + +lemma diskStoppedGreen_nonneg {Ω : Type*} {κ θ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (ω : Ω) (x : ℂ) : 0 ≤ diskStoppedGreen κ θ B γ G u ω x := + mul_nonneg (Real.rpow_nonneg (norm_nonneg _) _) (stoppedGreen_nonneg (hw ω) κ _ _) + +lemma measurable_diskStoppedGreen + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ θ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) : + Measurable (Function.uncurry (diskStoppedGreen κ θ B γ G u)) := by + have hi : Measurable (fun p : Ω × ℂ => Target.diskMapInv θ p.2) := + (Target.measurable_diskMapInv θ).comp measurable_snd + have hd : Measurable (fun p : Ω × ℂ => + ‖deriv (Target.diskMap θ) (Target.diskMapInv θ p.2)‖^(-codimension κ)) := + ((measurable_deriv (Target.diskMap θ)).comp hi).norm.pow_const _ + exact hd.mul ((actual_stoppedGreen_joint_measurable hm hc κ u hu γ G F hw).comp + (measurable_fst.prodMk hi)) + +lemma measurableSet_diskCenterHit + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (e : ℝ) (he : 0 Target.diskMapInv θ 0))) + have heq : diskCenterHit γ G θ e = + {ω | radiusLevelTime (γ ω) (G ω) (Target.diskMapInv θ 0) (Real.log (1/e)) < ∞} := by + ext ω + exact physical_radius_hit_iff_finite_level ((hc ω).const_mul _) (hw ω) + (Target.diskMap θ) (Target.diskMapInv θ 0) 1 e zero_lt_one he he1 + (Target.initialRadius_zero hθ).symm + rw [heq] + exact measurableSet_lt hτ measurable_const + +theorem diskStoppedGreen_centered_product_bound + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (m : ℕ) + (x : Fin m → ℂ) (hx : ∀ i, x i ∈ ball 0 1) + (hinj : Function.Injective (Fin.cons (0:ℂ) x)) + (b : ℝ) (hb : 0 ENNReal.ofReal (∏ i, diskStoppedGreen κ θ B γ G u ω (x i))) ω ∂P) ≤ + ENNReal.ofReal (multipointConstant κ (m+1)*e^codimension κ) * + cappedDistanceKernel (codimension κ) b (Fin.cons 0 x) := by + let y : Fin (m+1) → ℂ := Fin.cons (0:ℂ) x + let z := fun i => Target.diskMapInv θ (y i) + have hy (i) : y i ∈ ball 0 1 := Fin.cases (by change (0:ℂ)∈ball 0 1; simp) hx i + have hz (i) : z i ∈ upperHalfPlane := Target.diskMapInv_maps hθ (hy i) + have heq (i) : Target.diskMap θ (z i)=y i := Target.diskMap_diskMapInv hθ (hy i) + have hzi : Function.Injective z := fun i j h => hinj (by + simpa only [heq] using congrArg (Target.diskMap θ) h) + have hbR (i) : b≤2*(z i).im*‖deriv (Target.diskMap θ) (z i)‖ := by + rw [Target.diskMap_initial_radius hθ (hz i),heq] + exact Fin.cases (by simpa only [y,Fin.cons_zero,norm_zero,pow_two,mul_zero,sub_zero] using hb1) hbx i + let es : Fin (m+1) → ℝ := Fin.cons e + (fun i => (2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖)*Real.exp (-u)) + have hes (i) : 0 + mul_pos (hb.trans_le (hbR i.succ)) (Real.exp_pos _)) i + have hu (i : Fin m) : Real.log + (2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖/es i.succ)=u := by + change Real.log ((2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖)/ + ((2*(z i.succ).im*‖deriv (Target.diskMap θ) (z i.succ)‖)*Real.exp (-u)))=u + rw [div_mul_eq_div_div,div_self (hb.trans_le (hbR i.succ)).ne',one_div,← Real.exp_neg, + neg_neg,Real.log_exp] + have hbound := physical_stoppedGreen_centered_product_bound hB hm hBm hc h0 hκ hκ8 + γ G F hw hgrowth (Target.diskMap θ) (Target.diskMap_differentiableOn hθ) + (Target.diskMap_injOn hθ) m z hz hzi (cappedDistance b y) es hes + (fun i => cap_pos hb hinj i) (fun i => (cap_le_fixed b y i).trans (hbR i)) + (fun i j hij => by simpa only [heq] using cap_le_dist b y i j hij) + simp only [hu,es,Fin.cons_zero] at hbound + apply hbound.trans_eq + rw [ENNReal.ofReal_mul (mul_nonneg (multipointConstant_pos _ _).le + (Real.rpow_nonneg he.le _))] + congr 1 + exact ENNReal.ofReal_prod_of_nonneg (fun i _ => Real.rpow_nonneg (cap_pos hb hinj i).le _) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DiskLevelTests.lean b/lean/OAI/Probability/SLE/Mass/DiskLevelTests.lean index 3aac025aaa04a7a4d2f9cad38c0ea873f0e824b0..8970bc2fd0db0fc51ebfadda18a4e7bb7a0a1765 100644 --- a/lean/OAI/Probability/SLE/Mass/DiskLevelTests.lean +++ b/lean/OAI/Probability/SLE/Mass/DiskLevelTests.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.Gauge.DiskIntegratedMoments -import OAI.Probability.SLE.Mass.DiskLevelTransport -import OAI.Probability.SLE.Mass.CanonicalImageMass -import OAI.Probability.SLE.Mass.WeightedLevelMoments - -namespace OAI - -/-! Positive compact disk tests are dominated by physical level mass, -with the exact conformal derivative weight. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -lemma ofReal_disk_level_test_le - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (K : Set ℂ) (hK : IsCompact K) (hKd : K ⊆ ball 0 1) - (f : C_c((ball (0:ℂ) 1),ℝ)) (hf : ∀ x, 0≤f x) (hf1 : ∀ x, f x≤1) - (hfs : ∀ x : (ball (0:ℂ) 1), (x:ℂ)∉K → f x=0) - (u : ℝ) (hu : 0≤u) (t : ℝ≥0∞) (ω : Ω) : - ENNReal.ofReal (∫ x, f x ∂conformalMass (Target.diskHomeomorph θ hθ) - (Target.diskMassWeight κ θ hθ) (halfPlaneLevelMeasure κ B γ G u t ω)) ≤ - diskLevelMass κ θ B γ G u K ω := by - let g := conformalMassTest (Target.diskHomeomorph θ hθ) (Target.diskMassWeight κ θ hθ) f - let S := Target.diskMapInv θ '' K - have hS : IsCompact S := Target.diskCompactPreimage_compact θ hK hKd - have hSH : S ⊆ upperHalfPlane := Target.diskCompactPreimage_subset hθ hKd - have hg : ∀ z, 0≤g z := fun z => mul_nonneg (NNReal.coe_nonneg _) (hf _) - have hgz (z : ℂ) (hz : z∉S) : extendHalfPlaneTest g z=0 := by - by_cases hzH : z∈upperHalfPlane - · rw [show z=(⟨z,hzH⟩:upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] - have hn : ((Target.diskHomeomorph θ hθ ⟨z,hzH⟩):ℂ)∉K := by - intro h - exact hz ⟨_,h,Target.diskMapInv_diskMap hθ hzH⟩ - change (Target.diskMassWeight κ θ hθ ⟨z,hzH⟩:ℝ)* - f (Target.diskHomeomorph θ hθ ⟨z,hzH⟩)=0 - rw [hfs _ hn,mul_zero] - · exact extendHalfPlaneTest_of_not_mem g z hzH - rw [conformalMass_integral] - change ENNReal.ofReal (∫ z, g z ∂halfPlaneLevelMeasure κ B γ G u t ω) ≤ _ - rw [ofReal_integral_halfPlaneLevelMeasure hκ hκ8 γ G F ω (hc ω) (hw ω) u t g hg] - apply (levelApproximationTest_le_weightedLevelMass hm hc κ u hu γ G F hw S hS.measurableSet - (fun z => ENNReal.ofReal (extendHalfPlaneTest g z)) - (extendHalfPlaneTest_measurable g g.continuous.measurable).ennreal_ofReal - (fun z hz => by rw [hgz z hz,ENNReal.ofReal_zero]) t ω).trans - calc - weightedLevelMass κ B γ G u S (fun z => ENNReal.ofReal (extendHalfPlaneTest g z)) ω ≤ - ∫⁻ z in S, ENNReal.ofReal (‖deriv (Target.diskMap θ) z‖^dimension κ)* - ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u) := by - apply setLIntegral_mono' hS.measurableSet - intro z hz - apply mul_le_mul' ?_ le_rfl - apply ENNReal.ofReal_le_ofReal - rw [show z=(⟨z,hSH hz⟩:upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] - change (Target.diskMassWeight κ θ hθ ⟨z,hSH hz⟩:ℝ)* - f (Target.diskHomeomorph θ hθ ⟨z,hSH hz⟩) ≤ _ - rw [Target.diskMassWeight_coe] - exact mul_le_of_le_one_right (Real.rpow_nonneg (norm_nonneg _) _) (hf1 _) - _ = diskLevelMass κ θ B γ G u K ω := by - rw [← Target.disk_level_integral_transport κ θ hθ K hK hKd] - apply setLIntegral_congr_fun hK.measurableSet - intro x hx - exact (ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _)).symm - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Gauge.DiskIntegratedMoments +import OAI.Probability.SLE.Mass.DiskLevelTransport +import OAI.Probability.SLE.Mass.CanonicalImageMass +import OAI.Probability.SLE.Mass.WeightedLevelMoments + +namespace OAI + +/-! Positive compact disk tests are dominated by physical level mass, +with the exact conformal derivative weight. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +lemma ofReal_disk_level_test_le + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + {κ : ℝ} (hκ : 0<κ) (hκ8 : κ<8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (K : Set ℂ) (hK : IsCompact K) (hKd : K ⊆ ball 0 1) + (f : C_c((ball (0:ℂ) 1),ℝ)) (hf : ∀ x, 0≤f x) (hf1 : ∀ x, f x≤1) + (hfs : ∀ x : (ball (0:ℂ) 1), (x:ℂ)∉K → f x=0) + (u : ℝ) (hu : 0≤u) (t : ℝ≥0∞) (ω : Ω) : + ENNReal.ofReal (∫ x, f x ∂conformalMass (Target.diskHomeomorph θ hθ) + (Target.diskMassWeight κ θ hθ) (halfPlaneLevelMeasure κ B γ G u t ω)) ≤ + diskLevelMass κ θ B γ G u K ω := by + let g := conformalMassTest (Target.diskHomeomorph θ hθ) (Target.diskMassWeight κ θ hθ) f + let S := Target.diskMapInv θ '' K + have hS : IsCompact S := Target.diskCompactPreimage_compact θ hK hKd + have hSH : S ⊆ upperHalfPlane := Target.diskCompactPreimage_subset hθ hKd + have hg : ∀ z, 0≤g z := fun z => mul_nonneg (NNReal.coe_nonneg _) (hf _) + have hgz (z : ℂ) (hz : z∉S) : extendHalfPlaneTest g z=0 := by + by_cases hzH : z∈upperHalfPlane + · rw [show z=(⟨z,hzH⟩:upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] + have hn : ((Target.diskHomeomorph θ hθ ⟨z,hzH⟩):ℂ)∉K := by + intro h + exact hz ⟨_,h,Target.diskMapInv_diskMap hθ hzH⟩ + change (Target.diskMassWeight κ θ hθ ⟨z,hzH⟩:ℝ)* + f (Target.diskHomeomorph θ hθ ⟨z,hzH⟩)=0 + rw [hfs _ hn,mul_zero] + · exact extendHalfPlaneTest_of_not_mem g z hzH + rw [conformalMass_integral] + change ENNReal.ofReal (∫ z, g z ∂halfPlaneLevelMeasure κ B γ G u t ω) ≤ _ + rw [ofReal_integral_halfPlaneLevelMeasure hκ hκ8 γ G F ω (hc ω) (hw ω) u t g hg] + apply (levelApproximationTest_le_weightedLevelMass hm hc κ u hu γ G F hw S hS.measurableSet + (fun z => ENNReal.ofReal (extendHalfPlaneTest g z)) + (extendHalfPlaneTest_measurable g g.continuous.measurable).ennreal_ofReal + (fun z hz => by rw [hgz z hz,ENNReal.ofReal_zero]) t ω).trans + calc + weightedLevelMass κ B γ G u S (fun z => ENNReal.ofReal (extendHalfPlaneTest g z)) ω ≤ + ∫⁻ z in S, ENNReal.ofReal (‖deriv (Target.diskMap θ) z‖^dimension κ)* + ENNReal.ofReal (stoppedGreen κ (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) z u) := by + apply setLIntegral_mono' hS.measurableSet + intro z hz + apply mul_le_mul' ?_ le_rfl + apply ENNReal.ofReal_le_ofReal + rw [show z=(⟨z,hSH hz⟩:upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] + change (Target.diskMassWeight κ θ hθ ⟨z,hSH hz⟩:ℝ)* + f (Target.diskHomeomorph θ hθ ⟨z,hSH hz⟩) ≤ _ + rw [Target.diskMassWeight_coe] + exact mul_le_of_le_one_right (Real.rpow_nonneg (norm_nonneg _) _) (hf1 _) + _ = diskLevelMass κ θ B γ G u K ω := by + rw [← Target.disk_level_integral_transport κ θ hθ K hK hKd] + apply setLIntegral_congr_fun hK.measurableSet + intro x hx + exact (ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _)).symm + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DiskLevelTransport.lean b/lean/OAI/Probability/SLE/Mass/DiskLevelTransport.lean index 0a3f24ed91bee4f46145f39163a6acca818a6048..49859fec8d57b2abcffd6b259ebf14270a4426de 100644 --- a/lean/OAI/Probability/SLE/Mass/DiskLevelTransport.lean +++ b/lean/OAI/Probability/SLE/Mass/DiskLevelTransport.lean @@ -1,69 +1,69 @@ -import OAI.Probability.SLE.Angle.TargetDiskMass - -namespace OAI - -/-! Area transport for the disk Green weights at individual radius -levels. The derivative squared from area cancels the Green codimension -factor, leaving precisely the canonical mass weight. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Target - -lemma diskCompactPreimage_subset - {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - {K : Set ℂ} (hK : K ⊆ ball 0 1) : - diskMapInv θ '' K ⊆ upperHalfPlane := by - rintro z ⟨x,hx,rfl⟩ - exact diskMapInv_maps hθ (hK hx) - -lemma diskCompactPreimage_compact (θ : ℝ) - {K : Set ℂ} (hc : IsCompact K) (hK : K ⊆ ball 0 1) : - IsCompact (diskMapInv θ '' K) := by - apply hc.image_of_continuousOn - change ContinuousOn (fun x : ℂ => - (Complex.exp (θ*Complex.I)-x*Complex.exp (-θ*Complex.I))/(1-x)) K - exact (continuous_const.continuousOn.sub - (continuous_id.mul continuous_const).continuousOn).div - (continuous_const.sub continuous_id).continuousOn (fun x hx => disk_denominator_ne (hK hx)) - -theorem disk_level_integral_transport - (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (K : Set ℂ) (hc : IsCompact K) (hK : K ⊆ ball 0 1) - (g : ℂ → ℝ≥0∞) : - (∫⁻ x in K, ENNReal.ofReal - (‖deriv (diskMap θ) (diskMapInv θ x)‖^(-codimension κ))*g (diskMapInv θ x)) = - ∫⁻ z in diskMapInv θ '' K, - ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ)*g z := by - have hs := diskCompactPreimage_subset hθ hK - have hcomp := diskCompactPreimage_compact θ hc hK - have himage : diskMap θ '' (diskMapInv θ '' K)=K := by - ext x - constructor - · rintro ⟨z,⟨y,hy,rfl⟩,rfl⟩ - simpa only [diskMap_diskMapInv hθ (hK hy)] using hy - · intro hx - exact ⟨diskMapInv θ x,⟨x,hx,rfl⟩,diskMap_diskMapInv hθ (hK hx)⟩ - have hd (z : ℂ) (hz : z ∈ diskMapInv θ '' K) := diskMap_hasDerivAt hθ (hs hz) - conv_lhs => rw [← himage] - rw [lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hcomp.measurableSet - (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) - ((diskMap_injOn hθ).mono hs)] - apply setLIntegral_congr_fun hcomp.measurableSet - intro z hz - dsimp only - rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _), - diskMapInv_diskMap hθ (hs hz),← mul_assoc,← ENNReal.ofReal_mul (sq_nonneg _)] - congr 1 - rw [← (hd z hz).deriv,← Real.rpow_natCast,← Real.rpow_add (diskMap_deriv_pos hθ (hs hz))] - congr 1 - unfold codimension dimension - ring_nf - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.Angle.TargetDiskMass + +namespace OAI + +/-! Area transport for the disk Green weights at individual radius +levels. The derivative squared from area cancels the Green codimension +factor, leaving precisely the canonical mass weight. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Target + +lemma diskCompactPreimage_subset + {θ : ℝ} (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + {K : Set ℂ} (hK : K ⊆ ball 0 1) : + diskMapInv θ '' K ⊆ upperHalfPlane := by + rintro z ⟨x,hx,rfl⟩ + exact diskMapInv_maps hθ (hK hx) + +lemma diskCompactPreimage_compact (θ : ℝ) + {K : Set ℂ} (hc : IsCompact K) (hK : K ⊆ ball 0 1) : + IsCompact (diskMapInv θ '' K) := by + apply hc.image_of_continuousOn + change ContinuousOn (fun x : ℂ => + (Complex.exp (θ*Complex.I)-x*Complex.exp (-θ*Complex.I))/(1-x)) K + exact (continuous_const.continuousOn.sub + (continuous_id.mul continuous_const).continuousOn).div + (continuous_const.sub continuous_id).continuousOn (fun x hx => disk_denominator_ne (hK hx)) + +theorem disk_level_integral_transport + (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (K : Set ℂ) (hc : IsCompact K) (hK : K ⊆ ball 0 1) + (g : ℂ → ℝ≥0∞) : + (∫⁻ x in K, ENNReal.ofReal + (‖deriv (diskMap θ) (diskMapInv θ x)‖^(-codimension κ))*g (diskMapInv θ x)) = + ∫⁻ z in diskMapInv θ '' K, + ENNReal.ofReal (‖deriv (diskMap θ) z‖^dimension κ)*g z := by + have hs := diskCompactPreimage_subset hθ hK + have hcomp := diskCompactPreimage_compact θ hc hK + have himage : diskMap θ '' (diskMapInv θ '' K)=K := by + ext x + constructor + · rintro ⟨z,⟨y,hy,rfl⟩,rfl⟩ + simpa only [diskMap_diskMapInv hθ (hK hy)] using hy + · intro hx + exact ⟨diskMapInv θ x,⟨x,hx,rfl⟩,diskMap_diskMapInv hθ (hK hx)⟩ + have hd (z : ℂ) (hz : z ∈ diskMapInv θ '' K) := diskMap_hasDerivAt hθ (hs hz) + conv_lhs => rw [← himage] + rw [lintegral_image_eq_lintegral_abs_det_fderiv_mul volume hcomp.measurableSet + (fun z hz => ((hd z hz).hasFDerivAt.restrictScalars ℝ).hasFDerivWithinAt) + ((diskMap_injOn hθ).mono hs)] + apply setLIntegral_congr_fun hcomp.measurableSet + intro z hz + dsimp only + rw [complex_derivative_real_det,abs_of_nonneg (sq_nonneg _), + diskMapInv_diskMap hθ (hs hz),← mul_assoc,← ENNReal.ofReal_mul (sq_nonneg _)] + congr 1 + rw [← (hd z hz).deriv,← Real.rpow_natCast,← Real.rpow_add (diskMap_deriv_pos hθ (hs hz))] + congr 1 + unfold codimension dimension + ring_nf + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DiskMassMomentBound.lean b/lean/OAI/Probability/SLE/Mass/DiskMassMomentBound.lean index 4c9f1fa7eb67a8b130e9cf128ea407c6d7fac4fa..cd6487cbf9f98763cc4287a2ef2a7cf2c215f3e6 100644 --- a/lean/OAI/Probability/SLE/Mass/DiskMassMomentBound.lean +++ b/lean/OAI/Probability/SLE/Mass/DiskMassMomentBound.lean @@ -1,120 +1,120 @@ -import OAI.Probability.SLE.Gauge.DiskCanonicalMoments -import OAI.Probability.SLE.Geometry.CanonicalDiskPotential - -namespace OAI - -/-! Uniform moments of the exact quarter-disk terminal mass, retaining -the small centered-target factor before the change of law. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Target - -local instance : LocallyCompactSpace (ball (0:ℂ) 1) := isOpen_ball.locallyCompactSpace - -lemma exists_diskMass_cutoff : - ∃ f : C_c((ball (0:ℂ) 1),ℝ), (∀ x, 0≤f x) ∧ (∀ x, f x≤1) ∧ - (∀ x : (ball (0:ℂ) 1), (x:ℂ)∈diskK → 1≤f x) ∧ - (∀ x : (ball (0:ℂ) 1), (x:ℂ)∉closedBall 0 (1/2:ℝ) → f x=0) := by - let K : Set (ball (0:ℂ) 1) := Subtype.val ⁻¹' diskK - let V : Set (ball (0:ℂ) 1) := Subtype.val ⁻¹' ball 0 (1/2:ℝ) - have hK : IsCompact K := by - apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr - have he : Subtype.val '' K = diskK := by - ext x - constructor - · rintro ⟨y,hy,rfl⟩ - exact hy - · intro hx - exact ⟨⟨x,diskK_subset_unit hx⟩,hx,rfl⟩ - rw [he] - exact isCompact_closedBall _ _ - have hV : IsOpen V := isOpen_ball.preimage continuous_subtype_val - have hKV : K⊆V := by - intro x hx - have hn : ‖(x:ℂ)‖≤(1/4:ℝ) := mem_closedBall_zero_iff.mp hx - exact mem_ball_zero_iff.mpr (hn.trans_lt (by norm_num)) - obtain ⟨f,hf,hfc,hfs,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK hV hKV - refine ⟨⟨f,hfc⟩,fun x => (h01 x).1,fun x => (h01 x).2,?_,?_⟩ - · intro x hx - change 1≤f x - rw [hf hx] - rfl - · intro x hx - apply image_eq_zero_of_notMem_tsupport - intro hs - exact hx (mem_closedBall_zero_iff.mpr (mem_ball_zero_iff.mp (hfs hs)).le) - -end SLEExactGauge.Target - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -lemma diskTerminalScalar_le_test - (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) - (f : C_c((ball (0:ℂ) 1),ℝ)) (hf : ∀ x, 0≤f x) - (hKf : ∀ x : (ball (0:ℂ) 1), (x:ℂ)∈Target.diskK → 1≤f x) (ω : Ω) : - C.diskTerminalScalar θ hθ ω ≤ - ∫ x, f x ∂conformalMass (Target.diskHomeomorph θ hθ) - (Target.diskMassWeight κ θ hθ) (C.rational.terminal ω) := by - let : IsFiniteMeasureOnCompacts (C.rational.terminal ω) := (C.rational.terminal_radon ω).2 - let w := Target.diskMassWeight κ θ hθ - let g := conformalMassTest (Target.diskHomeomorph θ hθ) w f - have hK := Target.diskSourceHalfPlane_compact hθ - rw [conformalMass_integral] - change (∫ z in Target.diskSourceHalfPlane θ, (w z:ℝ) ∂C.rational.terminal ω) ≤ - ∫ z, g z ∂C.rational.terminal ω - rw [← integral_indicator hK.measurableSet] - apply integral_mono ((integrable_indicator_iff hK.measurableSet).mpr - (compact_weight_integrable w _ hK _)) g.integrable - intro z - by_cases hz : z∈Target.diskSourceHalfPlane θ - · rw [indicator_of_mem hz] - have hmap : ((Target.diskHomeomorph θ hθ z):ℂ)∈Target.diskK := by - obtain ⟨x,hx,he⟩ := hz - have heq := congrArg (Target.diskMap θ) he - have heq' : x=Target.diskMap θ z := by - simpa only [Target.diskMap_diskMapInv hθ (Target.diskK_subset_unit hx)] using heq - change Target.diskMap θ z∈Target.diskK - rw [← heq'] - exact hx - change (w z:ℝ)≤(w z:ℝ)*f (Target.diskHomeomorph θ hθ z) - exact le_mul_of_one_le_right (NNReal.coe_nonneg _) (hKf _ hmap) - · rw [indicator_of_notMem hz] - exact mul_nonneg (NNReal.coe_nonneg _) (hf _) - -theorem uniform_diskTerminalScalar_centered_moments - (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ) : - ∃ M : ℝ≥0∞, M<∞ ∧ ∀ (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi), ∀ e : ℝ, 0 pow_le_pow_left' (ENNReal.ofReal_le_ofReal - (C.diskTerminalScalar_le_test θ hθ f hf hKf ω)) m)).trans - exact hbound θ hθ f hf hf1 hfs e he he1 - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Gauge.DiskCanonicalMoments +import OAI.Probability.SLE.Geometry.CanonicalDiskPotential + +namespace OAI + +/-! Uniform moments of the exact quarter-disk terminal mass, retaining +the small centered-target factor before the change of law. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Target + +local instance : LocallyCompactSpace (ball (0:ℂ) 1) := isOpen_ball.locallyCompactSpace + +lemma exists_diskMass_cutoff : + ∃ f : C_c((ball (0:ℂ) 1),ℝ), (∀ x, 0≤f x) ∧ (∀ x, f x≤1) ∧ + (∀ x : (ball (0:ℂ) 1), (x:ℂ)∈diskK → 1≤f x) ∧ + (∀ x : (ball (0:ℂ) 1), (x:ℂ)∉closedBall 0 (1/2:ℝ) → f x=0) := by + let K : Set (ball (0:ℂ) 1) := Subtype.val ⁻¹' diskK + let V : Set (ball (0:ℂ) 1) := Subtype.val ⁻¹' ball 0 (1/2:ℝ) + have hK : IsCompact K := by + apply Topology.IsEmbedding.subtypeVal.isCompact_iff.mpr + have he : Subtype.val '' K = diskK := by + ext x + constructor + · rintro ⟨y,hy,rfl⟩ + exact hy + · intro hx + exact ⟨⟨x,diskK_subset_unit hx⟩,hx,rfl⟩ + rw [he] + exact isCompact_closedBall _ _ + have hV : IsOpen V := isOpen_ball.preimage continuous_subtype_val + have hKV : K⊆V := by + intro x hx + have hn : ‖(x:ℂ)‖≤(1/4:ℝ) := mem_closedBall_zero_iff.mp hx + exact mem_ball_zero_iff.mpr (hn.trans_lt (by norm_num)) + obtain ⟨f,hf,hfc,hfs,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen hK hV hKV + refine ⟨⟨f,hfc⟩,fun x => (h01 x).1,fun x => (h01 x).2,?_,?_⟩ + · intro x hx + change 1≤f x + rw [hf hx] + rfl + · intro x hx + apply image_eq_zero_of_notMem_tsupport + intro hs + exact hx (mem_closedBall_zero_iff.mpr (mem_ball_zero_iff.mp (hfs hs)).le) + +end SLEExactGauge.Target + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +lemma diskTerminalScalar_le_test + (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi) + (f : C_c((ball (0:ℂ) 1),ℝ)) (hf : ∀ x, 0≤f x) + (hKf : ∀ x : (ball (0:ℂ) 1), (x:ℂ)∈Target.diskK → 1≤f x) (ω : Ω) : + C.diskTerminalScalar θ hθ ω ≤ + ∫ x, f x ∂conformalMass (Target.diskHomeomorph θ hθ) + (Target.diskMassWeight κ θ hθ) (C.rational.terminal ω) := by + let : IsFiniteMeasureOnCompacts (C.rational.terminal ω) := (C.rational.terminal_radon ω).2 + let w := Target.diskMassWeight κ θ hθ + let g := conformalMassTest (Target.diskHomeomorph θ hθ) w f + have hK := Target.diskSourceHalfPlane_compact hθ + rw [conformalMass_integral] + change (∫ z in Target.diskSourceHalfPlane θ, (w z:ℝ) ∂C.rational.terminal ω) ≤ + ∫ z, g z ∂C.rational.terminal ω + rw [← integral_indicator hK.measurableSet] + apply integral_mono ((integrable_indicator_iff hK.measurableSet).mpr + (compact_weight_integrable w _ hK _)) g.integrable + intro z + by_cases hz : z∈Target.diskSourceHalfPlane θ + · rw [indicator_of_mem hz] + have hmap : ((Target.diskHomeomorph θ hθ z):ℂ)∈Target.diskK := by + obtain ⟨x,hx,he⟩ := hz + have heq := congrArg (Target.diskMap θ) he + have heq' : x=Target.diskMap θ z := by + simpa only [Target.diskMap_diskMapInv hθ (Target.diskK_subset_unit hx)] using heq + change Target.diskMap θ z∈Target.diskK + rw [← heq'] + exact hx + change (w z:ℝ)≤(w z:ℝ)*f (Target.diskHomeomorph θ hθ z) + exact le_mul_of_one_le_right (NNReal.coe_nonneg _) (hKf _ hmap) + · rw [indicator_of_notMem hz] + exact mul_nonneg (NNReal.coe_nonneg _) (hf _) + +theorem uniform_diskTerminalScalar_centered_moments + (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (hκ : 0<κ) (hκ8 : κ<8) (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) (m : ℕ) : + ∃ M : ℝ≥0∞, M<∞ ∧ ∀ (θ : ℝ) (hθ : θ∈Ioo (0:ℝ) Real.pi), ∀ e : ℝ, 0 pow_le_pow_left' (ENNReal.ofReal_le_ofReal + (C.diskTerminalScalar_le_test θ hθ f hf hKf ω)) m)).trans + exact hbound θ hθ f hf hf1 hfs e he he1 + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DominatedMassTestContinuity.lean b/lean/OAI/Probability/SLE/Mass/DominatedMassTestContinuity.lean index a4e651298cbed5221f3b29cdfad717f8d0ca0f5e..bd15f44288e6cdd4651c762c956d3c50dd3d65eb 100644 --- a/lean/OAI/Probability/SLE/Mass/DominatedMassTestContinuity.lean +++ b/lean/OAI/Probability/SLE/Mass/DominatedMassTestContinuity.lean @@ -1,76 +1,76 @@ -import OAI.Probability.SLE.LoewnerCharts -import Mathlib.MeasureTheory.Function.ContinuousMapDense - -namespace OAI - -/-! Compact-test continuity extends to every terminal-integrable weight. -This permits weighted inverse-chart tests whose supports approach the old -boundary, using domination by the Radon terminal mass. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge - -local instance dominatedTestLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma integral_test_distance_le_of_domination - (μ ν : Measure upperHalfPlane) (hμ : μ ≤ ν) - (f g : upperHalfPlane → ℝ) (hf : Integrable f ν) (hg : Integrable g ν) : - dist (∫ z, f z ∂μ) (∫ z, g z ∂μ) ≤ ∫ z, ‖f z-g z‖ ∂ν := by - rw [Real.dist_eq,← integral_sub (hf.mono_measure hμ) (hg.mono_measure hμ)] - calc - |∫ z, f z-g z ∂μ| ≤ ∫ z, ‖f z-g z‖ ∂μ := by - simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun z => f z-g z) - _ ≤ _ := integral_mono_measure hμ (ae_of_all _ (fun _ => norm_nonneg _)) (hf.sub hg).norm - -/-- Monotonicity is not needed: domination and compact-test continuity suffice. -/ -theorem continuous_integral_of_radon_domination - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] - (hdom : ∀ t, N t ≤ ν) - (hcont : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) - (g : upperHalfPlane → ℝ) (hg : Integrable g ν) : - Continuous (fun t => ∫ z, g z ∂N t) := by - apply continuous_iff_continuousAt.mpr - intro t - apply Metric.tendsto_nhds.mpr - intro ε hε - obtain ⟨f,hfs,herr,hfc,hfi⟩ := hg.exists_hasCompactSupport_integral_sub_le - (show 0 < ε/3 by positivity) - let φ : C_c(upperHalfPlane,ℝ) := ⟨⟨f,hfc⟩,hfs⟩ - have hct : ContinuousAt (fun u => ∫ z, φ z ∂N u) t := (hcont φ).continuousAt - have hnear := (Metric.tendsto_nhds.mp hct) (ε/3) - (show 0 < ε/3 by positivity) - filter_upwards [hnear] with s hs - have hb (u : ℝ≥0) : dist (∫ z, g z ∂N u) (∫ z, f z ∂N u) ≤ ε/3 := - (integral_test_distance_le_of_domination (N u) ν (hdom u) g f hg hfi).trans herr - have h1 := dist_triangle (∫ z, g z ∂N s) (∫ z, f z ∂N s) (∫ z, g z ∂N t) - have h2 := dist_triangle (∫ z, f z ∂N s) (∫ z, f z ∂N t) (∫ z, g z ∂N t) - change dist (∫ z, f z ∂N s) (∫ z, f z ∂N t) < ε/3 at hs - rw [dist_comm (∫ z, f z ∂N t) (∫ z, g z ∂N t)] at h2 - linarith [hb s,hb t] - -theorem continuous_lintegral_of_radon_domination - (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] - (hdom : ∀ t, N t ≤ ν) - (hcont : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) - (g : upperHalfPlane → ℝ≥0∞) (hg : Measurable g) (hfin : ∫⁻ z, g z ∂ν ≠ ∞) : - Continuous (fun t => ∫⁻ z, g z ∂N t) := by - have hreal : Integrable (fun z => (g z).toReal) ν := - integrable_toReal_of_lintegral_ne_top hg.aemeasurable hfin - have hc := continuous_integral_of_radon_domination N ν hdom hcont _ hreal - have he (t : ℝ≥0) : ENNReal.ofReal (∫ z, (g z).toReal ∂N t) = ∫⁻ z, g z ∂N t := by - rw [ofReal_integral_eq_lintegral_ofReal (hreal.mono_measure (hdom t)) - (ae_of_all _ (fun _ => ENNReal.toReal_nonneg))] - apply lintegral_congr_ae - have hfinite : ∀ᵐ z ∂ν, g z ≠ ∞ := (ae_lt_top hg hfin).mono (fun _ h => h.ne) - filter_upwards [ae_mono (hdom t) hfinite] with z hz - exact ENNReal.ofReal_toReal hz - simpa only [Function.comp_def,he] using ENNReal.continuous_ofReal.comp hc - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.LoewnerCharts +import Mathlib.MeasureTheory.Function.ContinuousMapDense + +namespace OAI + +/-! Compact-test continuity extends to every terminal-integrable weight. +This permits weighted inverse-chart tests whose supports approach the old +boundary, using domination by the Radon terminal mass. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge + +local instance dominatedTestLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma integral_test_distance_le_of_domination + (μ ν : Measure upperHalfPlane) (hμ : μ ≤ ν) + (f g : upperHalfPlane → ℝ) (hf : Integrable f ν) (hg : Integrable g ν) : + dist (∫ z, f z ∂μ) (∫ z, g z ∂μ) ≤ ∫ z, ‖f z-g z‖ ∂ν := by + rw [Real.dist_eq,← integral_sub (hf.mono_measure hμ) (hg.mono_measure hμ)] + calc + |∫ z, f z-g z ∂μ| ≤ ∫ z, ‖f z-g z‖ ∂μ := by + simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun z => f z-g z) + _ ≤ _ := integral_mono_measure hμ (ae_of_all _ (fun _ => norm_nonneg _)) (hf.sub hg).norm + +/-- Monotonicity is not needed: domination and compact-test continuity suffice. -/ +theorem continuous_integral_of_radon_domination + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] + (hdom : ∀ t, N t ≤ ν) + (hcont : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) + (g : upperHalfPlane → ℝ) (hg : Integrable g ν) : + Continuous (fun t => ∫ z, g z ∂N t) := by + apply continuous_iff_continuousAt.mpr + intro t + apply Metric.tendsto_nhds.mpr + intro ε hε + obtain ⟨f,hfs,herr,hfc,hfi⟩ := hg.exists_hasCompactSupport_integral_sub_le + (show 0 < ε/3 by positivity) + let φ : C_c(upperHalfPlane,ℝ) := ⟨⟨f,hfc⟩,hfs⟩ + have hct : ContinuousAt (fun u => ∫ z, φ z ∂N u) t := (hcont φ).continuousAt + have hnear := (Metric.tendsto_nhds.mp hct) (ε/3) + (show 0 < ε/3 by positivity) + filter_upwards [hnear] with s hs + have hb (u : ℝ≥0) : dist (∫ z, g z ∂N u) (∫ z, f z ∂N u) ≤ ε/3 := + (integral_test_distance_le_of_domination (N u) ν (hdom u) g f hg hfi).trans herr + have h1 := dist_triangle (∫ z, g z ∂N s) (∫ z, f z ∂N s) (∫ z, g z ∂N t) + have h2 := dist_triangle (∫ z, f z ∂N s) (∫ z, f z ∂N t) (∫ z, g z ∂N t) + change dist (∫ z, f z ∂N s) (∫ z, f z ∂N t) < ε/3 at hs + rw [dist_comm (∫ z, f z ∂N t) (∫ z, g z ∂N t)] at h2 + linarith [hb s,hb t] + +theorem continuous_lintegral_of_radon_domination + (N : ℝ≥0 → Measure upperHalfPlane) (ν : Measure upperHalfPlane) [ν.Regular] + (hdom : ∀ t, N t ≤ ν) + (hcont : ∀ f : C_c(upperHalfPlane,ℝ), Continuous (fun t => ∫ z, f z ∂N t)) + (g : upperHalfPlane → ℝ≥0∞) (hg : Measurable g) (hfin : ∫⁻ z, g z ∂ν ≠ ∞) : + Continuous (fun t => ∫⁻ z, g z ∂N t) := by + have hreal : Integrable (fun z => (g z).toReal) ν := + integrable_toReal_of_lintegral_ne_top hg.aemeasurable hfin + have hc := continuous_integral_of_radon_domination N ν hdom hcont _ hreal + have he (t : ℝ≥0) : ENNReal.ofReal (∫ z, (g z).toReal ∂N t) = ∫⁻ z, g z ∂N t := by + rw [ofReal_integral_eq_lintegral_ofReal (hreal.mono_measure (hdom t)) + (ae_of_all _ (fun _ => ENNReal.toReal_nonneg))] + apply lintegral_congr_ae + have hfinite : ∀ᵐ z ∂ν, g z ≠ ∞ := (ae_lt_top hg hfin).mono (fun _ h => h.ne) + filter_upwards [ae_mono (hdom t) hfinite] with z hz + exact ENNReal.ofReal_toReal hz + simpa only [Function.comp_def,he] using ENNReal.continuous_ofReal.comp hc + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/DominatedMassTestExhaustion.lean b/lean/OAI/Probability/SLE/Mass/DominatedMassTestExhaustion.lean index 9fd77656f48379dec3772f8453f0bf45be098892..85a16201b3cff59e2c57169eeb919787ca0e393b 100644 --- a/lean/OAI/Probability/SLE/Mass/DominatedMassTestExhaustion.lean +++ b/lean/OAI/Probability/SLE/Mass/DominatedMassTestExhaustion.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Mass.DominatedMassTestContinuity -import OAI.Probability.SLE.Mass.TerminalMassTestLimit - -namespace OAI - -/-! Terminal-integrable weights pass through local exhaustion. This supplies -terminal exhaustion after conformal weighting, including weights whose -support approaches the boundary. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge - -local instance exhaustedTestLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem terminal_integrable_test_tendsto - (ν : Measure upperHalfPlane) [ν.Regular] - (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular) - (hdom : ∀ t, N t ≤ ν) - (hlim : ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) - (g : upperHalfPlane → ℝ) (hg : Integrable g ν) : - Tendsto (fun t : ℝ≥0 => ∫ z, g z ∂N t) atTop (𝓝 (∫ z, g z ∂ν)) := by - apply Metric.tendsto_nhds.mpr - intro ε hε - obtain ⟨f,hfs,herr,hfc,hfi⟩ := hg.exists_hasCompactSupport_integral_sub_le - (show 0 < ε/3 by positivity) - let φ : C_c(upperHalfPlane,ℝ) := ⟨⟨f,hfc⟩,hfs⟩ - have hnear := (Metric.tendsto_nhds.mp (terminal_mass_test_tendsto ν N hNr hdom hlim φ)) - (ε/3) (show 0 < ε/3 by positivity) - filter_upwards [hnear] with t ht - have ha : dist (∫ z, g z ∂N t) (∫ z, f z ∂N t) ≤ ε/3 := - (integral_test_distance_le_of_domination (N t) ν (hdom t) g f hg hfi).trans herr - have hb : dist (∫ z, g z ∂ν) (∫ z, f z ∂ν) ≤ ε/3 := - (integral_test_distance_le_of_domination ν ν le_rfl g f hg hfi).trans herr - have h1 := dist_triangle (∫ z, g z ∂N t) (∫ z, f z ∂N t) (∫ z, g z ∂ν) - have h2 := dist_triangle (∫ z, f z ∂N t) (∫ z, f z ∂ν) (∫ z, g z ∂ν) - change dist (∫ z, f z ∂N t) (∫ z, f z ∂ν) < ε/3 at ht - rw [dist_comm (∫ z, f z ∂ν) (∫ z, g z ∂ν)] at h2 - linarith - -theorem remaining_integrable_test_tendsto_zero - (ν : Measure upperHalfPlane) [ν.Regular] - (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular) - (hdom : ∀ t, N t ≤ ν) - (hlim : ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) - (g : upperHalfPlane → ℝ) (hg : Integrable g ν) : - Tendsto (fun t : ℝ≥0 => ∫ z, g z ∂(ν-N t)) atTop (𝓝 0) := by - have he (t : ℝ≥0) : (∫ z, g z ∂(ν-N t))=(∫ z, g z ∂ν)-(∫ z, g z ∂N t) := by - let := hNr t - have hi := integral_add_measure (hg.mono_measure (Measure.sub_le : ν-N t ≤ ν)) - (hg.mono_measure (hdom t)) - rw [sigmaFinite_sub_add_cancel_of_le (hdom t)] at hi - linarith - simpa only [he,sub_self] using - (tendsto_const_nhds (x := ∫ z, g z ∂ν)).sub - (terminal_integrable_test_tendsto ν N hNr hdom hlim g hg) - -theorem remaining_lintegral_tendsto_zero - (ν : Measure upperHalfPlane) [ν.Regular] - (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular) - (hdom : ∀ t, N t ≤ ν) - (hlim : ∀ K : Set upperHalfPlane, IsCompact K → - Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) - (g : upperHalfPlane → ℝ≥0∞) (hg : Measurable g) (hfin : ∫⁻ z, g z ∂ν ≠ ∞) : - Tendsto (fun t : ℝ≥0 => ∫⁻ z, g z ∂(ν-N t)) atTop (𝓝 0) := by - have hreal : Integrable (fun z => (g z).toReal) ν := - integrable_toReal_of_lintegral_ne_top hg.aemeasurable hfin - have hfinite : ∀ᵐ z ∂ν, g z ≠ ∞ := (ae_lt_top hg hfin).mono (fun _ h => h.ne) - have he (t : ℝ≥0) : ENNReal.ofReal (∫ z, (g z).toReal ∂(ν-N t)) = - ∫⁻ z, g z ∂(ν-N t) := by - rw [ofReal_integral_eq_lintegral_ofReal (hreal.mono_measure (Measure.sub_le : ν-N t ≤ ν)) - (ae_of_all _ (fun _ => ENNReal.toReal_nonneg))] - apply lintegral_congr_ae - filter_upwards [ae_mono (Measure.sub_le : ν-N t ≤ ν) hfinite] with z hz - exact ENNReal.ofReal_toReal hz - have h := ENNReal.continuous_ofReal.continuousAt.tendsto.comp - (remaining_integrable_test_tendsto_zero ν N hNr hdom hlim _ hreal) - simpa only [Function.comp_def,he,ENNReal.ofReal_zero] using h - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.DominatedMassTestContinuity +import OAI.Probability.SLE.Mass.TerminalMassTestLimit + +namespace OAI + +/-! Terminal-integrable weights pass through local exhaustion. This supplies +terminal exhaustion after conformal weighting, including weights whose +support approaches the boundary. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge + +local instance exhaustedTestLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem terminal_integrable_test_tendsto + (ν : Measure upperHalfPlane) [ν.Regular] + (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular) + (hdom : ∀ t, N t ≤ ν) + (hlim : ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) + (g : upperHalfPlane → ℝ) (hg : Integrable g ν) : + Tendsto (fun t : ℝ≥0 => ∫ z, g z ∂N t) atTop (𝓝 (∫ z, g z ∂ν)) := by + apply Metric.tendsto_nhds.mpr + intro ε hε + obtain ⟨f,hfs,herr,hfc,hfi⟩ := hg.exists_hasCompactSupport_integral_sub_le + (show 0 < ε/3 by positivity) + let φ : C_c(upperHalfPlane,ℝ) := ⟨⟨f,hfc⟩,hfs⟩ + have hnear := (Metric.tendsto_nhds.mp (terminal_mass_test_tendsto ν N hNr hdom hlim φ)) + (ε/3) (show 0 < ε/3 by positivity) + filter_upwards [hnear] with t ht + have ha : dist (∫ z, g z ∂N t) (∫ z, f z ∂N t) ≤ ε/3 := + (integral_test_distance_le_of_domination (N t) ν (hdom t) g f hg hfi).trans herr + have hb : dist (∫ z, g z ∂ν) (∫ z, f z ∂ν) ≤ ε/3 := + (integral_test_distance_le_of_domination ν ν le_rfl g f hg hfi).trans herr + have h1 := dist_triangle (∫ z, g z ∂N t) (∫ z, f z ∂N t) (∫ z, g z ∂ν) + have h2 := dist_triangle (∫ z, f z ∂N t) (∫ z, f z ∂ν) (∫ z, g z ∂ν) + change dist (∫ z, f z ∂N t) (∫ z, f z ∂ν) < ε/3 at ht + rw [dist_comm (∫ z, f z ∂ν) (∫ z, g z ∂ν)] at h2 + linarith + +theorem remaining_integrable_test_tendsto_zero + (ν : Measure upperHalfPlane) [ν.Regular] + (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular) + (hdom : ∀ t, N t ≤ ν) + (hlim : ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) + (g : upperHalfPlane → ℝ) (hg : Integrable g ν) : + Tendsto (fun t : ℝ≥0 => ∫ z, g z ∂(ν-N t)) atTop (𝓝 0) := by + have he (t : ℝ≥0) : (∫ z, g z ∂(ν-N t))=(∫ z, g z ∂ν)-(∫ z, g z ∂N t) := by + let := hNr t + have hi := integral_add_measure (hg.mono_measure (Measure.sub_le : ν-N t ≤ ν)) + (hg.mono_measure (hdom t)) + rw [sigmaFinite_sub_add_cancel_of_le (hdom t)] at hi + linarith + simpa only [he,sub_self] using + (tendsto_const_nhds (x := ∫ z, g z ∂ν)).sub + (terminal_integrable_test_tendsto ν N hNr hdom hlim g hg) + +theorem remaining_lintegral_tendsto_zero + (ν : Measure upperHalfPlane) [ν.Regular] + (N : ℝ≥0 → Measure upperHalfPlane) (hNr : ∀ t, (N t).Regular) + (hdom : ∀ t, N t ≤ ν) + (hlim : ∀ K : Set upperHalfPlane, IsCompact K → + Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) + (g : upperHalfPlane → ℝ≥0∞) (hg : Measurable g) (hfin : ∫⁻ z, g z ∂ν ≠ ∞) : + Tendsto (fun t : ℝ≥0 => ∫⁻ z, g z ∂(ν-N t)) atTop (𝓝 0) := by + have hreal : Integrable (fun z => (g z).toReal) ν := + integrable_toReal_of_lintegral_ne_top hg.aemeasurable hfin + have hfinite : ∀ᵐ z ∂ν, g z ≠ ∞ := (ae_lt_top hg hfin).mono (fun _ h => h.ne) + have he (t : ℝ≥0) : ENNReal.ofReal (∫ z, (g z).toReal ∂(ν-N t)) = + ∫⁻ z, g z ∂(ν-N t) := by + rw [ofReal_integral_eq_lintegral_ofReal (hreal.mono_measure (Measure.sub_le : ν-N t ≤ ν)) + (ae_of_all _ (fun _ => ENNReal.toReal_nonneg))] + apply lintegral_congr_ae + filter_upwards [ae_mono (Measure.sub_le : ν-N t ≤ ν) hfinite] with z hz + exact ENNReal.ofReal_toReal hz + have h := ENNReal.continuous_ofReal.continuousAt.tendsto.comp + (remaining_integrable_test_tendsto_zero ν N hNr hdom hlim _ hreal) + simpa only [Function.comp_def,he,ENNReal.ofReal_zero] using h + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/GreenMassProcessUniqueness.lean b/lean/OAI/Probability/SLE/Mass/GreenMassProcessUniqueness.lean index 45f01fb06ca430caaf490405d713cc91c70773ec..3b44992bc4bbd110e3dc76dd72b1f53c77dd573b 100644 --- a/lean/OAI/Probability/SLE/Mass/GreenMassProcessUniqueness.lean +++ b/lean/OAI/Probability/SLE/Mass/GreenMassProcessUniqueness.lean @@ -1,94 +1,94 @@ -import OAI.Probability.SLE.Geometry.PositiveKernelTests -import OAI.Probability.SLE.Loewner.StoppedInversePaths -import OAI.Probability.SLE.Mass.RadonProcessAEUniqueness - -namespace OAI - -/-! The uniqueness step used to compare canonical mass with its spatial -and temporal dilation. Only the positive compact tests of the two Green -remainder kernels are required. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance greenMassUniqueLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {ℱ : Filtration ℝ≥0 mΩ} - -omit [IsProbabilityMeasure P] in -lemma restricted_kernel_mean_le - (μ : Ω → Measure upperHalfPlane) (hm : Measurable μ) - (hb : P.bind μ ≤ initialGreenMeasure κ) (E : Set Ω) : - (P.restrict E).bind μ ≤ initialGreenMeasure κ := - (restricted_bind_le_of_ae_le P μ μ hm hm (ae_of_all _ (fun _ => le_rfl)) E).trans hb - -lemma positive_mass_process_eq_of_cross_kernel - (A B : ℝ≥0 → Ω → Measure upperHalfPlane) (ZA ZB : Ω → Measure upperHalfPlane) - (hA : Adapted ℱ A) (hB : Adapted ℱ B) (hZA : Measurable ZA) (hZB : Measurable ZB) - (hAb : ∀ t, P.bind (A t) ≤ initialGreenMeasure κ) - (hBb : ∀ t, P.bind (B t) ≤ initialGreenMeasure κ) - (hZAb : P.bind ZA ≤ initialGreenMeasure κ) (hZBb : P.bind ZB ≤ initialGreenMeasure κ) - (hpA : ∀ᵐ ω ∂P, PositiveMassPaths (fun t => A t ω) (ZA ω)) - (hpB : ∀ᵐ ω ∂P, PositiveMassPaths (fun t => B t ω) (ZB ω)) - (hcross : ∀ t E, MeasurableSet[ℱ t] E → - (P.restrict E).bind ZA+(P.restrict E).bind (B t)= - (P.restrict E).bind ZB+(P.restrict E).bind (A t)) : - ∀ᵐ ω ∂P, ∀ t, A t ω=B t ω := by - apply random_radon_process_eq_of_positive_tests_ae P (localTestFamily upperHalfPlane) A B - (hpA.mono (fun _ h => fun t => (h.radon t).1)) - (hpB.mono (fun _ h => fun t => (h.radon t).1)) - intro f hf - have hAm (t : ℝ≥0) := hA.measurable (i:=t) - have hBm (t : ℝ≥0) := hB.measurable (i:=t) - have hAI (t : ℝ≥0) := positive_kernel_test_integrable P κ (A t) (hAm t) (hAb t) f hf - have hBI (t : ℝ≥0) := positive_kernel_test_integrable P κ (B t) (hBm t) (hBb t) f hf - have hZAI := positive_kernel_test_integrable P κ ZA hZA hZAb f hf - have hZBI := positive_kernel_test_integrable P κ ZB hZB hZBb f hf - have hAA : StronglyAdapted ℱ (fun t ω => ∫ z, f z ∂A t ω) := fun t => - (@positive_test_integral_measurable Ω (ℱ t) (A t) (hA t) f hf).stronglyMeasurable - have hBA : StronglyAdapted ℱ (fun t ω => ∫ z, f z ∂B t ω) := fun t => - (@positive_test_integral_measurable Ω (ℱ t) (B t) (hB t) f hf).stronglyMeasurable - apply increasing_mass_unique_of_terminal_mean - (fun t ω => ∫ z, f z ∂A t ω) (fun t ω => ∫ z, f z ∂B t ω) - (fun ω => (∫ z, f z ∂ZA ω)-(∫ z, f z ∂ZB ω)) (hZAI.sub hZBI) - hAA hBA hAI hBI - · intro t E hE - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have ha := restricted_kernel_mean_le (A t) (hAm t) (hAb t) E - have hb := restricted_kernel_mean_le (B t) (hBm t) (hBb t) E - have hza := restricted_kernel_mean_le ZA hZA hZAb E - have hzb := restricted_kernel_mean_le ZB hZB hZBb E - have h := congrArg (fun μ : Measure upperHalfPlane => ∫ z, f z ∂μ) (hcross t E hE) - rw [integral_add_measure (f.integrable.mono_measure hza) (f.integrable.mono_measure hb), - integral_add_measure (f.integrable.mono_measure hzb) (f.integrable.mono_measure ha), - ← positive_kernel_integral_bind (P.restrict E) κ ZA hZA hza f hf, - ← positive_kernel_integral_bind (P.restrict E) κ (B t) (hBm t) hb f hf, - ← positive_kernel_integral_bind (P.restrict E) κ ZB hZB hzb f hf, - ← positive_kernel_integral_bind (P.restrict E) κ (A t) (hAm t) ha f hf] at h - rw [integral_sub (hAI t).integrableOn (hBI t).integrableOn, - integral_sub hZAI.integrableOn hZBI.integrableOn] - linarith - · filter_upwards [hpA] with ω hp - intro s t hst - let := (hp.radon t).1 - exact integral_mono_measure (hp.monotone hst) (ae_of_all _ hf) f.integrable - · filter_upwards [hpB] with ω hp - intro s t hst - let := (hp.radon t).1 - exact integral_mono_measure (hp.monotone hst) (ae_of_all _ hf) f.integrable - · filter_upwards [hpA] with ω hp - simp only [hp.zero,integral_zero_measure,Pi.zero_apply] - · filter_upwards [hpB] with ω hp - simp only [hp.zero,integral_zero_measure,Pi.zero_apply] - · filter_upwards [hpA,hpB] with ω ha hb - exact (ha.continuous_tests f hf).sub (hb.continuous_tests f hf) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.PositiveKernelTests +import OAI.Probability.SLE.Loewner.StoppedInversePaths +import OAI.Probability.SLE.Mass.RadonProcessAEUniqueness + +namespace OAI + +/-! The uniqueness step used to compare canonical mass with its spatial +and temporal dilation. Only the positive compact tests of the two Green +remainder kernels are required. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance greenMassUniqueLocallyCompact : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {ℱ : Filtration ℝ≥0 mΩ} + +omit [IsProbabilityMeasure P] in +lemma restricted_kernel_mean_le + (μ : Ω → Measure upperHalfPlane) (hm : Measurable μ) + (hb : P.bind μ ≤ initialGreenMeasure κ) (E : Set Ω) : + (P.restrict E).bind μ ≤ initialGreenMeasure κ := + (restricted_bind_le_of_ae_le P μ μ hm hm (ae_of_all _ (fun _ => le_rfl)) E).trans hb + +lemma positive_mass_process_eq_of_cross_kernel + (A B : ℝ≥0 → Ω → Measure upperHalfPlane) (ZA ZB : Ω → Measure upperHalfPlane) + (hA : Adapted ℱ A) (hB : Adapted ℱ B) (hZA : Measurable ZA) (hZB : Measurable ZB) + (hAb : ∀ t, P.bind (A t) ≤ initialGreenMeasure κ) + (hBb : ∀ t, P.bind (B t) ≤ initialGreenMeasure κ) + (hZAb : P.bind ZA ≤ initialGreenMeasure κ) (hZBb : P.bind ZB ≤ initialGreenMeasure κ) + (hpA : ∀ᵐ ω ∂P, PositiveMassPaths (fun t => A t ω) (ZA ω)) + (hpB : ∀ᵐ ω ∂P, PositiveMassPaths (fun t => B t ω) (ZB ω)) + (hcross : ∀ t E, MeasurableSet[ℱ t] E → + (P.restrict E).bind ZA+(P.restrict E).bind (B t)= + (P.restrict E).bind ZB+(P.restrict E).bind (A t)) : + ∀ᵐ ω ∂P, ∀ t, A t ω=B t ω := by + apply random_radon_process_eq_of_positive_tests_ae P (localTestFamily upperHalfPlane) A B + (hpA.mono (fun _ h => fun t => (h.radon t).1)) + (hpB.mono (fun _ h => fun t => (h.radon t).1)) + intro f hf + have hAm (t : ℝ≥0) := hA.measurable (i:=t) + have hBm (t : ℝ≥0) := hB.measurable (i:=t) + have hAI (t : ℝ≥0) := positive_kernel_test_integrable P κ (A t) (hAm t) (hAb t) f hf + have hBI (t : ℝ≥0) := positive_kernel_test_integrable P κ (B t) (hBm t) (hBb t) f hf + have hZAI := positive_kernel_test_integrable P κ ZA hZA hZAb f hf + have hZBI := positive_kernel_test_integrable P κ ZB hZB hZBb f hf + have hAA : StronglyAdapted ℱ (fun t ω => ∫ z, f z ∂A t ω) := fun t => + (@positive_test_integral_measurable Ω (ℱ t) (A t) (hA t) f hf).stronglyMeasurable + have hBA : StronglyAdapted ℱ (fun t ω => ∫ z, f z ∂B t ω) := fun t => + (@positive_test_integral_measurable Ω (ℱ t) (B t) (hB t) f hf).stronglyMeasurable + apply increasing_mass_unique_of_terminal_mean + (fun t ω => ∫ z, f z ∂A t ω) (fun t ω => ∫ z, f z ∂B t ω) + (fun ω => (∫ z, f z ∂ZA ω)-(∫ z, f z ∂ZB ω)) (hZAI.sub hZBI) + hAA hBA hAI hBI + · intro t E hE + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have ha := restricted_kernel_mean_le (A t) (hAm t) (hAb t) E + have hb := restricted_kernel_mean_le (B t) (hBm t) (hBb t) E + have hza := restricted_kernel_mean_le ZA hZA hZAb E + have hzb := restricted_kernel_mean_le ZB hZB hZBb E + have h := congrArg (fun μ : Measure upperHalfPlane => ∫ z, f z ∂μ) (hcross t E hE) + rw [integral_add_measure (f.integrable.mono_measure hza) (f.integrable.mono_measure hb), + integral_add_measure (f.integrable.mono_measure hzb) (f.integrable.mono_measure ha), + ← positive_kernel_integral_bind (P.restrict E) κ ZA hZA hza f hf, + ← positive_kernel_integral_bind (P.restrict E) κ (B t) (hBm t) hb f hf, + ← positive_kernel_integral_bind (P.restrict E) κ ZB hZB hzb f hf, + ← positive_kernel_integral_bind (P.restrict E) κ (A t) (hAm t) ha f hf] at h + rw [integral_sub (hAI t).integrableOn (hBI t).integrableOn, + integral_sub hZAI.integrableOn hZBI.integrableOn] + linarith + · filter_upwards [hpA] with ω hp + intro s t hst + let := (hp.radon t).1 + exact integral_mono_measure (hp.monotone hst) (ae_of_all _ hf) f.integrable + · filter_upwards [hpB] with ω hp + intro s t hst + let := (hp.radon t).1 + exact integral_mono_measure (hp.monotone hst) (ae_of_all _ hf) f.integrable + · filter_upwards [hpA] with ω hp + simp only [hp.zero,integral_zero_measure,Pi.zero_apply] + · filter_upwards [hpB] with ω hp + simp only [hp.zero,integral_zero_measure,Pi.zero_apply] + · filter_upwards [hpA,hpB] with ω ha hb + exact (ha.continuous_tests f hf).sub (hb.continuous_tests f hf) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/GreenMeasureSupport.lean b/lean/OAI/Probability/SLE/Mass/GreenMeasureSupport.lean index 7e12bf24f68b18d1d9afb9f254aa835a90fb24d4..329b129199552e5f22e475fea1cabdaedb896873 100644 --- a/lean/OAI/Probability/SLE/Mass/GreenMeasureSupport.lean +++ b/lean/OAI/Probability/SLE/Mass/GreenMeasureSupport.lean @@ -1,71 +1,71 @@ -import OAI.Probability.SLE.Mass.LevelCurrentLimit -import OAI.Probability.SLE.Mass.RadonTestFamily -import OAI.Probability.SLE.Geometry.SpatialPastKernels - -namespace OAI - -/-! The actual killed Green measure vanishes off the surviving domain and -is almost surely locally finite, as follows from its initial-Green mean bound. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma halfPlaneGreenMeasure_off_surviving - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : - halfPlaneGreenMeasure κ B γ G t ω {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain (γ ω) t} = 0 := by - have hA : MeasurableSet {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain (γ ω) t} := - ((isOpen_survivingDomain hw.1 t).preimage continuous_subtype_val).measurableSet.compl - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - rw [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - apply setLIntegral_eq_zero hAI - rintro z ⟨x,hx,rfl⟩ - simp only [greenDensity,ite_eq_right hx,ENNReal.ofReal_zero,Pi.zero_apply] - -lemma ae_halfPlaneGreenMeasure_compact_finite - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (K : Set upperHalfPlane) (hK : IsCompact K) : - ∀ᵐ ω ∂P, halfPlaneGreenMeasure κ B γ G t ω K < ⊤ := by - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hb := halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t univ K - rw [Measure.restrict_univ,Measure.bind_apply hK.measurableSet - (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable] at hb - exact ae_lt_top ((Measure.measurable_coe hK.measurableSet).comp - (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t)) - (lt_of_le_of_lt hb hK.measure_lt_top).ne - -theorem ae_halfPlaneGreenMeasure_finite_on_compacts - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (halfPlaneGreenMeasure κ B γ G t ω) := by - let D := localTestFamily upperHalfPlane - have ha (n : ℕ) := ae_halfPlaneGreenMeasure_compact_finite hB hBm hBc hB0 hκ hκ8 γ G F hw - t (D.compactSet n) (D.isCompact n) - filter_upwards [ae_all_iff.mpr ha] with ω hω - constructor - intro K hK - obtain ⟨n,hn⟩ := D.exhaustive K hK - exact lt_of_le_of_lt (measure_mono hn) (hω n) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelCurrentLimit +import OAI.Probability.SLE.Mass.RadonTestFamily +import OAI.Probability.SLE.Geometry.SpatialPastKernels + +namespace OAI + +/-! The actual killed Green measure vanishes off the surviving domain and +is almost surely locally finite, as follows from its initial-Green mean bound. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma halfPlaneGreenMeasure_off_surviving + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : + halfPlaneGreenMeasure κ B γ G t ω {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain (γ ω) t} = 0 := by + have hA : MeasurableSet {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain (γ ω) t} := + ((isOpen_survivingDomain hw.1 t).preimage continuous_subtype_val).measurableSet.compl + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + rw [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + apply setLIntegral_eq_zero hAI + rintro z ⟨x,hx,rfl⟩ + simp only [greenDensity,ite_eq_right hx,ENNReal.ofReal_zero,Pi.zero_apply] + +lemma ae_halfPlaneGreenMeasure_compact_finite + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (K : Set upperHalfPlane) (hK : IsCompact K) : + ∀ᵐ ω ∂P, halfPlaneGreenMeasure κ B γ G t ω K < ⊤ := by + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hb := halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t univ K + rw [Measure.restrict_univ,Measure.bind_apply hK.measurableSet + (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable] at hb + exact ae_lt_top ((Measure.measurable_coe hK.measurableSet).comp + (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t)) + (lt_of_le_of_lt hb hK.measure_lt_top).ne + +theorem ae_halfPlaneGreenMeasure_finite_on_compacts + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : ∀ᵐ ω ∂P, IsFiniteMeasureOnCompacts (halfPlaneGreenMeasure κ B γ G t ω) := by + let D := localTestFamily upperHalfPlane + have ha (n : ℕ) := ae_halfPlaneGreenMeasure_compact_finite hB hBm hBc hB0 hκ hκ8 γ G F hw + t (D.compactSet n) (D.isCompact n) + filter_upwards [ae_all_iff.mpr ha] with ω hω + constructor + intro K hK + obtain ⟨n,hn⟩ := D.exhaustive K hK + exact lt_of_le_of_lt (measure_mono hn) (hω n) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/GridMassOverlap.lean b/lean/OAI/Probability/SLE/Mass/GridMassOverlap.lean index 620669f8ce45f8be7b37026552cbd646b9811eb6..749b321ee3fcd284a26ff25e799a39e317608b05 100644 --- a/lean/OAI/Probability/SLE/Mass/GridMassOverlap.lean +++ b/lean/OAI/Probability/SLE/Mass/GridMassOverlap.lean @@ -1,57 +1,57 @@ -import OAI.Probability.SLE.Mass.MassGridOverlap - -namespace OAI - -/-! Summing physical grid-cell masses costs only the fixed overlap constant, -independently of the radius and of the containing compact set. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma grid_indicator_sum_le {r : ℝ} (hr : 0 < r) (x : ℂ) : - (∑' i : ℤ×ℤ, (closedBall (coverGridCenter r i) (2*r)).indicator (fun _ => (1:ℝ≥0∞)) x) ≤ 81 := by - classical - rw [ENNReal.tsum_eq_iSup_sum] - apply iSup_le - intro I - let f := fun (i : ℤ×ℤ) (z : ℂ) => - (closedBall (coverGridCenter r i) (2*r)).indicator (fun _ => (1:ℝ≥0∞)) z - have hbound : ∀ i z, f i z ≤ 1 := by - intro i z - by_cases hz : z ∈ closedBall (coverGridCenter r i) (2*r) <;> simp [f,hz] - have hs : ∀ i z, f i z ≠ 0 → dist z (coverGridCenter r i) ≤ 3*r := by - intro i z hz - by_cases hm : z ∈ closedBall (coverGridCenter r i) (2*r) - · have hh : dist z (coverGridCenter r i) ≤ 2*r := hm - linarith - · exact (hz (by simp [f,hm])).elim - have hh := massGrid_cutoff_product_sum_le hr I f hbound hs x x - have hdist : dist x x ≤ 6*r := by simp; positivity - rw [ite_eq_left hdist] at hh - have he : (∑ i∈I, f i x*f i x) = ∑ i∈I, f i x := by - apply Finset.sum_congr rfl - intro i hi - by_cases hx : x ∈ closedBall (coverGridCenter r i) (2*r) <;> simp [f,hx] - rwa [he] at hh - -lemma grid_mass_sum_le (μ : Measure ℂ) {r : ℝ} (hr : 0 < r) : - (∑' i : ℤ×ℤ, μ (closedBall (coverGridCenter r i) (2*r))) ≤ 81*μ univ := by - calc - _ = ∫⁻ x, ∑' i : ℤ×ℤ, - (closedBall (coverGridCenter r i) (2*r)).indicator (fun _ => (1:ℝ≥0∞)) x ∂μ := by - rw [lintegral_tsum (fun i => (measurable_const.indicator measurableSet_closedBall).aemeasurable)] - apply tsum_congr - intro i - rw [lintegral_indicator measurableSet_closedBall] - simp - _ ≤ ∫⁻ _x, (81:ℝ≥0∞) ∂μ := lintegral_mono (grid_indicator_sum_le hr) - _ = _ := by simp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassGridOverlap + +namespace OAI + +/-! Summing physical grid-cell masses costs only the fixed overlap constant, +independently of the radius and of the containing compact set. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma grid_indicator_sum_le {r : ℝ} (hr : 0 < r) (x : ℂ) : + (∑' i : ℤ×ℤ, (closedBall (coverGridCenter r i) (2*r)).indicator (fun _ => (1:ℝ≥0∞)) x) ≤ 81 := by + classical + rw [ENNReal.tsum_eq_iSup_sum] + apply iSup_le + intro I + let f := fun (i : ℤ×ℤ) (z : ℂ) => + (closedBall (coverGridCenter r i) (2*r)).indicator (fun _ => (1:ℝ≥0∞)) z + have hbound : ∀ i z, f i z ≤ 1 := by + intro i z + by_cases hz : z ∈ closedBall (coverGridCenter r i) (2*r) <;> simp [f,hz] + have hs : ∀ i z, f i z ≠ 0 → dist z (coverGridCenter r i) ≤ 3*r := by + intro i z hz + by_cases hm : z ∈ closedBall (coverGridCenter r i) (2*r) + · have hh : dist z (coverGridCenter r i) ≤ 2*r := hm + linarith + · exact (hz (by simp [f,hm])).elim + have hh := massGrid_cutoff_product_sum_le hr I f hbound hs x x + have hdist : dist x x ≤ 6*r := by simp; positivity + rw [ite_eq_left hdist] at hh + have he : (∑ i∈I, f i x*f i x) = ∑ i∈I, f i x := by + apply Finset.sum_congr rfl + intro i hi + by_cases hx : x ∈ closedBall (coverGridCenter r i) (2*r) <;> simp [f,hx] + rwa [he] at hh + +lemma grid_mass_sum_le (μ : Measure ℂ) {r : ℝ} (hr : 0 < r) : + (∑' i : ℤ×ℤ, μ (closedBall (coverGridCenter r i) (2*r))) ≤ 81*μ univ := by + calc + _ = ∫⁻ x, ∑' i : ℤ×ℤ, + (closedBall (coverGridCenter r i) (2*r)).indicator (fun _ => (1:ℝ≥0∞)) x ∂μ := by + rw [lintegral_tsum (fun i => (measurable_const.indicator measurableSet_closedBall).aemeasurable)] + apply tsum_congr + intro i + rw [lintegral_indicator measurableSet_closedBall] + simp + _ ≤ ∫⁻ _x, (81:ℝ≥0∞) ∂μ := lintegral_mono (grid_indicator_sum_le hr) + _ = _ := by simp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/HalfPlaneHerglotzMass.lean b/lean/OAI/Probability/SLE/Mass/HalfPlaneHerglotzMass.lean index 0a24f5c69d95c63bff1c2a522a9311e3ac29a8c2..73fcd35e08033076e6df6425dbd4cc095cf0e9f0 100644 --- a/lean/OAI/Probability/SLE/Mass/HalfPlaneHerglotzMass.lean +++ b/lean/OAI/Probability/SLE/Mass/HalfPlaneHerglotzMass.lean @@ -1,68 +1,68 @@ -import OAI.Probability.SLE.Loewner.HydrodynamicComposition - -namespace OAI - -/-! The Schwarz–Pick bound for a half-plane map vanishing at infinity. -Only the total mass coefficient is used; no representation theorem is assumed. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology ComplexConjugate - -namespace SLEExactGauge - -lemma halfPlane_schwarz_pick_vanishing_scaled {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f upperHalfPlane) (hm : MapsTo f upperHalfPlane upperHalfPlane) - {z : ℂ} (hz : z ∈ upperHalfPlane) {y : ℝ} (hy : 0 f (verticalInfinity n)) atTop (𝓝 0)) - (hc : Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 c)) - {z : ℂ} (hz : z ∈ upperHalfPlane) : z.im*normSq (f z) ≤ c*(f z).im := by - have hi := tendsto_inv_atTop_zero.comp verticalHeights_tendsto - have hsmall : Tendsto (fun n : ℕ => z/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 0) := by - have hh := (tendsto_const_nhds (x := z) (f := atTop)).mul - (Complex.continuous_ofReal.continuousAt.tendsto.comp hi) - simpa only [Function.comp_apply,ofReal_zero,mul_zero,ofReal_inv,div_eq_mul_inv] using hh - have hleft := (tendsto_const_nhds (x := z.im) (f := atTop)).mul - (continuous_normSq.tendsto (f z-conj 0) |>.comp - ((tendsto_const_nhds (x := f z) (f := atTop)).sub - (Complex.continuous_conj.tendsto 0 |>.comp hzero))) - have hright := ((tendsto_const_nhds (x := (f z).im) (f := atTop)).mul hc).mul - (continuous_normSq.tendsto ((0:ℂ)+I) |>.comp (hsmall.add_const I)) - have hh := le_of_tendsto_of_tendsto hleft hright - (Eventually.of_forall (fun n => halfPlane_schwarz_pick_vanishing_scaled hf hm hz - (by positivity : 0<(n:ℝ)+1))) - simpa only [map_zero,sub_zero,zero_add,normSq_I,mul_one,mul_comm] using hh - -lemma norm_le_mass_div_height {w : ℂ} {c y : ℝ} (hy : 0 f (verticalInfinity n)) atTop (𝓝 0)) + (hc : Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 c)) + {z : ℂ} (hz : z ∈ upperHalfPlane) : z.im*normSq (f z) ≤ c*(f z).im := by + have hi := tendsto_inv_atTop_zero.comp verticalHeights_tendsto + have hsmall : Tendsto (fun n : ℕ => z/(((n:ℝ)+1:ℝ):ℂ)) atTop (𝓝 0) := by + have hh := (tendsto_const_nhds (x := z) (f := atTop)).mul + (Complex.continuous_ofReal.continuousAt.tendsto.comp hi) + simpa only [Function.comp_apply,ofReal_zero,mul_zero,ofReal_inv,div_eq_mul_inv] using hh + have hleft := (tendsto_const_nhds (x := z.im) (f := atTop)).mul + (continuous_normSq.tendsto (f z-conj 0) |>.comp + ((tendsto_const_nhds (x := f z) (f := atTop)).sub + (Complex.continuous_conj.tendsto 0 |>.comp hzero))) + have hright := ((tendsto_const_nhds (x := (f z).im) (f := atTop)).mul hc).mul + (continuous_normSq.tendsto ((0:ℂ)+I) |>.comp (hsmall.add_const I)) + have hh := le_of_tendsto_of_tendsto hleft hright + (Eventually.of_forall (fun n => halfPlane_schwarz_pick_vanishing_scaled hf hm hz + (by positivity : 0<(n:ℝ)+1))) + simpa only [map_zero,sub_zero,zero_add,normSq_I,mul_one,mul_comm] using hh + +lemma norm_le_mass_div_height {w : ℂ} {c y : ℝ} (hy : 0 B t ω)) - (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) (u : ℝ) : - levelMass κ B γ G u K ω < ⊤ := by - obtain ⟨b,hb,hbK⟩ := compact_positive_radius_cap hK hKH - let C := (b * Real.exp (-u)) ^ (-codimension κ) - have hW (z : ℂ) (hz : z ∈ K) : stoppedGreen κ (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) z u ≤ C := by - have he : 0 < 2*z.im*Real.exp (-u) := mul_pos (mul_pos (by norm_num) (hKH hz)) (Real.exp_pos _) - have hs := stoppedGreen_level_bound (hBc.const_mul _) hw hκ hκ8 z (hKH hz) - (2*z.im*Real.exp (-u)) he - rw [common_level_log_target z (hKH hz) u] at hs - exact hs.trans (Real.rpow_le_rpow_of_nonpos (mul_pos hb (Real.exp_pos _)) - (mul_le_mul_of_nonneg_right (hbK z hz) (Real.exp_pos _).le) - (neg_nonpos.mpr (codimension_pos κ hκ8).le)) - have hbound : levelMass κ B γ G u K ω ≤ ENNReal.ofReal C * volume K := by - apply (setLIntegral_mono' hK.measurableSet (fun z hz => ENNReal.ofReal_le_ofReal (hW z hz))).trans_eq - exact setLIntegral_const K (ENNReal.ofReal C) - exact hbound.trans_lt (ENNReal.mul_lt_top ENNReal.ofReal_lt_top hK.measure_lt_top) - -def halfPlaneLevelMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := - (levelApproximation κ B γ G u t ω).comap Subtype.val - -theorem halfPlaneLevelMeasure_finite_on_compacts - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (ω : Ω) (hBc : Continuous (fun t => B t ω)) - (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (t : ℝ≥0∞) : IsFiniteMeasureOnCompacts (halfPlaneLevelMeasure κ B γ G u t ω) := by - constructor - intro K hK - have hKI : IsCompact ((Subtype.val : upperHalfPlane → ℂ) '' K) := hK.image continuous_subtype_val - have hKH : (Subtype.val : upperHalfPlane → ℂ) '' K ⊆ upperHalfPlane := by - rintro _ ⟨x,_,rfl⟩ - exact x.property - rw [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet] - exact (levelApproximation_apply_le κ B γ G u t ω hKI.measurableSet).trans_lt - (levelMass_lt_top_of_compact hκ hκ8 γ G F ω hBc hw _ hKI hKH u) - -def extendHalfPlaneTest (f : upperHalfPlane → ℝ) : ℂ → ℝ := - Function.extend (Subtype.val : upperHalfPlane → ℂ) f (fun _ => 0) - -lemma extendHalfPlaneTest_apply (f : upperHalfPlane → ℝ) (x : upperHalfPlane) : - extendHalfPlaneTest f x = f x := Subtype.val_injective.extend_apply _ _ _ - -lemma extendHalfPlaneTest_of_not_mem (f : upperHalfPlane → ℝ) (z : ℂ) (hz : z ∉ upperHalfPlane) : - extendHalfPlaneTest f z = 0 := by - apply Function.extend_apply' - rintro ⟨x,rfl⟩ - exact hz x.property - -lemma extendHalfPlaneTest_measurable (f : upperHalfPlane → ℝ) (hf : Measurable f) : - Measurable (extendHalfPlaneTest f) := - (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurable_extend hf measurable_const - -lemma extendHalfPlaneTest_support (f : C_c(upperHalfPlane,ℝ)) (z : ℂ) - (hz : z ∉ (Subtype.val : upperHalfPlane → ℂ) '' tsupport f) : extendHalfPlaneTest f z = 0 := by - by_cases hH : z ∈ upperHalfPlane - · let x : upperHalfPlane := ⟨z,hH⟩ - change extendHalfPlaneTest f x = 0 - rw [extendHalfPlaneTest_apply] - exact image_eq_zero_of_notMem_tsupport (fun hx => hz ⟨x,hx,rfl⟩) - · exact extendHalfPlaneTest_of_not_mem f z hH - -lemma extendHalfPlaneTest_norm_le (f : C_c(upperHalfPlane,ℝ)) (z : ℂ) : - ‖extendHalfPlaneTest f z‖ ≤ ‖f.toBoundedContinuousFunction‖ := by - by_cases hH : z ∈ upperHalfPlane - · let x : upperHalfPlane := ⟨z,hH⟩ - change ‖extendHalfPlaneTest f x‖ ≤ _ - rw [extendHalfPlaneTest_apply] - exact f.toBoundedContinuousFunction.norm_coe_le_norm x - · rw [extendHalfPlaneTest_of_not_mem f z hH,norm_zero] - exact norm_nonneg _ - -lemma integral_halfPlaneLevelMeasure - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0∞) (ω : Ω) (f : upperHalfPlane → ℝ) : - (∫ x, f x ∂halfPlaneLevelMeasure κ B γ G u t ω) = - levelApproximationRealTest κ B γ G u t (extendHalfPlaneTest f) ω := by - change (∫ x : upperHalfPlane, f x ∂(levelApproximation κ B γ G u t ω).comap Subtype.val) = _ - calc - _ = ∫ x : upperHalfPlane, extendHalfPlaneTest f x - ∂(levelApproximation κ B γ G u t ω).comap Subtype.val := by - congr 1 - funext x - exact (extendHalfPlaneTest_apply f x).symm - _ = ∫ z in upperHalfPlane, extendHalfPlaneTest f z ∂levelApproximation κ B γ G u t ω := - integral_subtype_comap isOpen_upperHalfPlane.measurableSet _ - _ = _ := setIntegral_eq_integral_of_forall_compl_eq_zero (extendHalfPlaneTest_of_not_mem f) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.SignedLevelTests + +namespace OAI + +/-! The finite-level approximants as locally finite measures on the open +half-plane, and their signed compact tests viewed on the ambient plane. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +lemma levelMass_lt_top_of_compact + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (ω : Ω) (hBc : Continuous (fun t => B t ω)) + (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) (u : ℝ) : + levelMass κ B γ G u K ω < ⊤ := by + obtain ⟨b,hb,hbK⟩ := compact_positive_radius_cap hK hKH + let C := (b * Real.exp (-u)) ^ (-codimension κ) + have hW (z : ℂ) (hz : z ∈ K) : stoppedGreen κ (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) z u ≤ C := by + have he : 0 < 2*z.im*Real.exp (-u) := mul_pos (mul_pos (by norm_num) (hKH hz)) (Real.exp_pos _) + have hs := stoppedGreen_level_bound (hBc.const_mul _) hw hκ hκ8 z (hKH hz) + (2*z.im*Real.exp (-u)) he + rw [common_level_log_target z (hKH hz) u] at hs + exact hs.trans (Real.rpow_le_rpow_of_nonpos (mul_pos hb (Real.exp_pos _)) + (mul_le_mul_of_nonneg_right (hbK z hz) (Real.exp_pos _).le) + (neg_nonpos.mpr (codimension_pos κ hκ8).le)) + have hbound : levelMass κ B γ G u K ω ≤ ENNReal.ofReal C * volume K := by + apply (setLIntegral_mono' hK.measurableSet (fun z hz => ENNReal.ofReal_le_ofReal (hW z hz))).trans_eq + exact setLIntegral_const K (ENNReal.ofReal C) + exact hbound.trans_lt (ENNReal.mul_lt_top ENNReal.ofReal_lt_top hK.measure_lt_top) + +def halfPlaneLevelMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := + (levelApproximation κ B γ G u t ω).comap Subtype.val + +theorem halfPlaneLevelMeasure_finite_on_compacts + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (ω : Ω) (hBc : Continuous (fun t => B t ω)) + (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (t : ℝ≥0∞) : IsFiniteMeasureOnCompacts (halfPlaneLevelMeasure κ B γ G u t ω) := by + constructor + intro K hK + have hKI : IsCompact ((Subtype.val : upperHalfPlane → ℂ) '' K) := hK.image continuous_subtype_val + have hKH : (Subtype.val : upperHalfPlane → ℂ) '' K ⊆ upperHalfPlane := by + rintro _ ⟨x,_,rfl⟩ + exact x.property + rw [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet] + exact (levelApproximation_apply_le κ B γ G u t ω hKI.measurableSet).trans_lt + (levelMass_lt_top_of_compact hκ hκ8 γ G F ω hBc hw _ hKI hKH u) + +def extendHalfPlaneTest (f : upperHalfPlane → ℝ) : ℂ → ℝ := + Function.extend (Subtype.val : upperHalfPlane → ℂ) f (fun _ => 0) + +lemma extendHalfPlaneTest_apply (f : upperHalfPlane → ℝ) (x : upperHalfPlane) : + extendHalfPlaneTest f x = f x := Subtype.val_injective.extend_apply _ _ _ + +lemma extendHalfPlaneTest_of_not_mem (f : upperHalfPlane → ℝ) (z : ℂ) (hz : z ∉ upperHalfPlane) : + extendHalfPlaneTest f z = 0 := by + apply Function.extend_apply' + rintro ⟨x,rfl⟩ + exact hz x.property + +lemma extendHalfPlaneTest_measurable (f : upperHalfPlane → ℝ) (hf : Measurable f) : + Measurable (extendHalfPlaneTest f) := + (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurable_extend hf measurable_const + +lemma extendHalfPlaneTest_support (f : C_c(upperHalfPlane,ℝ)) (z : ℂ) + (hz : z ∉ (Subtype.val : upperHalfPlane → ℂ) '' tsupport f) : extendHalfPlaneTest f z = 0 := by + by_cases hH : z ∈ upperHalfPlane + · let x : upperHalfPlane := ⟨z,hH⟩ + change extendHalfPlaneTest f x = 0 + rw [extendHalfPlaneTest_apply] + exact image_eq_zero_of_notMem_tsupport (fun hx => hz ⟨x,hx,rfl⟩) + · exact extendHalfPlaneTest_of_not_mem f z hH + +lemma extendHalfPlaneTest_norm_le (f : C_c(upperHalfPlane,ℝ)) (z : ℂ) : + ‖extendHalfPlaneTest f z‖ ≤ ‖f.toBoundedContinuousFunction‖ := by + by_cases hH : z ∈ upperHalfPlane + · let x : upperHalfPlane := ⟨z,hH⟩ + change ‖extendHalfPlaneTest f x‖ ≤ _ + rw [extendHalfPlaneTest_apply] + exact f.toBoundedContinuousFunction.norm_coe_le_norm x + · rw [extendHalfPlaneTest_of_not_mem f z hH,norm_zero] + exact norm_nonneg _ + +lemma integral_halfPlaneLevelMeasure + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0∞) (ω : Ω) (f : upperHalfPlane → ℝ) : + (∫ x, f x ∂halfPlaneLevelMeasure κ B γ G u t ω) = + levelApproximationRealTest κ B γ G u t (extendHalfPlaneTest f) ω := by + change (∫ x : upperHalfPlane, f x ∂(levelApproximation κ B γ G u t ω).comap Subtype.val) = _ + calc + _ = ∫ x : upperHalfPlane, extendHalfPlaneTest f x + ∂(levelApproximation κ B γ G u t ω).comap Subtype.val := by + congr 1 + funext x + exact (extendHalfPlaneTest_apply f x).symm + _ = ∫ z in upperHalfPlane, extendHalfPlaneTest f z ∂levelApproximation κ B γ G u t ω := + integral_subtype_comap isOpen_upperHalfPlane.measurableSet _ + _ = _ := setIntegral_eq_integral_of_forall_compl_eq_zero (extendHalfPlaneTest_of_not_mem f) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/HalfPlaneZeroMass.lean b/lean/OAI/Probability/SLE/Mass/HalfPlaneZeroMass.lean index cfb3cf8d3cb45f6b43e99b2cb6cebd75c66ffbd7..34dfcd3277199eced099d52f80e597eb5f440b76 100644 --- a/lean/OAI/Probability/SLE/Mass/HalfPlaneZeroMass.lean +++ b/lean/OAI/Probability/SLE/Mass/HalfPlaneZeroMass.lean @@ -1,89 +1,89 @@ -import OAI.Probability.SLE.Mass.HalfPlaneHerglotzMass -import Mathlib.Analysis.Complex.OpenMapping - -namespace OAI - -/-! A holomorphic map with nonnegative imaginary part, vanishing value -and zero mass at infinity is identically zero. -/ -noncomputable section -open Set Filter Complex Function Metric -open scoped Topology - -namespace SLEExactGauge - -lemma holomorphic_nonnegative_im_dichotomy {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f upperHalfPlane) - (hm : ∀ z ∈ upperHalfPlane, 0≤(f z).im) : - (∃ c : ℂ, ∀ z ∈ upperHalfPlane, f z=c) ∨ MapsTo f upperHalfPlane upperHalfPlane := by - rcases (hf.analyticOnNhd isOpen_upperHalfPlane).is_constant_or_isOpen - (convex_halfSpace_im_gt (0:ℝ)).isPreconnected with hc|ho - · exact Or.inl hc - right - intro z hz - change 0 < (f z).im - by_contra hn - have hzero : (f z).im=0 := le_antisymm (le_of_not_gt hn) (hm z hz) - have hi := (ho upperHalfPlane Subset.rfl isOpen_upperHalfPlane).mem_nhds - (mem_image_of_mem f hz) - obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp hi - have hw : f z-((ε/2:ℝ):ℂ)*I ∈ ball (f z) ε := by - rw [mem_ball,dist_eq_norm,sub_sub_cancel_left,norm_neg,norm_mul,norm_I,mul_one, - Complex.norm_real,Real.norm_eq_abs,abs_of_pos (half_pos hε)] - linarith - obtain ⟨w,hwH,he⟩ := hball hw - have him := hm w hwH - rw [he] at him - simp only [sub_im,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero,hzero] at him - linarith - -theorem halfPlane_zero_mass_eq_zero {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f upperHalfPlane) - (hm : ∀ z ∈ upperHalfPlane, 0≤(f z).im) - (hzero : Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0)) - (hc : Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 0)) : - ∀ z ∈ upperHalfPlane, f z=0 := by - rcases holomorphic_nonnegative_im_dichotomy hf hm with ⟨c,hc'⟩|hpos - · have ht : Tendsto (fun _ : ℕ => c) atTop (𝓝 0) := by - apply hzero.congr - intro n - exact hc' _ (verticalInfinity_mem n) - have he : c=0 := tendsto_nhds_unique tendsto_const_nhds ht - exact fun z hz => (hc' z hz).trans he - · have hi : I ∈ upperHalfPlane := by simp [upperHalfPlane] - have hb := halfPlane_vanishing_mass_bound hf hpos hzero hc hi - simp only [I_im,one_mul,zero_mul] at hb - have he : f I=0 := normSq_eq_zero.mp (le_antisymm hb (normSq_nonneg _)) - have hp := hpos hi - simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hp - -theorem holomorphic_zero_hydrodynamic_mass {f : ℂ → ℂ} - (hf : DifferentiableOn ℂ f upperHalfPlane) - (hm : ∀ z ∈ upperHalfPlane, 0≤(f z).im) - (hcoef : Tendsto (fun z : ℂ => z*f z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0)) : - ∀ z ∈ upperHalfPlane, f z=0 := by - have hp := hcoef.comp verticalInfinity_tendsto - have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by - simpa only [Metric.cobounded_eq_cocompact] using - (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) - have hh := hp.mul (hi.comp (verticalInfinity_tendsto.mono_right inf_le_left)) - have hzero : Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0) := by - simp only [mul_zero] at hh - apply hh.congr - intro n - have hn : verticalInfinity n ≠ 0 := by - intro he - have hv := verticalInfinity_mem n - simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hv - simp only [Function.comp_apply] - field_simp - have hmass : Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 0) := by - have ht := (Complex.continuous_re.tendsto (0:ℂ) |>.comp hp).neg - simpa only [Function.comp_apply,verticalInfinity,mul_re,mul_im,ofReal_re,ofReal_im, - I_re,I_im,mul_zero,zero_mul,add_zero,mul_one,zero_re,neg_zero,zero_sub,neg_neg] using ht - exact halfPlane_zero_mass_eq_zero hf hm hzero hmass - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.HalfPlaneHerglotzMass +import Mathlib.Analysis.Complex.OpenMapping + +namespace OAI + +/-! A holomorphic map with nonnegative imaginary part, vanishing value +and zero mass at infinity is identically zero. -/ +noncomputable section +open Set Filter Complex Function Metric +open scoped Topology + +namespace SLEExactGauge + +lemma holomorphic_nonnegative_im_dichotomy {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f upperHalfPlane) + (hm : ∀ z ∈ upperHalfPlane, 0≤(f z).im) : + (∃ c : ℂ, ∀ z ∈ upperHalfPlane, f z=c) ∨ MapsTo f upperHalfPlane upperHalfPlane := by + rcases (hf.analyticOnNhd isOpen_upperHalfPlane).is_constant_or_isOpen + (convex_halfSpace_im_gt (0:ℝ)).isPreconnected with hc|ho + · exact Or.inl hc + right + intro z hz + change 0 < (f z).im + by_contra hn + have hzero : (f z).im=0 := le_antisymm (le_of_not_gt hn) (hm z hz) + have hi := (ho upperHalfPlane Subset.rfl isOpen_upperHalfPlane).mem_nhds + (mem_image_of_mem f hz) + obtain ⟨ε,hε,hball⟩ := Metric.mem_nhds_iff.mp hi + have hw : f z-((ε/2:ℝ):ℂ)*I ∈ ball (f z) ε := by + rw [mem_ball,dist_eq_norm,sub_sub_cancel_left,norm_neg,norm_mul,norm_I,mul_one, + Complex.norm_real,Real.norm_eq_abs,abs_of_pos (half_pos hε)] + linarith + obtain ⟨w,hwH,he⟩ := hball hw + have him := hm w hwH + rw [he] at him + simp only [sub_im,mul_im,ofReal_re,I_im,mul_one,ofReal_im,I_re,mul_zero,add_zero,hzero] at him + linarith + +theorem halfPlane_zero_mass_eq_zero {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f upperHalfPlane) + (hm : ∀ z ∈ upperHalfPlane, 0≤(f z).im) + (hzero : Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0)) + (hc : Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 0)) : + ∀ z ∈ upperHalfPlane, f z=0 := by + rcases holomorphic_nonnegative_im_dichotomy hf hm with ⟨c,hc'⟩|hpos + · have ht : Tendsto (fun _ : ℕ => c) atTop (𝓝 0) := by + apply hzero.congr + intro n + exact hc' _ (verticalInfinity_mem n) + have he : c=0 := tendsto_nhds_unique tendsto_const_nhds ht + exact fun z hz => (hc' z hz).trans he + · have hi : I ∈ upperHalfPlane := by simp [upperHalfPlane] + have hb := halfPlane_vanishing_mass_bound hf hpos hzero hc hi + simp only [I_im,one_mul,zero_mul] at hb + have he : f I=0 := normSq_eq_zero.mp (le_antisymm hb (normSq_nonneg _)) + have hp := hpos hi + simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hp + +theorem holomorphic_zero_hydrodynamic_mass {f : ℂ → ℂ} + (hf : DifferentiableOn ℂ f upperHalfPlane) + (hm : ∀ z ∈ upperHalfPlane, 0≤(f z).im) + (hcoef : Tendsto (fun z : ℂ => z*f z) (cocompact ℂ ⊓ 𝓟 upperHalfPlane) (𝓝 0)) : + ∀ z ∈ upperHalfPlane, f z=0 := by + have hp := hcoef.comp verticalInfinity_tendsto + have hi : Tendsto (fun z : ℂ => z⁻¹) (cocompact ℂ) (𝓝 0) := by + simpa only [Metric.cobounded_eq_cocompact] using + (tendsto_inv₀_cobounded : Tendsto (fun z : ℂ => z⁻¹) (Bornology.cobounded ℂ) (𝓝 0)) + have hh := hp.mul (hi.comp (verticalInfinity_tendsto.mono_right inf_le_left)) + have hzero : Tendsto (fun n : ℕ => f (verticalInfinity n)) atTop (𝓝 0) := by + simp only [mul_zero] at hh + apply hh.congr + intro n + have hn : verticalInfinity n ≠ 0 := by + intro he + have hv := verticalInfinity_mem n + simp only [upperHalfPlane,mem_ofPred_eq,he,zero_im,lt_self_iff_false] at hv + simp only [Function.comp_apply] + field_simp + have hmass : Tendsto (fun n : ℕ => ((n:ℝ)+1)*(f (verticalInfinity n)).im) atTop (𝓝 0) := by + have ht := (Complex.continuous_re.tendsto (0:ℂ) |>.comp hp).neg + simpa only [Function.comp_apply,verticalInfinity,mul_re,mul_im,ofReal_re,ofReal_im, + I_re,I_im,mul_zero,zero_mul,add_zero,mul_one,zero_re,neg_zero,zero_sub,neg_neg] using ht + exact halfPlane_zero_mass_eq_zero hf hm hzero hmass + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/HilbertConvexTails.lean b/lean/OAI/Probability/SLE/Mass/HilbertConvexTails.lean index 3175a678366ccc134dbad4097d89412418a475b6..1060c1164878ea2be3796d9856de658331cd8f44 100644 --- a/lean/OAI/Probability/SLE/Mass/HilbertConvexTails.lean +++ b/lean/OAI/Probability/SLE/Mass/HilbertConvexTails.lean @@ -1,125 +1,125 @@ -import Mathlib.Analysis.InnerProductSpace.Projection.Minimal -import Mathlib.Analysis.Convex.Combination -import Mathlib.Analysis.Convex.Topology -import Mathlib.Analysis.Normed.Module.Convex -import Mathlib.Topology.Order.MonotoneConvergence - -namespace OAI - -/-! Deterministic convex extraction in a Hilbert space. Nested closed -convex tail hulls have a common point, and finite convex combinations -of the original tails converge to that point. -/ - -noncomputable section -open Set Filter Metric -open scoped Topology - -namespace SLEExactGauge - -variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] - -lemma nested_closed_convex_intersection (K : ℕ → Set E) - (hne : ∀ n, (K n).Nonempty) (hclosed : ∀ n, IsClosed (K n)) - (hconvex : ∀ n, Convex ℝ (K n)) (hanti : Antitone K) - (B : ℝ) (hbound : ∀ n x, x ∈ K n → ‖x‖ ≤ B) : - ∃ x, ∀ n, x ∈ K n := by - have hex (n : ℕ) := exists_norm_eq_iInf_of_complete_convex (hne n) - (hclosed n).isComplete (hconvex n) (0 : E) - choose p hp hmin using hex - have hleast (n : ℕ) (w : E) (hw : w ∈ K n) : ‖p n‖ ≤ ‖w‖ := by - have hm : ‖p n‖ = ⨅ v : K n, ‖(v : E)‖ := by simpa only [zero_sub,norm_neg] using hmin n - rw [hm] - exact ciInf_le (show BddBelow (range (fun v : K n => ‖(v : E)‖)) from - ⟨0,by rintro y ⟨v,rfl⟩; exact norm_nonneg _⟩) (⟨w,hw⟩ : K n) - have hnormmono : Monotone (fun n => ‖p n‖^2) := by - intro n m hnm - exact pow_le_pow_left₀ (norm_nonneg _) (hleast n (p m) (hanti hnm (hp m))) 2 - have hnormbdd : BddAbove (range (fun n => ‖p n‖^2)) := by - refine ⟨B^2,?_⟩ - rintro _ ⟨n,rfl⟩ - exact pow_le_pow_left₀ (norm_nonneg _) (hbound n (p n) (hp n)) 2 - have hsquare : CauchySeq (fun n => ‖p n‖^2) := - (tendsto_atTop_ciSup hnormmono hnormbdd).cauchySeq - have hdist (n m : ℕ) (hnm : n ≤ m) : - dist (p m) (p n)^2 ≤ ‖p m‖^2-‖p n‖^2 := by - have hproj := (norm_eq_iInf_iff_real_inner_le_zero (hconvex n) (hp n)).mp - (hmin n) (p m) (hanti hnm (hp m)) - simp only [zero_sub,inner_neg_left,inner_sub_right, - real_inner_self_eq_norm_sq] at hproj - rw [dist_eq_norm,norm_sub_sq_real,real_inner_comm] - linarith - have hcauchy : CauchySeq p := by - apply Metric.cauchySeq_iff'.mpr - intro ε hε - obtain ⟨N,hN⟩ := Metric.cauchySeq_iff.mp hsquare (ε^2) (sq_pos_of_pos hε) - refine ⟨N,fun n hn => ?_⟩ - have hh := hN n hn N le_rfl - rw [Real.dist_eq,abs_of_nonneg (sub_nonneg.mpr (hnormmono hn))] at hh - have hsq := hdist N n hn - nlinarith [dist_nonneg (x := p n) (y := p N)] - obtain ⟨x,hx⟩ := cauchySeq_tendsto_of_complete hcauchy - refine ⟨x,fun n => ?_⟩ - apply (hclosed n).mem_of_tendsto hx - filter_upwards [eventually_ge_atTop n] with m hm - exact hanti hm (hp m) - -/-- The combinations use indices `n+i`, so every term is in the `n`-th -original tail. No weak-convergence or separability hypothesis is needed. -/ -theorem exists_convex_tail_combinations (u : ℕ → E) - (B : ℝ) (hB : ∀ n, ‖u n‖ ≤ B) : - ∃ x : E, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ - (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ n, dist (∑ i ∈ s n, w n i • u (n+i)) x < (1/2:ℝ)^n) ∧ - Tendsto (fun n => ∑ i ∈ s n, w n i • u (n+i)) atTop (𝓝 x) := by - let K : ℕ → Set E := fun n => closedConvexHull ℝ (range (fun i => u (n+i))) - have hne (n : ℕ) : (K n).Nonempty := - ⟨u (n+0),subset_closedConvexHull ⟨0,rfl⟩⟩ - have hclosed (n : ℕ) : IsClosed (K n) := isClosed_closedConvexHull - have hconvex (n : ℕ) : Convex ℝ (K n) := convex_closedConvexHull - have hanti : Antitone K := by - intro n m hnm - apply (closedConvexHull ℝ).monotone - rintro _ ⟨i,rfl⟩ - refine ⟨m-n+i,?_⟩ - exact congrArg u (by omega) - have hbound (n : ℕ) (x : E) (hx : x ∈ K n) : ‖x‖ ≤ B := by - have hh : K n ⊆ closedBall 0 B := by - apply closedConvexHull_min _ (convex_closedBall 0 B) isClosed_closedBall - rintro _ ⟨i,rfl⟩ - simpa only [mem_closedBall,dist_zero_right] using hB (n+i) - simpa only [mem_closedBall,dist_zero_right] using hh hx - obtain ⟨x,hx⟩ := nested_closed_convex_intersection K hne hclosed hconvex hanti B hbound - have happ (n : ℕ) : ∃ y ∈ convexHull ℝ (range (fun i => u (n+i))), - dist y x < (1/2:ℝ)^n := by - have hmem : x ∈ closure (convexHull ℝ (range (fun i => u (n+i)))) := by - simpa only [K,closedConvexHull_eq_closure_convexHull] using hx n - obtain ⟨y,hy,hyx⟩ := Metric.mem_closure_iff.mp hmem ((1/2:ℝ)^n) (by positivity) - exact ⟨y,hy,by simpa only [dist_comm] using hyx⟩ - choose y hy hdist using happ - have hcomb (n : ℕ) : ∃ s : Finset ℕ, ∃ w : ℕ → ℝ, - (∀ i ∈ s, 0 ≤ w i) ∧ (∑ i ∈ s, w i = 1) ∧ - (∑ i ∈ s, w i • u (n+i)) = y n := by - have hn := hy n - rw [convexHull_range_eq_exists_affineCombination] at hn - obtain ⟨s,w,hw,hs,hv⟩ := hn - refine ⟨s,w,hw,hs,?_⟩ - simpa only [Finset.affineCombination_eq_linear_combination _ _ _ hs] using hv - choose s w hw hs hsum using hcomb - refine ⟨x,s,w,hw,hs,?_,?_⟩ - · intro n - simpa only [hsum] using hdist n - have hyconv : Tendsto y atTop (𝓝 x) := by - apply Metric.tendsto_atTop.mpr - intro ε hε - have he := ((tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2) - (by norm_num : (1/2:ℝ) < 1)).eventually (gt_mem_nhds hε)) - obtain ⟨N,hN⟩ := eventually_atTop.mp he - exact ⟨N,fun n hn => (hdist n).trans (hN n hn)⟩ - simpa only [hsum] using hyconv - -end SLEExactGauge - -end - -end OAI +import Mathlib.Analysis.InnerProductSpace.Projection.Minimal +import Mathlib.Analysis.Convex.Combination +import Mathlib.Analysis.Convex.Topology +import Mathlib.Analysis.Normed.Module.Convex +import Mathlib.Topology.Order.MonotoneConvergence + +namespace OAI + +/-! Deterministic convex extraction in a Hilbert space. Nested closed +convex tail hulls have a common point, and finite convex combinations +of the original tails converge to that point. -/ + +noncomputable section +open Set Filter Metric +open scoped Topology + +namespace SLEExactGauge + +variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] + +lemma nested_closed_convex_intersection (K : ℕ → Set E) + (hne : ∀ n, (K n).Nonempty) (hclosed : ∀ n, IsClosed (K n)) + (hconvex : ∀ n, Convex ℝ (K n)) (hanti : Antitone K) + (B : ℝ) (hbound : ∀ n x, x ∈ K n → ‖x‖ ≤ B) : + ∃ x, ∀ n, x ∈ K n := by + have hex (n : ℕ) := exists_norm_eq_iInf_of_complete_convex (hne n) + (hclosed n).isComplete (hconvex n) (0 : E) + choose p hp hmin using hex + have hleast (n : ℕ) (w : E) (hw : w ∈ K n) : ‖p n‖ ≤ ‖w‖ := by + have hm : ‖p n‖ = ⨅ v : K n, ‖(v : E)‖ := by simpa only [zero_sub,norm_neg] using hmin n + rw [hm] + exact ciInf_le (show BddBelow (range (fun v : K n => ‖(v : E)‖)) from + ⟨0,by rintro y ⟨v,rfl⟩; exact norm_nonneg _⟩) (⟨w,hw⟩ : K n) + have hnormmono : Monotone (fun n => ‖p n‖^2) := by + intro n m hnm + exact pow_le_pow_left₀ (norm_nonneg _) (hleast n (p m) (hanti hnm (hp m))) 2 + have hnormbdd : BddAbove (range (fun n => ‖p n‖^2)) := by + refine ⟨B^2,?_⟩ + rintro _ ⟨n,rfl⟩ + exact pow_le_pow_left₀ (norm_nonneg _) (hbound n (p n) (hp n)) 2 + have hsquare : CauchySeq (fun n => ‖p n‖^2) := + (tendsto_atTop_ciSup hnormmono hnormbdd).cauchySeq + have hdist (n m : ℕ) (hnm : n ≤ m) : + dist (p m) (p n)^2 ≤ ‖p m‖^2-‖p n‖^2 := by + have hproj := (norm_eq_iInf_iff_real_inner_le_zero (hconvex n) (hp n)).mp + (hmin n) (p m) (hanti hnm (hp m)) + simp only [zero_sub,inner_neg_left,inner_sub_right, + real_inner_self_eq_norm_sq] at hproj + rw [dist_eq_norm,norm_sub_sq_real,real_inner_comm] + linarith + have hcauchy : CauchySeq p := by + apply Metric.cauchySeq_iff'.mpr + intro ε hε + obtain ⟨N,hN⟩ := Metric.cauchySeq_iff.mp hsquare (ε^2) (sq_pos_of_pos hε) + refine ⟨N,fun n hn => ?_⟩ + have hh := hN n hn N le_rfl + rw [Real.dist_eq,abs_of_nonneg (sub_nonneg.mpr (hnormmono hn))] at hh + have hsq := hdist N n hn + nlinarith [dist_nonneg (x := p n) (y := p N)] + obtain ⟨x,hx⟩ := cauchySeq_tendsto_of_complete hcauchy + refine ⟨x,fun n => ?_⟩ + apply (hclosed n).mem_of_tendsto hx + filter_upwards [eventually_ge_atTop n] with m hm + exact hanti hm (hp m) + +/-- The combinations use indices `n+i`, so every term is in the `n`-th +original tail. No weak-convergence or separability hypothesis is needed. -/ +theorem exists_convex_tail_combinations (u : ℕ → E) + (B : ℝ) (hB : ∀ n, ‖u n‖ ≤ B) : + ∃ x : E, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ + (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ n, dist (∑ i ∈ s n, w n i • u (n+i)) x < (1/2:ℝ)^n) ∧ + Tendsto (fun n => ∑ i ∈ s n, w n i • u (n+i)) atTop (𝓝 x) := by + let K : ℕ → Set E := fun n => closedConvexHull ℝ (range (fun i => u (n+i))) + have hne (n : ℕ) : (K n).Nonempty := + ⟨u (n+0),subset_closedConvexHull ⟨0,rfl⟩⟩ + have hclosed (n : ℕ) : IsClosed (K n) := isClosed_closedConvexHull + have hconvex (n : ℕ) : Convex ℝ (K n) := convex_closedConvexHull + have hanti : Antitone K := by + intro n m hnm + apply (closedConvexHull ℝ).monotone + rintro _ ⟨i,rfl⟩ + refine ⟨m-n+i,?_⟩ + exact congrArg u (by omega) + have hbound (n : ℕ) (x : E) (hx : x ∈ K n) : ‖x‖ ≤ B := by + have hh : K n ⊆ closedBall 0 B := by + apply closedConvexHull_min _ (convex_closedBall 0 B) isClosed_closedBall + rintro _ ⟨i,rfl⟩ + simpa only [mem_closedBall,dist_zero_right] using hB (n+i) + simpa only [mem_closedBall,dist_zero_right] using hh hx + obtain ⟨x,hx⟩ := nested_closed_convex_intersection K hne hclosed hconvex hanti B hbound + have happ (n : ℕ) : ∃ y ∈ convexHull ℝ (range (fun i => u (n+i))), + dist y x < (1/2:ℝ)^n := by + have hmem : x ∈ closure (convexHull ℝ (range (fun i => u (n+i)))) := by + simpa only [K,closedConvexHull_eq_closure_convexHull] using hx n + obtain ⟨y,hy,hyx⟩ := Metric.mem_closure_iff.mp hmem ((1/2:ℝ)^n) (by positivity) + exact ⟨y,hy,by simpa only [dist_comm] using hyx⟩ + choose y hy hdist using happ + have hcomb (n : ℕ) : ∃ s : Finset ℕ, ∃ w : ℕ → ℝ, + (∀ i ∈ s, 0 ≤ w i) ∧ (∑ i ∈ s, w i = 1) ∧ + (∑ i ∈ s, w i • u (n+i)) = y n := by + have hn := hy n + rw [convexHull_range_eq_exists_affineCombination] at hn + obtain ⟨s,w,hw,hs,hv⟩ := hn + refine ⟨s,w,hw,hs,?_⟩ + simpa only [Finset.affineCombination_eq_linear_combination _ _ _ hs] using hv + choose s w hw hs hsum using hcomb + refine ⟨x,s,w,hw,hs,?_,?_⟩ + · intro n + simpa only [hsum] using hdist n + have hyconv : Tendsto y atTop (𝓝 x) := by + apply Metric.tendsto_atTop.mpr + intro ε hε + have he := ((tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0:ℝ) ≤ 1/2) + (by norm_num : (1/2:ℝ) < 1)).eventually (gt_mem_nhds hε)) + obtain ⟨N,hN⟩ := eventually_atTop.mp he + exact ⟨N,fun n hn => (hdist n).trans (hN n hn)⟩ + simpa only [hsum] using hyconv + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/InverseMassDilation.lean b/lean/OAI/Probability/SLE/Mass/InverseMassDilation.lean index b364699ed7ab252c12a8822ca403a359ede63e5e..41cd8a6a87f44e4e6304b268b894217f17ef2898 100644 --- a/lean/OAI/Probability/SLE/Mass/InverseMassDilation.lean +++ b/lean/OAI/Probability/SLE/Mass/InverseMassDilation.lean @@ -1,73 +1,73 @@ -import OAI.Probability.SLE.Mass.PhysicalDiskMass -import OAI.Probability.SLE.Mass.DilationMass - -namespace OAI - -/-! Compatibility of the actual inverse restart mass with spatial normalization. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge -open Multipoint - -lemma IsCapacityTwoWitness.inverse_mass_dilation - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) - (κ : ℝ) (hκ : 0 < κ) (t : ℝ≥0) (a : ℝ≥0) (ha : 0 < a) - (μ : Measure upperHalfPlane) : - weightedImageMass (fun z : upperHalfPlane => F t ((z:ℂ)+(U t:ℂ))) - (fun z : upperHalfPlane => ENNReal.ofReal - (‖deriv (F t) ((z:ℂ)+(U t:ℂ))‖^dimension κ)) (dilationMass κ a ha μ) = - weightedImageMass (fun z : upperHalfPlane => F t ((U t:ℂ)+(a:ℂ)*z)) - (fun z : upperHalfPlane => ENNReal.ofReal - (‖deriv (fun v : ℂ => F t ((U t:ℂ)+(a:ℂ)*v)) z‖^dimension κ)) μ := by - let J := fun z : upperHalfPlane => F t ((z:ℂ)+(U t:ℂ)) - let w := fun z : upperHalfPlane => ENNReal.ofReal - (‖deriv (F t) ((z:ℂ)+(U t:ℂ))‖^dimension κ) - have hshift (z : upperHalfPlane) : (z:ℂ)+(U t:ℂ) ∈ upperHalfPlane := by - simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero] - using z.property - have hJ : Measurable J := - ((hw.2.2.2.2.1 t).continuousOn.comp_continuous - (continuous_subtype_val.add continuous_const) hshift).measurable - have hwM : Measurable w := by - have hh : w=(fun z : upperHalfPlane => (hw.restartMassWeight t κ hκ z:ℝ≥0∞)) := by - funext z - rw [← ENNReal.ofReal_coe_nnreal] - rfl - rw [hh] - exact (hw.restartMassWeight t κ hκ).continuous.measurable.coe_nnreal_ennreal - change weightedImageMass J w (weightedImageMass (halfPlaneDilation a ha) - (fun _ : upperHalfPlane => ENNReal.ofReal ((a:ℝ)^dimension κ)) μ)=_ - rw [weightedImageMass_comp J hJ w hwM _ (halfPlaneDilation a ha).continuous.measurable _ - measurable_const] - have hmap : J ∘ halfPlaneDilation a ha = - (fun z : upperHalfPlane => F t ((U t:ℂ)+(a:ℂ)*z)) := by - funext z - simp only [Function.comp_apply,J,halfPlaneDilation_apply,add_comm] - rw [hmap] - congr 1 - funext z - change ENNReal.ofReal ((a:ℝ)^dimension κ) * - ENNReal.ofReal (‖deriv (F t) ((a:ℂ)*z+(U t:ℂ))‖^dimension κ)=_ - have haz : (U t:ℂ)+(a:ℂ)*z ∈ upperHalfPlane := by - change 0 < ((U t:ℂ)+(a:ℂ)*z).im - simpa only [Complex.add_im,Complex.ofReal_im,zero_add,Complex.mul_im, - Complex.ofReal_re,zero_mul,add_zero] using mul_pos (show 0<(a:ℝ) from ha) z.property - have hFd := (hw.2.2.2.2.1 t).differentiableAt (isOpen_upperHalfPlane.mem_nhds haz) - have had : HasDerivAt (fun v : ℂ => (U t:ℂ)+(a:ℂ)*v) (a:ℂ) (z:ℂ) := by - convert ((hasDerivAt_id (z:ℂ)).const_mul (a:ℂ)).const_add (U t:ℂ) using 1 <;> simp - have hdcomp : deriv (fun v : ℂ => F t ((U t:ℂ)+(a:ℂ)*v)) (z:ℂ)= - deriv (F t) ((U t:ℂ)+(a:ℂ)*z)*(a:ℂ) := (hFd.hasDerivAt.comp (z:ℂ) had).deriv - rw [hdcomp,norm_mul,Real.mul_rpow (norm_nonneg _) (norm_nonneg _), - ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _),Complex.norm_real, - Real.norm_eq_abs,abs_of_pos (show 0<(a:ℝ) from ha),add_comm ((a:ℂ)*z) (U t:ℂ)] - exact mul_comm _ _ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.PhysicalDiskMass +import OAI.Probability.SLE.Mass.DilationMass + +namespace OAI + +/-! Compatibility of the actual inverse restart mass with spatial normalization. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Multipoint + +lemma IsCapacityTwoWitness.inverse_mass_dilation + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) + (κ : ℝ) (hκ : 0 < κ) (t : ℝ≥0) (a : ℝ≥0) (ha : 0 < a) + (μ : Measure upperHalfPlane) : + weightedImageMass (fun z : upperHalfPlane => F t ((z:ℂ)+(U t:ℂ))) + (fun z : upperHalfPlane => ENNReal.ofReal + (‖deriv (F t) ((z:ℂ)+(U t:ℂ))‖^dimension κ)) (dilationMass κ a ha μ) = + weightedImageMass (fun z : upperHalfPlane => F t ((U t:ℂ)+(a:ℂ)*z)) + (fun z : upperHalfPlane => ENNReal.ofReal + (‖deriv (fun v : ℂ => F t ((U t:ℂ)+(a:ℂ)*v)) z‖^dimension κ)) μ := by + let J := fun z : upperHalfPlane => F t ((z:ℂ)+(U t:ℂ)) + let w := fun z : upperHalfPlane => ENNReal.ofReal + (‖deriv (F t) ((z:ℂ)+(U t:ℂ))‖^dimension κ) + have hshift (z : upperHalfPlane) : (z:ℂ)+(U t:ℂ) ∈ upperHalfPlane := by + simpa only [upperHalfPlane,mem_ofPred_eq,Complex.add_im,Complex.ofReal_im,add_zero] + using z.property + have hJ : Measurable J := + ((hw.2.2.2.2.1 t).continuousOn.comp_continuous + (continuous_subtype_val.add continuous_const) hshift).measurable + have hwM : Measurable w := by + have hh : w=(fun z : upperHalfPlane => (hw.restartMassWeight t κ hκ z:ℝ≥0∞)) := by + funext z + rw [← ENNReal.ofReal_coe_nnreal] + rfl + rw [hh] + exact (hw.restartMassWeight t κ hκ).continuous.measurable.coe_nnreal_ennreal + change weightedImageMass J w (weightedImageMass (halfPlaneDilation a ha) + (fun _ : upperHalfPlane => ENNReal.ofReal ((a:ℝ)^dimension κ)) μ)=_ + rw [weightedImageMass_comp J hJ w hwM _ (halfPlaneDilation a ha).continuous.measurable _ + measurable_const] + have hmap : J ∘ halfPlaneDilation a ha = + (fun z : upperHalfPlane => F t ((U t:ℂ)+(a:ℂ)*z)) := by + funext z + simp only [Function.comp_apply,J,halfPlaneDilation_apply,add_comm] + rw [hmap] + congr 1 + funext z + change ENNReal.ofReal ((a:ℝ)^dimension κ) * + ENNReal.ofReal (‖deriv (F t) ((a:ℂ)*z+(U t:ℂ))‖^dimension κ)=_ + have haz : (U t:ℂ)+(a:ℂ)*z ∈ upperHalfPlane := by + change 0 < ((U t:ℂ)+(a:ℂ)*z).im + simpa only [Complex.add_im,Complex.ofReal_im,zero_add,Complex.mul_im, + Complex.ofReal_re,zero_mul,add_zero] using mul_pos (show 0<(a:ℝ) from ha) z.property + have hFd := (hw.2.2.2.2.1 t).differentiableAt (isOpen_upperHalfPlane.mem_nhds haz) + have had : HasDerivAt (fun v : ℂ => (U t:ℂ)+(a:ℂ)*v) (a:ℂ) (z:ℂ) := by + convert ((hasDerivAt_id (z:ℂ)).const_mul (a:ℂ)).const_add (U t:ℂ) using 1 <;> simp + have hdcomp : deriv (fun v : ℂ => F t ((U t:ℂ)+(a:ℂ)*v)) (z:ℂ)= + deriv (F t) ((U t:ℂ)+(a:ℂ)*z)*(a:ℂ) := (hFd.hasDerivAt.comp (z:ℂ) had).deriv + rw [hdcomp,norm_mul,Real.mul_rpow (norm_nonneg _) (norm_nonneg _), + ENNReal.ofReal_mul (Real.rpow_nonneg (norm_nonneg _) _),Complex.norm_real, + Real.norm_eq_abs,abs_of_pos (show 0<(a:ℝ) from ha),add_comm ((a:ℂ)*z) (U t:ℂ)] + exact mul_comm _ _ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/JointDiskMass.lean b/lean/OAI/Probability/SLE/Mass/JointDiskMass.lean index 0e26951478f489b66969ef38cd3cb8c169acf65c..8755c0d707141f4d4a8126777b1e9485ab209a71 100644 --- a/lean/OAI/Probability/SLE/Mass/JointDiskMass.lean +++ b/lean/OAI/Probability/SLE/Mass/JointDiskMass.lean @@ -1,84 +1,84 @@ -import OAI.Probability.SLE.Mass.DiskChartMass -import OAI.Probability.SLE.Mass.RadonRandomTesting -import OAI.Probability.SLE.Stochastic.ProgressiveParameterTime -import OAI.Probability.SLE.Geometry.CanonicalRawTerminal - -namespace OAI - -/-! Measurable disk mass for a varying angle and raw path. -The defining integral agrees with the already constructed compact disk scalar. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge - -/-- A formula for the derivative weight which is measurable at every real -angle and agrees with the disk chart derivative on the natural angle interval. -/ -def diskDerivativeWeight (κ θ : ℝ) (z : upperHalfPlane) : ℝ≥0∞ := - ENNReal.ofReal (‖(Complex.exp (θ*Complex.I)-Complex.exp (-θ*Complex.I))/ - ((z:ℂ)-Complex.exp (-θ*Complex.I))^2‖^dimension κ) - -def diskMassValue (κ θ : ℝ) (μ : Measure upperHalfPlane) : ℝ≥0∞ := - ∫⁻ z, (if Target.diskMap θ z ∈ Target.diskK then diskDerivativeWeight κ θ z else 0) ∂μ - -lemma diskMassValue_measurable - {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ) (θ : Ω → ℝ) (hθ : Measurable θ) - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) - (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω)) : - Measurable (fun ω => diskMassValue κ (θ ω) (μ ω)) := by - unfold diskMassValue - apply halfPlane_random_test_measurable μ hμ hfin - (fun q => if Target.diskMap (θ q.1) q.2 ∈ Target.diskK then - diskDerivativeWeight κ (θ q.1) q.2 else 0) - have hm : Measurable (fun q : Ω × upperHalfPlane => Target.diskMap (θ q.1) q.2) := by - unfold Target.diskMap - fun_prop - have hw : Measurable (fun q : Ω × upperHalfPlane => diskDerivativeWeight κ (θ q.1) q.2) := by - unfold diskDerivativeWeight - fun_prop - exact hw.ite ((isClosed_closedBall.measurableSet).preimage hm) measurable_const - -lemma diskMassValue_eq (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (μ : Measure upperHalfPlane) : - diskMassValue κ θ μ = ∫⁻ z in Target.diskSourceHalfPlane θ, - (Target.diskMassWeight κ θ hθ z : ℝ≥0∞) ∂μ := by - rw [diskMassValue,← lintegral_indicator (Target.diskSourceHalfPlane_compact hθ).measurableSet] - apply lintegral_congr - intro z - by_cases hz : z ∈ Target.diskSourceHalfPlane θ - · rw [indicator_of_mem hz,ite_eq_left ((Target.mem_diskSourceHalfPlane_iff hθ z).mp hz), - Target.diskMassWeight_ennreal,diskDerivativeWeight,(Target.diskMap_hasDerivAt hθ z.property).deriv] - · rw [indicator_of_notMem hz,ite_eq_right (fun h => hz ((Target.mem_diskSourceHalfPlane_iff hθ z).mpr h))] - -lemma diskMassValue_toReal_eq (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (μ : Measure upperHalfPlane) : - (diskMassValue κ θ μ).toReal = Multipoint.compactWeightedMass - (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) μ := by - rw [diskMassValue_eq κ θ hθ μ,Multipoint.compactWeightedMass_eq_lintegral] - -namespace Multipoint - -local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ - -lemma RawCanonicalMass.disk_at_parameter_measurable - {Ξ : Type*} [MeasurableSpace Ξ] (R : RawCanonicalMass) (κ : ℝ) - (θ : Ξ → ℝ) (hθ : Measurable θ) (g : Ξ → C(ℝ≥0,ℝ)) (hg : Measurable g) - (T : Ξ → ℝ≥0∞) (hT : Measurable T) : - Measurable (fun x => diskMassValue κ (θ x) - (extendedCanonicalMass R.process R.terminal (T x) (g x))) := by - apply diskMassValue_measurable κ θ hθ - · exact ((_root_.OAI.MeasureTheory.IsProgressive.parameter_time_measurable R.progressive g hg _ hT.ennreal_toNNReal).ite - (measurableSet_lt hT measurable_const) (R.terminal_measurable.comp hg)) - · intro x - exact (extendedCanonicalMass_radon R.process R.terminal R.radon R.terminal_radon (T x) (g x)).2 - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.DiskChartMass +import OAI.Probability.SLE.Mass.RadonRandomTesting +import OAI.Probability.SLE.Stochastic.ProgressiveParameterTime +import OAI.Probability.SLE.Geometry.CanonicalRawTerminal + +namespace OAI + +/-! Measurable disk mass for a varying angle and raw path. +The defining integral agrees with the already constructed compact disk scalar. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Metric +open scoped Topology NNReal ENNReal +attribute [local instance] Classical.propDecidable + +namespace SLEExactGauge + +/-- A formula for the derivative weight which is measurable at every real +angle and agrees with the disk chart derivative on the natural angle interval. -/ +def diskDerivativeWeight (κ θ : ℝ) (z : upperHalfPlane) : ℝ≥0∞ := + ENNReal.ofReal (‖(Complex.exp (θ*Complex.I)-Complex.exp (-θ*Complex.I))/ + ((z:ℂ)-Complex.exp (-θ*Complex.I))^2‖^dimension κ) + +def diskMassValue (κ θ : ℝ) (μ : Measure upperHalfPlane) : ℝ≥0∞ := + ∫⁻ z, (if Target.diskMap θ z ∈ Target.diskK then diskDerivativeWeight κ θ z else 0) ∂μ + +lemma diskMassValue_measurable + {Ω : Type*} [MeasurableSpace Ω] (κ : ℝ) (θ : Ω → ℝ) (hθ : Measurable θ) + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) + (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω)) : + Measurable (fun ω => diskMassValue κ (θ ω) (μ ω)) := by + unfold diskMassValue + apply halfPlane_random_test_measurable μ hμ hfin + (fun q => if Target.diskMap (θ q.1) q.2 ∈ Target.diskK then + diskDerivativeWeight κ (θ q.1) q.2 else 0) + have hm : Measurable (fun q : Ω × upperHalfPlane => Target.diskMap (θ q.1) q.2) := by + unfold Target.diskMap + fun_prop + have hw : Measurable (fun q : Ω × upperHalfPlane => diskDerivativeWeight κ (θ q.1) q.2) := by + unfold diskDerivativeWeight + fun_prop + exact hw.ite ((isClosed_closedBall.measurableSet).preimage hm) measurable_const + +lemma diskMassValue_eq (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (μ : Measure upperHalfPlane) : + diskMassValue κ θ μ = ∫⁻ z in Target.diskSourceHalfPlane θ, + (Target.diskMassWeight κ θ hθ z : ℝ≥0∞) ∂μ := by + rw [diskMassValue,← lintegral_indicator (Target.diskSourceHalfPlane_compact hθ).measurableSet] + apply lintegral_congr + intro z + by_cases hz : z ∈ Target.diskSourceHalfPlane θ + · rw [indicator_of_mem hz,ite_eq_left ((Target.mem_diskSourceHalfPlane_iff hθ z).mp hz), + Target.diskMassWeight_ennreal,diskDerivativeWeight,(Target.diskMap_hasDerivAt hθ z.property).deriv] + · rw [indicator_of_notMem hz,ite_eq_right (fun h => hz ((Target.mem_diskSourceHalfPlane_iff hθ z).mpr h))] + +lemma diskMassValue_toReal_eq (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (μ : Measure upperHalfPlane) : + (diskMassValue κ θ μ).toReal = Multipoint.compactWeightedMass + (Target.diskMassWeight κ θ hθ) (Target.diskSourceHalfPlane θ) μ := by + rw [diskMassValue_eq κ θ hθ μ,Multipoint.compactWeightedMass_eq_lintegral] + +namespace Multipoint + +local instance : MeasurableSpace C(ℝ≥0,ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0,ℝ) := ⟨rfl⟩ + +lemma RawCanonicalMass.disk_at_parameter_measurable + {Ξ : Type*} [MeasurableSpace Ξ] (R : RawCanonicalMass) (κ : ℝ) + (θ : Ξ → ℝ) (hθ : Measurable θ) (g : Ξ → C(ℝ≥0,ℝ)) (hg : Measurable g) + (T : Ξ → ℝ≥0∞) (hT : Measurable T) : + Measurable (fun x => diskMassValue κ (θ x) + (extendedCanonicalMass R.process R.terminal (T x) (g x))) := by + apply diskMassValue_measurable κ θ hθ + · exact ((_root_.OAI.MeasureTheory.IsProgressive.parameter_time_measurable R.progressive g hg _ hT.ennreal_toNNReal).ite + (measurableSet_lt hT measurable_const) (R.terminal_measurable.comp hg)) + · intro x + exact (extendedCanonicalMass_radon R.process R.terminal R.radon R.terminal_radon (T x) (g x)).2 + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/L2ConvexTails.lean b/lean/OAI/Probability/SLE/Mass/L2ConvexTails.lean index 9741cc7d5581ceadf2ae509fba553370a2a73a9f..d97e922646e5d7acc456a5f802b419b942b217d1 100644 --- a/lean/OAI/Probability/SLE/Mass/L2ConvexTails.lean +++ b/lean/OAI/Probability/SLE/Mass/L2ConvexTails.lean @@ -1,83 +1,83 @@ -import OAI.Probability.SLE.Mass.SimultaneousConvexTails -import Mathlib.MeasureTheory.Function.L2Space -import Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum - -namespace OAI - -/-! A single deterministic sequence of finite convex tail combinations -for a countable family of uniformly L²-bounded random variables. Both -L² convergence and simultaneous almost-sure convergence are obtained. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - -lemma ae_tendsto_Lp_of_summable_errors {E : Type*} [NormedAddCommGroup E] - {p : ℝ≥0∞} [Fact (1 ≤ p)] (u : ℕ → Lp E p P) (x : Lp E p P) - (h : Summable (fun n => ‖u n-x‖)) : - ∀ᵐ ω ∂P, Tendsto (fun n => u n ω) atTop (𝓝 (x ω)) := by - have hsum : ∑' n, eLpNorm (fun ω => (u n-x) ω) p P ≠ ⊤ := by - simpa only [← Lp.enorm_def] using - (tsum_enorm_ne_top_iff_summable_norm.mpr h) - have hae := summable_norm_of_tsum_eLpNorm_ne_top (show 1 ≤ p from Fact.out) hsum - have heq : ∀ᵐ ω ∂P, ∀ n, (u n-x) ω = u n ω-x ω := - ae_all_iff.mpr (fun n => Lp.coeFn_sub (u n) x) - filter_upwards [hae,heq] with ω hω heω - apply tendsto_iff_norm_sub_tendsto_zero.mpr - simpa only [heω] using hω.tendsto_atTop_zero - -lemma ae_coeFn_weighted_sum (s : Finset ℕ) (w : ℕ → ℝ) (u : ℕ → Lp ℝ 2 P) : - ∀ᵐ ω ∂P, (∑ i ∈ s, w i • u i) ω = ∑ i ∈ s, w i * u i ω := by - have hs := (Lp.coeFn_finsetSum s (fun i => w i • u i)).trans - (eventuallyEq_sum (fun i _ => Lp.coeFn_smul (w i) (u i))) - filter_upwards [hs] with ω hω - simpa only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul] using hω - -theorem exists_L2_convex_tail_limit (u : ℕ → ℕ → Lp ℝ 2 P) - (hbound : ∀ j, ∃ B : ℝ, ∀ n, ‖u n j‖ ≤ B) : - ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ - (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i • u (n+i) j) atTop (𝓝 (x j))) ∧ - (∀ᵐ ω ∂P, ∀ j, - Tendsto (fun n => ∑ i ∈ s n, w n i * u (n+i) j ω) atTop (𝓝 (x j ω))) := by - obtain ⟨x,s,w,hw,hs,hsum,hconv⟩ := - exists_simultaneous_convex_tail_combinations u hbound - refine ⟨x,s,w,hw,hs,hconv,ae_all_iff.mpr (fun j => ?_)⟩ - have hae := ae_tendsto_Lp_of_summable_errors - (fun n => ∑ i ∈ s n, w n i • u (n+i) j) (x j) (hsum j) - have heq : ∀ᵐ ω ∂P, ∀ n, - (∑ i ∈ s n, w n i • u (n+i) j) ω = ∑ i ∈ s n, w n i * u (n+i) j ω := - ae_all_iff.mpr (fun n => ae_coeFn_weighted_sum (s n) (w n) (fun i => u (n+i) j)) - filter_upwards [hae,heq] with ω hω heω - simpa only [heω] using hω - -/-- The same extraction stated for actual random variables, rather than -chosen representatives of their L² classes. -/ -theorem exists_randomVariable_convex_tail_limit (f : ℕ → ℕ → Ω → ℝ) - (hf : ∀ n j, MemLp (f n j) 2 P) - (hbound : ∀ j, ∃ B : ℝ, ∀ n, ‖(hf n j).toLp (f n j)‖ ≤ B) : - ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ - (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i • (hf (n+i) j).toLp (f (n+i) j)) - atTop (𝓝 (x j))) ∧ - (∀ᵐ ω ∂P, ∀ j, - Tendsto (fun n => ∑ i ∈ s n, w n i * f (n+i) j ω) atTop (𝓝 (x j ω))) := by - let u (n j : ℕ) : Lp ℝ 2 P := (hf n j).toLp (f n j) - obtain ⟨x,s,w,hw,hs,hconv,hae⟩ := exists_L2_convex_tail_limit u hbound - refine ⟨x,s,w,hw,hs,hconv,?_⟩ - have heq : ∀ᵐ ω ∂P, ∀ n j, u n j ω = f n j ω := - ae_all_iff.mpr (fun n => ae_all_iff.mpr (fun j => (hf n j).coeFn_toLp)) - filter_upwards [hae,heq] with ω hω heω - simpa only [heω] using hω - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.SimultaneousConvexTails +import Mathlib.MeasureTheory.Function.L2Space +import Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum + +namespace OAI + +/-! A single deterministic sequence of finite convex tail combinations +for a countable family of uniformly L²-bounded random variables. Both +L² convergence and simultaneous almost-sure convergence are obtained. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + +lemma ae_tendsto_Lp_of_summable_errors {E : Type*} [NormedAddCommGroup E] + {p : ℝ≥0∞} [Fact (1 ≤ p)] (u : ℕ → Lp E p P) (x : Lp E p P) + (h : Summable (fun n => ‖u n-x‖)) : + ∀ᵐ ω ∂P, Tendsto (fun n => u n ω) atTop (𝓝 (x ω)) := by + have hsum : ∑' n, eLpNorm (fun ω => (u n-x) ω) p P ≠ ⊤ := by + simpa only [← Lp.enorm_def] using + (tsum_enorm_ne_top_iff_summable_norm.mpr h) + have hae := summable_norm_of_tsum_eLpNorm_ne_top (show 1 ≤ p from Fact.out) hsum + have heq : ∀ᵐ ω ∂P, ∀ n, (u n-x) ω = u n ω-x ω := + ae_all_iff.mpr (fun n => Lp.coeFn_sub (u n) x) + filter_upwards [hae,heq] with ω hω heω + apply tendsto_iff_norm_sub_tendsto_zero.mpr + simpa only [heω] using hω.tendsto_atTop_zero + +lemma ae_coeFn_weighted_sum (s : Finset ℕ) (w : ℕ → ℝ) (u : ℕ → Lp ℝ 2 P) : + ∀ᵐ ω ∂P, (∑ i ∈ s, w i • u i) ω = ∑ i ∈ s, w i * u i ω := by + have hs := (Lp.coeFn_finsetSum s (fun i => w i • u i)).trans + (eventuallyEq_sum (fun i _ => Lp.coeFn_smul (w i) (u i))) + filter_upwards [hs] with ω hω + simpa only [Finset.sum_apply,Pi.smul_apply,smul_eq_mul] using hω + +theorem exists_L2_convex_tail_limit (u : ℕ → ℕ → Lp ℝ 2 P) + (hbound : ∀ j, ∃ B : ℝ, ∀ n, ‖u n j‖ ≤ B) : + ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ + (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i • u (n+i) j) atTop (𝓝 (x j))) ∧ + (∀ᵐ ω ∂P, ∀ j, + Tendsto (fun n => ∑ i ∈ s n, w n i * u (n+i) j ω) atTop (𝓝 (x j ω))) := by + obtain ⟨x,s,w,hw,hs,hsum,hconv⟩ := + exists_simultaneous_convex_tail_combinations u hbound + refine ⟨x,s,w,hw,hs,hconv,ae_all_iff.mpr (fun j => ?_)⟩ + have hae := ae_tendsto_Lp_of_summable_errors + (fun n => ∑ i ∈ s n, w n i • u (n+i) j) (x j) (hsum j) + have heq : ∀ᵐ ω ∂P, ∀ n, + (∑ i ∈ s n, w n i • u (n+i) j) ω = ∑ i ∈ s n, w n i * u (n+i) j ω := + ae_all_iff.mpr (fun n => ae_coeFn_weighted_sum (s n) (w n) (fun i => u (n+i) j)) + filter_upwards [hae,heq] with ω hω heω + simpa only [heω] using hω + +/-- The same extraction stated for actual random variables, rather than +chosen representatives of their L² classes. -/ +theorem exists_randomVariable_convex_tail_limit (f : ℕ → ℕ → Ω → ℝ) + (hf : ∀ n j, MemLp (f n j) 2 P) + (hbound : ∀ j, ∃ B : ℝ, ∀ n, ‖(hf n j).toLp (f n j)‖ ≤ B) : + ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ + (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i • (hf (n+i) j).toLp (f (n+i) j)) + atTop (𝓝 (x j))) ∧ + (∀ᵐ ω ∂P, ∀ j, + Tendsto (fun n => ∑ i ∈ s n, w n i * f (n+i) j ω) atTop (𝓝 (x j ω))) := by + let u (n j : ℕ) : Lp ℝ 2 P := (hf n j).toLp (f n j) + obtain ⟨x,s,w,hw,hs,hconv,hae⟩ := exists_L2_convex_tail_limit u hbound + refine ⟨x,s,w,hw,hs,hconv,?_⟩ + have heq : ∀ᵐ ω ∂P, ∀ n j, u n j ω = f n j ω := + ae_all_iff.mpr (fun n => ae_all_iff.mpr (fun j => (hf n j).coeFn_toLp)) + filter_upwards [hae,heq] with ω hω heω + simpa only [heω] using hω + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/L2MeanConvergence.lean b/lean/OAI/Probability/SLE/Mass/L2MeanConvergence.lean index 7b4031368b917e23a1ffbca98a71fb645be2b050..f094dc866ae15c121cda39472d117a3d268f3147 100644 --- a/lean/OAI/Probability/SLE/Mass/L2MeanConvergence.lean +++ b/lean/OAI/Probability/SLE/Mass/L2MeanConvergence.lean @@ -1,79 +1,79 @@ -import OAI.Probability.SLE.Mass.L2ConvexTails - -namespace OAI - -/-! Passing expectations through the common L² convex extraction. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology ENNReal - -namespace SLEExactGauge - -lemma continuous_L2_integral - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] : - Continuous (fun x : Lp ℝ 2 P => ∫ ω, x ω ∂P) := by - let h1 : MemLp (fun _ : Ω => (1 : ℝ)) 2 P := memLp_const 1 - let one : Lp ℝ 2 P := h1.toLp (fun _ => 1) - have he (x : Lp ℝ 2 P) : inner ℝ one x = ∫ ω, x ω ∂P := by - rw [L2.inner_def] - apply integral_congr_ae - filter_upwards [h1.coeFn_toLp] with ω hω - simp only [Real.inner_apply,one,hω,one_mul] - simpa only [he] using (continuous_const.inner continuous_id : - Continuous (fun x : Lp ℝ 2 P => inner ℝ one x)) - -lemma weighted_randomVariable_integral_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] - (f : ℕ → Ω → ℝ) (hf : ∀ n, MemLp (f n) 2 P) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (x : Lp ℝ 2 P) - (h : Tendsto (fun n => ∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) atTop (𝓝 x)) : - Tendsto (fun n => ∫ ω, ∑ i ∈ s n, w n i * f (n+i) ω ∂P) - atTop (𝓝 (∫ ω, x ω ∂P)) := by - have he (n : ℕ) : (∫ ω, ∑ i ∈ s n, w n i * f (n+i) ω ∂P) = - ∫ ω, (∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) ω ∂P := by - apply integral_congr_ae - have hc := ae_coeFn_weighted_sum (s n) (w n) (fun i => (hf (n+i)).toLp (f (n+i))) - have hr : ∀ᵐ ω ∂P, ∀ i, (hf (n+i)).toLp (f (n+i)) ω = f (n+i) ω := - ae_all_iff.mpr (fun i => (hf (n+i)).coeFn_toLp) - filter_upwards [hc,hr] with ω hc hr - simpa only [hr] using hc.symm - simpa only [he,Function.comp_def] using (continuous_L2_integral P).continuousAt.tendsto.comp h - -lemma continuous_L2_pairing - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (a : Ω → ℝ) (ha : MemLp a 2 P) : - Continuous (fun x : Lp ℝ 2 P => ∫ ω, a ω * x ω ∂P) := by - let y : Lp ℝ 2 P := ha.toLp a - have he (x : Lp ℝ 2 P) : inner ℝ y x = ∫ ω, a ω * x ω ∂P := by - rw [L2.inner_def] - apply integral_congr_ae - filter_upwards [ha.coeFn_toLp] with ω hω - simp only [Real.inner_apply,y,hω] - simpa only [he] using (continuous_const.inner continuous_id : - Continuous (fun x : Lp ℝ 2 P => inner ℝ y x)) - -lemma weighted_randomVariable_pairing_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (f : ℕ → Ω → ℝ) (hf : ∀ n, MemLp (f n) 2 P) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (x : Lp ℝ 2 P) - (h : Tendsto (fun n => ∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) atTop (𝓝 x)) - (a : Ω → ℝ) (ha : MemLp a 2 P) : - Tendsto (fun n => ∫ ω, a ω * (∑ i ∈ s n, w n i * f (n+i) ω) ∂P) - atTop (𝓝 (∫ ω, a ω * x ω ∂P)) := by - have he (n : ℕ) : (∫ ω, a ω * (∑ i ∈ s n, w n i * f (n+i) ω) ∂P) = - ∫ ω, a ω * (∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) ω ∂P := by - apply integral_congr_ae - have hc := ae_coeFn_weighted_sum (s n) (w n) (fun i => (hf (n+i)).toLp (f (n+i))) - have hr : ∀ᵐ ω ∂P, ∀ i, (hf (n+i)).toLp (f (n+i)) ω = f (n+i) ω := - ae_all_iff.mpr (fun i => (hf (n+i)).coeFn_toLp) - filter_upwards [hc,hr] with ω hc hr - apply congrArg (fun v : ℝ => a ω * v) - simpa only [hr] using hc.symm - simpa only [he,Function.comp_def] using (continuous_L2_pairing P a ha).continuousAt.tendsto.comp h - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.L2ConvexTails + +namespace OAI + +/-! Passing expectations through the common L² convex extraction. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology ENNReal + +namespace SLEExactGauge + +lemma continuous_L2_integral + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] : + Continuous (fun x : Lp ℝ 2 P => ∫ ω, x ω ∂P) := by + let h1 : MemLp (fun _ : Ω => (1 : ℝ)) 2 P := memLp_const 1 + let one : Lp ℝ 2 P := h1.toLp (fun _ => 1) + have he (x : Lp ℝ 2 P) : inner ℝ one x = ∫ ω, x ω ∂P := by + rw [L2.inner_def] + apply integral_congr_ae + filter_upwards [h1.coeFn_toLp] with ω hω + simp only [Real.inner_apply,one,hω,one_mul] + simpa only [he] using (continuous_const.inner continuous_id : + Continuous (fun x : Lp ℝ 2 P => inner ℝ one x)) + +lemma weighted_randomVariable_integral_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] + (f : ℕ → Ω → ℝ) (hf : ∀ n, MemLp (f n) 2 P) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (x : Lp ℝ 2 P) + (h : Tendsto (fun n => ∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) atTop (𝓝 x)) : + Tendsto (fun n => ∫ ω, ∑ i ∈ s n, w n i * f (n+i) ω ∂P) + atTop (𝓝 (∫ ω, x ω ∂P)) := by + have he (n : ℕ) : (∫ ω, ∑ i ∈ s n, w n i * f (n+i) ω ∂P) = + ∫ ω, (∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) ω ∂P := by + apply integral_congr_ae + have hc := ae_coeFn_weighted_sum (s n) (w n) (fun i => (hf (n+i)).toLp (f (n+i))) + have hr : ∀ᵐ ω ∂P, ∀ i, (hf (n+i)).toLp (f (n+i)) ω = f (n+i) ω := + ae_all_iff.mpr (fun i => (hf (n+i)).coeFn_toLp) + filter_upwards [hc,hr] with ω hc hr + simpa only [hr] using hc.symm + simpa only [he,Function.comp_def] using (continuous_L2_integral P).continuousAt.tendsto.comp h + +lemma continuous_L2_pairing + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (a : Ω → ℝ) (ha : MemLp a 2 P) : + Continuous (fun x : Lp ℝ 2 P => ∫ ω, a ω * x ω ∂P) := by + let y : Lp ℝ 2 P := ha.toLp a + have he (x : Lp ℝ 2 P) : inner ℝ y x = ∫ ω, a ω * x ω ∂P := by + rw [L2.inner_def] + apply integral_congr_ae + filter_upwards [ha.coeFn_toLp] with ω hω + simp only [Real.inner_apply,y,hω] + simpa only [he] using (continuous_const.inner continuous_id : + Continuous (fun x : Lp ℝ 2 P => inner ℝ y x)) + +lemma weighted_randomVariable_pairing_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (f : ℕ → Ω → ℝ) (hf : ∀ n, MemLp (f n) 2 P) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (x : Lp ℝ 2 P) + (h : Tendsto (fun n => ∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) atTop (𝓝 x)) + (a : Ω → ℝ) (ha : MemLp a 2 P) : + Tendsto (fun n => ∫ ω, a ω * (∑ i ∈ s n, w n i * f (n+i) ω) ∂P) + atTop (𝓝 (∫ ω, a ω * x ω ∂P)) := by + have he (n : ℕ) : (∫ ω, a ω * (∑ i ∈ s n, w n i * f (n+i) ω) ∂P) = + ∫ ω, a ω * (∑ i ∈ s n, w n i • (hf (n+i)).toLp (f (n+i))) ω ∂P := by + apply integral_congr_ae + have hc := ae_coeFn_weighted_sum (s n) (w n) (fun i => (hf (n+i)).toLp (f (n+i))) + have hr : ∀ᵐ ω ∂P, ∀ i, (hf (n+i)).toLp (f (n+i)) ω = f (n+i) ω := + ae_all_iff.mpr (fun i => (hf (n+i)).coeFn_toLp) + filter_upwards [hc,hr] with ω hc hr + apply congrArg (fun v : ℝ => a ω * v) + simpa only [hr] using hc.symm + simpa only [he,Function.comp_def] using (continuous_L2_pairing P a ha).continuousAt.tendsto.comp h + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelApproximants.lean b/lean/OAI/Probability/SLE/Mass/LevelApproximants.lean index b2a06b6c0116d4bc6ad6872679a285eaa1597c9a..91fe0f8b473fc7846ca951dde4d66b725ba10220 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelApproximants.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelApproximants.lean @@ -1,109 +1,109 @@ -import OAI.Probability.SLE.Multipoint.MultipointLevelMass - -namespace OAI - -/-! The actual finite-level random measures used to construct canonical -trace mass. Time truncation retains precisely levels already attained. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _ -local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ - -lemma actual_radiusLevel_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) : - Measurable (fun q : Ω × ℂ => radiusLevelTime (γ q.1) (G q.1) q.2 u) := by - have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm - have h : Measurable (fun q : Ω × ℂ => - canonicalRadiusLevel κ (u,q.2,sourceDriverPath B hBc q.1)) := - (canonicalRadiusLevel_measurable κ).comp - (measurable_const.prodMk (measurable_snd.prodMk (hp.comp measurable_fst))) - convert h using 1 - funext q - exact (canonicalRadiusLevel_eq κ (sourceDriverPath B hBc q.1) (hw q.1) u hu q.2).symm - -def levelApproximationDensity {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0∞) (ω : Ω) (z : ℂ) : ℝ≥0∞ := - if radiusLevelTime (γ ω) (G ω) z u ≤ t then - ENNReal.ofReal (stoppedGreen κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) z u) - else 0 - -def levelApproximation {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0∞) (ω : Ω) : Measure ℂ := - volume.withDensity (levelApproximationDensity κ B γ G u t ω) - -lemma levelApproximationDensity_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0∞) : Measurable (Function.uncurry (levelApproximationDensity κ B γ G u t)) := by - exact Measurable.ite - (measurableSet_le (actual_radiusLevel_joint_measurable hm hBc κ u hu γ G F hw) measurable_const) - (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal - measurable_const - -lemma levelApproximationDensity_mono - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (ω : Ω) (z : ℂ) : - Monotone (fun t => levelApproximationDensity κ B γ G u t ω z) := by - intro s t hst - dsimp only [levelApproximationDensity] - split_ifs with hs ht - · exact le_rfl - · exact False.elim (ht (hs.trans hst)) - · exact bot_le - · exact le_rfl - -lemma levelApproximation_mono - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (ω : Ω) : Monotone (fun t => levelApproximation κ B γ G u t ω) := by - intro s t hst - exact withDensity_mono (Filter.Eventually.of_forall (fun z => - levelApproximationDensity_mono κ B γ G u ω z hst)) - -lemma levelApproximation_apply_top - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (ω : Ω) {K : Set ℂ} (hK : MeasurableSet K) : - levelApproximation κ B γ G u ∞ ω K = levelMass κ B γ G u K ω := by - rw [levelApproximation,withDensity_apply _ hK] - simp only [levelApproximationDensity,le_top,ite_true,levelMass] - -lemma levelApproximation_apply_le - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0∞) (ω : Ω) {K : Set ℂ} (hK : MeasurableSet K) : - levelApproximation κ B γ G u t ω K ≤ levelMass κ B γ G u K ω := by - rw [← levelApproximation_apply_top κ B γ G u ω hK] - exact levelApproximation_mono κ B γ G u ω le_top K - -lemma levelApproximation_measurable_apply - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0∞) {K : Set ℂ} (hK : MeasurableSet K) : - Measurable (fun ω => levelApproximation κ B γ G u t ω K) := by - simp only [levelApproximation,withDensity_apply _ hK] - exact (levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t).lintegral_prod_right' - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Multipoint.MultipointLevelMass + +namespace OAI + +/-! The actual finite-level random measures used to construct canonical +trace mass. Time truncation retains precisely levels already attained. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +local instance : MeasurableSpace C(ℝ≥0, ℝ) := borel _ +local instance : BorelSpace C(ℝ≥0, ℝ) := ⟨rfl⟩ + +lemma actual_radiusLevel_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) : + Measurable (fun q : Ω × ℂ => radiusLevelTime (γ q.1) (G q.1) q.2 u) := by + have hp : Measurable (sourceDriverPath B hBc) := source_path_measurable _ hm + have h : Measurable (fun q : Ω × ℂ => + canonicalRadiusLevel κ (u,q.2,sourceDriverPath B hBc q.1)) := + (canonicalRadiusLevel_measurable κ).comp + (measurable_const.prodMk (measurable_snd.prodMk (hp.comp measurable_fst))) + convert h using 1 + funext q + exact (canonicalRadiusLevel_eq κ (sourceDriverPath B hBc q.1) (hw q.1) u hu q.2).symm + +def levelApproximationDensity {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0∞) (ω : Ω) (z : ℂ) : ℝ≥0∞ := + if radiusLevelTime (γ ω) (G ω) z u ≤ t then + ENNReal.ofReal (stoppedGreen κ (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) z u) + else 0 + +def levelApproximation {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0∞) (ω : Ω) : Measure ℂ := + volume.withDensity (levelApproximationDensity κ B γ G u t ω) + +lemma levelApproximationDensity_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0∞) : Measurable (Function.uncurry (levelApproximationDensity κ B γ G u t)) := by + exact Measurable.ite + (measurableSet_le (actual_radiusLevel_joint_measurable hm hBc κ u hu γ G F hw) measurable_const) + (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal + measurable_const + +lemma levelApproximationDensity_mono + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (ω : Ω) (z : ℂ) : + Monotone (fun t => levelApproximationDensity κ B γ G u t ω z) := by + intro s t hst + dsimp only [levelApproximationDensity] + split_ifs with hs ht + · exact le_rfl + · exact False.elim (ht (hs.trans hst)) + · exact bot_le + · exact le_rfl + +lemma levelApproximation_mono + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (ω : Ω) : Monotone (fun t => levelApproximation κ B γ G u t ω) := by + intro s t hst + exact withDensity_mono (Filter.Eventually.of_forall (fun z => + levelApproximationDensity_mono κ B γ G u ω z hst)) + +lemma levelApproximation_apply_top + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (ω : Ω) {K : Set ℂ} (hK : MeasurableSet K) : + levelApproximation κ B γ G u ∞ ω K = levelMass κ B γ G u K ω := by + rw [levelApproximation,withDensity_apply _ hK] + simp only [levelApproximationDensity,le_top,ite_true,levelMass] + +lemma levelApproximation_apply_le + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0∞) (ω : Ω) {K : Set ℂ} (hK : MeasurableSet K) : + levelApproximation κ B γ G u t ω K ≤ levelMass κ B γ G u K ω := by + rw [← levelApproximation_apply_top κ B γ G u ω hK] + exact levelApproximation_mono κ B γ G u ω le_top K + +lemma levelApproximation_measurable_apply + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0∞) {K : Set ℂ} (hK : MeasurableSet K) : + Measurable (fun ω => levelApproximation κ B γ G u t ω K) := by + simp only [levelApproximation,withDensity_apply _ hK] + exact (levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t).lintegral_prod_right' + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelApproximationAdapted.lean b/lean/OAI/Probability/SLE/Mass/LevelApproximationAdapted.lean index 2bddc3d4e7141a7dd535dbd70ea6615f65ac2b73..cba494f895591987493411009be5fed99fd750a7 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelApproximationAdapted.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelApproximationAdapted.lean @@ -1,307 +1,307 @@ -import OAI.Probability.SLE.Geometry.SpatialPastKernels -import OAI.Probability.SLE.Mass.LevelRandomRadon -import OAI.Probability.SLE.Mass.SubSigmaRadonLimit - -namespace OAI - -/-! The finite-level mass accumulated by a deterministic time is measurable -in the natural driver filtration. All spatial variables are retained in the -past-measurable kernel before integration. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Classical -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge - -lemma radiusLevelTime_le_iff_prefix - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F) - (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) : - radiusLevelTime γ G z u ≤ (T : ℝ≥0∞) ↔ - (z ∈ survivingDomain γ T ∧ fullRadius G T z ≤ (2*z.im)*Real.exp (-u)) ∨ - radiusLevelTime γ G z u < (T : ℝ≥0∞) := by - constructor - · intro h - rcases h.eq_or_lt with he | ht - · left - obtain ⟨s,hs,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u - (h.trans_lt (by simp)) - have hτ := radiusLevelTime_eq_of_live_level hU hw z u s hs hR - have hsT : s = T := by exact_mod_cast hτ.symm.trans he - subst s - exact ⟨hs,hR.le⟩ - · exact Or.inr ht - · rintro (⟨hT,hR⟩ | ht) - · rcases hR.eq_or_lt with he | ht - · rw [radiusLevelTime_eq_of_live_level hU hw z u T hT he] - · exact (radiusLevelTime_lt_of_live_radius_lt hU hw z u hu T hT ht).le - · exact ht.le - -lemma capped_radiusLevelTime_dense - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F) - (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) : - min (radiusLevelTime γ G z u) (T : ℝ≥0∞) = - ⨅ n : ℕ, if denseSeq ℝ≥0 n < T ∧ - z ∈ survivingDomain γ (denseSeq ℝ≥0 n) ∧ - fullRadius G (denseSeq ℝ≥0 n) z < (2*z.im)*Real.exp (-u) - then (denseSeq ℝ≥0 n : ℝ≥0∞) else T := by - apply eq_of_forall_gt_iff - intro x - by_cases hTx : (T : ℝ≥0∞) < x - · have hl : min (radiusLevelTime γ G z u) (T : ℝ≥0∞) < x := - (min_le_right _ _).trans_lt hTx - have hr : (⨅ n : ℕ, if denseSeq ℝ≥0 n < T ∧ - z ∈ survivingDomain γ (denseSeq ℝ≥0 n) ∧ - fullRadius G (denseSeq ℝ≥0 n) z < (2*z.im)*Real.exp (-u) - then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) < x := by - apply (iInf_le_of_le 0 ?_).trans_lt hTx - split_ifs with h - · exact_mod_cast h.1.le - · exact le_rfl - exact iff_of_true hl hr - · rw [min_lt_iff, or_iff_left hTx, iInf_lt_iff, - radiusLevelTime_lt_iff_dense hU hw z u hu x] - constructor - · rintro ⟨n,hnx,hn,hR⟩ - have hnT : denseSeq ℝ≥0 n < T := by - exact_mod_cast hnx.trans_le (le_of_not_gt hTx) - refine ⟨n,?_⟩ - have hh : denseSeq ℝ≥0 n < T ∧ z ∈ survivingDomain γ (denseSeq ℝ≥0 n) ∧ - fullRadius G (denseSeq ℝ≥0 n) z < (2*z.im)*Real.exp (-u) := ⟨hnT,hn,hR⟩ - rwa [ite_eq_left hh] - · rintro ⟨n,hn⟩ - split_ifs at hn with h - · exact ⟨n,hn,h.2.1,h.2.2⟩ - · exact False.elim (hTx hn) - -private lemma prefix_time_measurable - {Y : Type*} [MeasurableSpace Y] (τ : Y → ℝ≥0∞) (T : ℝ≥0) - (D : ℕ → Y → Prop) (hD : ∀ n, MeasurableSet {p | D n p}) - (E : Y → Prop) (hE : MeasurableSet {p | E p}) - (hcap : ∀ p, min (τ p) (T : ℝ≥0∞) = - ⨅ n : ℕ, if D n p then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) - (hlt : ∀ p, τ p < (T : ℝ≥0∞) ↔ ∃ n, D n p) - (hle : ∀ p, τ p ≤ (T : ℝ≥0∞) ↔ E p ∨ τ p < (T : ℝ≥0∞)) : - Measurable (fun p => min (τ p) (T : ℝ≥0∞)) ∧ - MeasurableSet {p | τ p ≤ (T : ℝ≥0∞)} := by - classical - have hmLess : MeasurableSet {p | τ p < (T : ℝ≥0∞)} := by - have he : {p | τ p < (T : ℝ≥0∞)} = ⋃ n, {p | D n p} := by - ext p - simpa only [mem_iUnion,mem_ofPred_eq] using hlt p - rw [he] - exact MeasurableSet.iUnion hD - constructor - · have hh : Measurable (fun p => ⨅ n : ℕ, - if D n p then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) := - Measurable.iInf fun n => Measurable.ite (hD n) measurable_const measurable_const - convert hh using 1 - exact funext hcap - · convert hE.union hmLess using 1 - ext p - exact hle p - -private lemma prefix_level_time_measurable - {Y : Type*} [MeasurableSpace Y] (τ : Y → ℝ≥0∞) (T : ℝ≥0) - (R : ℝ≥0 → Y → ℝ) (L : ℝ≥0 → Y → Prop) (a : Y → ℝ) - (hR : Measurable (fun p : Iic T × Y => if L p.1 p.2 then R p.1 p.2 else 0)) - (hL : MeasurableSet {p : Iic T × Y | L p.1 p.2}) (ha : Measurable a) - (hcap : ∀ p, min (τ p) (T : ℝ≥0∞) = - ⨅ n : ℕ, if denseSeq ℝ≥0 n < T ∧ L (denseSeq ℝ≥0 n) p ∧ - R (denseSeq ℝ≥0 n) p < a p then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) - (hlt : ∀ p, τ p < (T : ℝ≥0∞) ↔ ∃ n, denseSeq ℝ≥0 n < T ∧ - L (denseSeq ℝ≥0 n) p ∧ R (denseSeq ℝ≥0 n) p < a p) - (hle : ∀ p, τ p ≤ (T : ℝ≥0∞) ↔ - (L T p ∧ R T p ≤ a p) ∨ τ p < (T : ℝ≥0∞)) : - Measurable (fun p => min (τ p) (T : ℝ≥0∞)) ∧ - MeasurableSet {p | τ p ≤ (T : ℝ≥0∞)} := by - classical - have hmR (s : Iic T) : Measurable (fun p => if L s p then R s p else 0) := - hR.comp (measurable_const.prodMk measurable_id) - have hmL (s : Iic T) : MeasurableSet {p | L s p} := - hL.preimage (measurable_const.prodMk measurable_id) - have hmD (n : ℕ) : MeasurableSet {p : Y | - denseSeq ℝ≥0 n < T ∧ L (denseSeq ℝ≥0 n) p ∧ R (denseSeq ℝ≥0 n) p < a p} := by - by_cases hn : denseSeq ℝ≥0 n < T - · let s : Iic T := ⟨denseSeq ℝ≥0 n,hn.le⟩ - convert (hmL s).inter (measurableSet_lt (hmR s) ha) using 1 - ext p - simp only [mem_ofPred_eq,mem_inter_iff,s] - by_cases hl : L (denseSeq ℝ≥0 n) p <;> simp [hl,hn] - · simp only [hn,false_and,ofPred_false,MeasurableSet.empty] - have hEnd : MeasurableSet {p : Y | L T p ∧ R T p ≤ a p} := by - let s : Iic T := ⟨T,by simp⟩ - convert (hmL s).inter (measurableSet_le (hmR s) ha) using 1 - ext p - simp only [mem_inter_iff,mem_ofPred_eq,s] - by_cases hl : L T p <;> simp [hl] - apply prefix_time_measurable τ T _ hmD _ hEnd ?_ hlt hle - intro p - rw [hcap p] - apply iInf_congr - intro n - split_ifs <;> rfl - -theorem actual_radius_prefix_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) (T : ℝ≥0) : - let mT := (borel ℂ).prod (Filtration.natural B (fun t => (hm t).stronglyMeasurable) T) - @Measurable (ℂ × Ω) ℝ≥0∞ mT (borel ℝ≥0∞) - (fun p => min (radiusLevelTime (γ p.2) (G p.2) p.1 u) (T : ℝ≥0∞)) ∧ - @MeasurableSet (ℂ × Ω) mT - {p | radiusLevelTime (γ p.2) (G p.2) p.1 u ≤ (T : ℝ≥0∞)} := by - classical - obtain ⟨_,hR,hL⟩ := actual_spatial_past_kernels hm hc κ γ G F hw T - let : MeasurableSpace Ω := Filtration.natural B (fun t => (hm t).stronglyMeasurable) T - have htarget : Measurable (fun p : ℂ × Ω => (2*p.1.im)*Real.exp (-u)) := - (Complex.measurable_im.comp measurable_fst).const_mul 2 |>.mul_const _ - apply prefix_level_time_measurable - (fun p : ℂ × Ω => radiusLevelTime (γ p.2) (G p.2) p.1 u) T - (fun t p => fullRadius (G p.2) t p.1) - (fun t p => p.1 ∈ survivingDomain (γ p.2) t) - (fun p => (2*p.1.im)*Real.exp (-u)) hR hL htarget - · intro p - exact capped_radiusLevelTime_dense ((hc p.2).const_mul _) (hw p.2) p.1 u hu T - · intro p - simpa only [ENNReal.coe_lt_coe] using - radiusLevelTime_lt_iff_dense ((hc p.2).const_mul _) (hw p.2) p.1 u hu (T : ℝ≥0∞) - · intro p - exact radiusLevelTime_le_iff_prefix ((hc p.2).const_mul _) (hw p.2) p.1 u hu T - -namespace Multipoint - -private lemma measurable_capped_green - {Y : Type*} [MeasurableSpace Y] (τ : Y → ℝ≥0∞) (T : ℝ≥0) - (g : Iic T × Y → ℝ) - (hτ : Measurable (fun p => min (τ p) (T : ℝ≥0∞))) - (hA : MeasurableSet {p | τ p ≤ (T : ℝ≥0∞)}) (hg : Measurable g) : - Measurable (fun p => if τ p ≤ (T : ℝ≥0∞) then ENNReal.ofReal - (g (⟨(min (τ p) (T : ℝ≥0∞)).toNNReal, by - exact_mod_cast ENNReal.toNNReal_mono - (by simp : (T : ℝ≥0∞) ≠ ∞) (min_le_right (τ p) (T : ℝ≥0∞))⟩,p)) else 0) := by - classical - have hb (p : Y) : (min (τ p) (T : ℝ≥0∞)).toNNReal ∈ Iic T := by - exact_mod_cast ENNReal.toNNReal_mono - (by simp : (T : ℝ≥0∞) ≠ ∞) (min_le_right (τ p) (T : ℝ≥0∞)) - have hm := (ENNReal.measurable_toNNReal.comp hτ).subtype_mk (h := hb) - exact Measurable.ite hA (hg.comp (hm.prodMk measurable_id)).ennreal_ofReal measurable_const - -lemma levelApproximationDensity_eq_cappedGreen - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (T : ℝ≥0) (ω : Ω) (z : ℂ) : - levelApproximationDensity κ B γ G u T ω z = - if radiusLevelTime (γ ω) (G ω) z u ≤ (T : ℝ≥0∞) then - ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) - (min (radiusLevelTime (γ ω) (G ω) z u) (T : ℝ≥0∞)).toNNReal z) else 0 := by - unfold levelApproximationDensity - by_cases h : radiusLevelTime (γ ω) (G ω) z u ≤ (T : ℝ≥0∞) - · have hf : radiusLevelTime (γ ω) (G ω) z u < ∞ := h.trans_lt (by simp) - simp only [ite_eq_left h,min_eq_left h,stoppedGreen,ite_eq_left hf] - · simp only [ite_eq_right h] - -theorem levelApproximationDensity_past_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) (T : ℝ≥0) : - @Measurable (Ω × ℂ) ℝ≥0∞ - ((Filtration.natural B (fun t => (hm t).stronglyMeasurable) T).prod (borel ℂ)) - (borel ℝ≥0∞) (fun p => levelApproximationDensity κ B γ G u T p.1 p.2) := by - classical - obtain ⟨hρ,hA⟩ := actual_radius_prefix_measurable hm hc κ u hu γ G F hw T - obtain ⟨hG,_,_⟩ := actual_spatial_past_kernels hm hc κ γ G F hw T - let : MeasurableSpace Ω := Filtration.natural B (fun t => (hm t).stronglyMeasurable) T - have hd := measurable_capped_green - (fun p : ℂ × Ω => radiusLevelTime (γ p.2) (G p.2) p.1 u) T - (fun p : Iic T × (ℂ × Ω) => greenDensity κ - (fun s => Real.sqrt κ * B s p.2.2) (γ p.2.2) (G p.2.2) p.1 p.2.1) hρ hA hG - simpa only [levelApproximationDensity_eq_cappedGreen,Function.comp_def, - Prod.fst_swap,Prod.snd_swap] using - hd.comp measurable_swap - -lemma levelApproximationRealTest_past_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) (T : ℝ≥0) (f : ℂ → ℝ) (hf : Measurable f) : - Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) T] - (levelApproximationRealTest κ B γ G u T f) := by - have hd := levelApproximationDensity_past_measurable hm hc κ u hu γ G F hw T - let : MeasurableSpace Ω := Filtration.natural B (fun t => (hm t).stronglyMeasurable) T - have he (ω : Ω) : levelApproximationRealTest κ B γ G u T f ω = - ∫ z, (levelApproximationDensity κ B γ G u T ω z).toReal * f z := by - apply integral_withDensity_eq_integral_toReal_smul hd.of_uncurry_left - filter_upwards [] with z - unfold levelApproximationDensity - split_ifs <;> simp - simp_rw [show levelApproximationRealTest κ B γ G u T f = - (fun ω => ∫ z, (levelApproximationDensity κ B γ G u T ω z).toReal * f z) - from funext he] - exact (hd.ennreal_toReal.mul (hf.comp measurable_snd)).stronglyMeasurable.integral_prod_right'.measurable - -lemma convexLevelMeasure_test_past_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (n : ℕ) (T : ℝ≥0) - (f : C_c(upperHalfPlane,ℝ)) : - Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) T] - (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n T ω) := by - have he (ω : Ω) := integral_convexLevelMeasure hκ hκ8 γ G F ω (hc ω) (hw ω) - u s w n T (hw₀ n) f - simp_rw [he] - apply Finset.measurable_sum - intro i hi - exact measurable_const.mul (levelApproximationRealTest_past_measurable hm hc κ - (u (n+i)) (hu (n+i)) γ G F hw T (extendHalfPlaneTest f) - (extendHalfPlaneTest_measurable f f.continuous.measurable)) - -/-- Every actual vague limit at a fixed time has a version measurable in -that time's uncompleted natural filtration. The convex sequence is unchanged. -/ -theorem level_radon_limit_has_past_measurable_version - {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) - (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (T : ℝ≥0) - (ν : Ω → Measure upperHalfPlane) - (hν : ∀ᵐ ω ∂P, (ν ω).Regular) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n T ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) : - ∃ η : Ω → Measure upperHalfPlane, - Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) T] η ∧ - (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ η =ᵐ[P] ν := by - let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - exact exists_subsigma_radon_version - (Filtration.natural B (fun t => (hm t).stronglyMeasurable) T) P _ ν - (fun n ω => convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hc ω) (hw ω) - u s w n T) - (fun n f => convexLevelMeasure_test_past_measurable hm hc hκ hκ8 γ G F hw - u hu s w hw₀ n T f) hν hlim - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.SpatialPastKernels +import OAI.Probability.SLE.Mass.LevelRandomRadon +import OAI.Probability.SLE.Mass.SubSigmaRadonLimit + +namespace OAI + +/-! The finite-level mass accumulated by a deterministic time is measurable +in the natural driver filtration. All spatial variables are retained in the +past-measurable kernel before integration. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory TopologicalSpace Classical +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge + +lemma radiusLevelTime_le_iff_prefix + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F) + (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) : + radiusLevelTime γ G z u ≤ (T : ℝ≥0∞) ↔ + (z ∈ survivingDomain γ T ∧ fullRadius G T z ≤ (2*z.im)*Real.exp (-u)) ∨ + radiusLevelTime γ G z u < (T : ℝ≥0∞) := by + constructor + · intro h + rcases h.eq_or_lt with he | ht + · left + obtain ⟨s,hs,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top γ G z u + (h.trans_lt (by simp)) + have hτ := radiusLevelTime_eq_of_live_level hU hw z u s hs hR + have hsT : s = T := by exact_mod_cast hτ.symm.trans he + subst s + exact ⟨hs,hR.le⟩ + · exact Or.inr ht + · rintro (⟨hT,hR⟩ | ht) + · rcases hR.eq_or_lt with he | ht + · rw [radiusLevelTime_eq_of_live_level hU hw z u T hT he] + · exact (radiusLevelTime_lt_of_live_radius_lt hU hw z u hu T hT ht).le + · exact ht.le + +lemma capped_radiusLevelTime_dense + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hU : Continuous U) (hw : IsCapacityTwoWitness U γ G F) + (z : ℂ) (u : ℝ) (hu : 0 ≤ u) (T : ℝ≥0) : + min (radiusLevelTime γ G z u) (T : ℝ≥0∞) = + ⨅ n : ℕ, if denseSeq ℝ≥0 n < T ∧ + z ∈ survivingDomain γ (denseSeq ℝ≥0 n) ∧ + fullRadius G (denseSeq ℝ≥0 n) z < (2*z.im)*Real.exp (-u) + then (denseSeq ℝ≥0 n : ℝ≥0∞) else T := by + apply eq_of_forall_gt_iff + intro x + by_cases hTx : (T : ℝ≥0∞) < x + · have hl : min (radiusLevelTime γ G z u) (T : ℝ≥0∞) < x := + (min_le_right _ _).trans_lt hTx + have hr : (⨅ n : ℕ, if denseSeq ℝ≥0 n < T ∧ + z ∈ survivingDomain γ (denseSeq ℝ≥0 n) ∧ + fullRadius G (denseSeq ℝ≥0 n) z < (2*z.im)*Real.exp (-u) + then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) < x := by + apply (iInf_le_of_le 0 ?_).trans_lt hTx + split_ifs with h + · exact_mod_cast h.1.le + · exact le_rfl + exact iff_of_true hl hr + · rw [min_lt_iff, or_iff_left hTx, iInf_lt_iff, + radiusLevelTime_lt_iff_dense hU hw z u hu x] + constructor + · rintro ⟨n,hnx,hn,hR⟩ + have hnT : denseSeq ℝ≥0 n < T := by + exact_mod_cast hnx.trans_le (le_of_not_gt hTx) + refine ⟨n,?_⟩ + have hh : denseSeq ℝ≥0 n < T ∧ z ∈ survivingDomain γ (denseSeq ℝ≥0 n) ∧ + fullRadius G (denseSeq ℝ≥0 n) z < (2*z.im)*Real.exp (-u) := ⟨hnT,hn,hR⟩ + rwa [ite_eq_left hh] + · rintro ⟨n,hn⟩ + split_ifs at hn with h + · exact ⟨n,hn,h.2.1,h.2.2⟩ + · exact False.elim (hTx hn) + +private lemma prefix_time_measurable + {Y : Type*} [MeasurableSpace Y] (τ : Y → ℝ≥0∞) (T : ℝ≥0) + (D : ℕ → Y → Prop) (hD : ∀ n, MeasurableSet {p | D n p}) + (E : Y → Prop) (hE : MeasurableSet {p | E p}) + (hcap : ∀ p, min (τ p) (T : ℝ≥0∞) = + ⨅ n : ℕ, if D n p then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) + (hlt : ∀ p, τ p < (T : ℝ≥0∞) ↔ ∃ n, D n p) + (hle : ∀ p, τ p ≤ (T : ℝ≥0∞) ↔ E p ∨ τ p < (T : ℝ≥0∞)) : + Measurable (fun p => min (τ p) (T : ℝ≥0∞)) ∧ + MeasurableSet {p | τ p ≤ (T : ℝ≥0∞)} := by + classical + have hmLess : MeasurableSet {p | τ p < (T : ℝ≥0∞)} := by + have he : {p | τ p < (T : ℝ≥0∞)} = ⋃ n, {p | D n p} := by + ext p + simpa only [mem_iUnion,mem_ofPred_eq] using hlt p + rw [he] + exact MeasurableSet.iUnion hD + constructor + · have hh : Measurable (fun p => ⨅ n : ℕ, + if D n p then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) := + Measurable.iInf fun n => Measurable.ite (hD n) measurable_const measurable_const + convert hh using 1 + exact funext hcap + · convert hE.union hmLess using 1 + ext p + exact hle p + +private lemma prefix_level_time_measurable + {Y : Type*} [MeasurableSpace Y] (τ : Y → ℝ≥0∞) (T : ℝ≥0) + (R : ℝ≥0 → Y → ℝ) (L : ℝ≥0 → Y → Prop) (a : Y → ℝ) + (hR : Measurable (fun p : Iic T × Y => if L p.1 p.2 then R p.1 p.2 else 0)) + (hL : MeasurableSet {p : Iic T × Y | L p.1 p.2}) (ha : Measurable a) + (hcap : ∀ p, min (τ p) (T : ℝ≥0∞) = + ⨅ n : ℕ, if denseSeq ℝ≥0 n < T ∧ L (denseSeq ℝ≥0 n) p ∧ + R (denseSeq ℝ≥0 n) p < a p then (denseSeq ℝ≥0 n : ℝ≥0∞) else T) + (hlt : ∀ p, τ p < (T : ℝ≥0∞) ↔ ∃ n, denseSeq ℝ≥0 n < T ∧ + L (denseSeq ℝ≥0 n) p ∧ R (denseSeq ℝ≥0 n) p < a p) + (hle : ∀ p, τ p ≤ (T : ℝ≥0∞) ↔ + (L T p ∧ R T p ≤ a p) ∨ τ p < (T : ℝ≥0∞)) : + Measurable (fun p => min (τ p) (T : ℝ≥0∞)) ∧ + MeasurableSet {p | τ p ≤ (T : ℝ≥0∞)} := by + classical + have hmR (s : Iic T) : Measurable (fun p => if L s p then R s p else 0) := + hR.comp (measurable_const.prodMk measurable_id) + have hmL (s : Iic T) : MeasurableSet {p | L s p} := + hL.preimage (measurable_const.prodMk measurable_id) + have hmD (n : ℕ) : MeasurableSet {p : Y | + denseSeq ℝ≥0 n < T ∧ L (denseSeq ℝ≥0 n) p ∧ R (denseSeq ℝ≥0 n) p < a p} := by + by_cases hn : denseSeq ℝ≥0 n < T + · let s : Iic T := ⟨denseSeq ℝ≥0 n,hn.le⟩ + convert (hmL s).inter (measurableSet_lt (hmR s) ha) using 1 + ext p + simp only [mem_ofPred_eq,mem_inter_iff,s] + by_cases hl : L (denseSeq ℝ≥0 n) p <;> simp [hl,hn] + · simp only [hn,false_and,ofPred_false,MeasurableSet.empty] + have hEnd : MeasurableSet {p : Y | L T p ∧ R T p ≤ a p} := by + let s : Iic T := ⟨T,by simp⟩ + convert (hmL s).inter (measurableSet_le (hmR s) ha) using 1 + ext p + simp only [mem_inter_iff,mem_ofPred_eq,s] + by_cases hl : L T p <;> simp [hl] + apply prefix_time_measurable τ T _ hmD _ hEnd ?_ hlt hle + intro p + rw [hcap p] + apply iInf_congr + intro n + split_ifs <;> rfl + +theorem actual_radius_prefix_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) (T : ℝ≥0) : + let mT := (borel ℂ).prod (Filtration.natural B (fun t => (hm t).stronglyMeasurable) T) + @Measurable (ℂ × Ω) ℝ≥0∞ mT (borel ℝ≥0∞) + (fun p => min (radiusLevelTime (γ p.2) (G p.2) p.1 u) (T : ℝ≥0∞)) ∧ + @MeasurableSet (ℂ × Ω) mT + {p | radiusLevelTime (γ p.2) (G p.2) p.1 u ≤ (T : ℝ≥0∞)} := by + classical + obtain ⟨_,hR,hL⟩ := actual_spatial_past_kernels hm hc κ γ G F hw T + let : MeasurableSpace Ω := Filtration.natural B (fun t => (hm t).stronglyMeasurable) T + have htarget : Measurable (fun p : ℂ × Ω => (2*p.1.im)*Real.exp (-u)) := + (Complex.measurable_im.comp measurable_fst).const_mul 2 |>.mul_const _ + apply prefix_level_time_measurable + (fun p : ℂ × Ω => radiusLevelTime (γ p.2) (G p.2) p.1 u) T + (fun t p => fullRadius (G p.2) t p.1) + (fun t p => p.1 ∈ survivingDomain (γ p.2) t) + (fun p => (2*p.1.im)*Real.exp (-u)) hR hL htarget + · intro p + exact capped_radiusLevelTime_dense ((hc p.2).const_mul _) (hw p.2) p.1 u hu T + · intro p + simpa only [ENNReal.coe_lt_coe] using + radiusLevelTime_lt_iff_dense ((hc p.2).const_mul _) (hw p.2) p.1 u hu (T : ℝ≥0∞) + · intro p + exact radiusLevelTime_le_iff_prefix ((hc p.2).const_mul _) (hw p.2) p.1 u hu T + +namespace Multipoint + +private lemma measurable_capped_green + {Y : Type*} [MeasurableSpace Y] (τ : Y → ℝ≥0∞) (T : ℝ≥0) + (g : Iic T × Y → ℝ) + (hτ : Measurable (fun p => min (τ p) (T : ℝ≥0∞))) + (hA : MeasurableSet {p | τ p ≤ (T : ℝ≥0∞)}) (hg : Measurable g) : + Measurable (fun p => if τ p ≤ (T : ℝ≥0∞) then ENNReal.ofReal + (g (⟨(min (τ p) (T : ℝ≥0∞)).toNNReal, by + exact_mod_cast ENNReal.toNNReal_mono + (by simp : (T : ℝ≥0∞) ≠ ∞) (min_le_right (τ p) (T : ℝ≥0∞))⟩,p)) else 0) := by + classical + have hb (p : Y) : (min (τ p) (T : ℝ≥0∞)).toNNReal ∈ Iic T := by + exact_mod_cast ENNReal.toNNReal_mono + (by simp : (T : ℝ≥0∞) ≠ ∞) (min_le_right (τ p) (T : ℝ≥0∞)) + have hm := (ENNReal.measurable_toNNReal.comp hτ).subtype_mk (h := hb) + exact Measurable.ite hA (hg.comp (hm.prodMk measurable_id)).ennreal_ofReal measurable_const + +lemma levelApproximationDensity_eq_cappedGreen + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (T : ℝ≥0) (ω : Ω) (z : ℂ) : + levelApproximationDensity κ B γ G u T ω z = + if radiusLevelTime (γ ω) (G ω) z u ≤ (T : ℝ≥0∞) then + ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) + (min (radiusLevelTime (γ ω) (G ω) z u) (T : ℝ≥0∞)).toNNReal z) else 0 := by + unfold levelApproximationDensity + by_cases h : radiusLevelTime (γ ω) (G ω) z u ≤ (T : ℝ≥0∞) + · have hf : radiusLevelTime (γ ω) (G ω) z u < ∞ := h.trans_lt (by simp) + simp only [ite_eq_left h,min_eq_left h,stoppedGreen,ite_eq_left hf] + · simp only [ite_eq_right h] + +theorem levelApproximationDensity_past_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) (T : ℝ≥0) : + @Measurable (Ω × ℂ) ℝ≥0∞ + ((Filtration.natural B (fun t => (hm t).stronglyMeasurable) T).prod (borel ℂ)) + (borel ℝ≥0∞) (fun p => levelApproximationDensity κ B γ G u T p.1 p.2) := by + classical + obtain ⟨hρ,hA⟩ := actual_radius_prefix_measurable hm hc κ u hu γ G F hw T + obtain ⟨hG,_,_⟩ := actual_spatial_past_kernels hm hc κ γ G F hw T + let : MeasurableSpace Ω := Filtration.natural B (fun t => (hm t).stronglyMeasurable) T + have hd := measurable_capped_green + (fun p : ℂ × Ω => radiusLevelTime (γ p.2) (G p.2) p.1 u) T + (fun p : Iic T × (ℂ × Ω) => greenDensity κ + (fun s => Real.sqrt κ * B s p.2.2) (γ p.2.2) (G p.2.2) p.1 p.2.1) hρ hA hG + simpa only [levelApproximationDensity_eq_cappedGreen,Function.comp_def, + Prod.fst_swap,Prod.snd_swap] using + hd.comp measurable_swap + +lemma levelApproximationRealTest_past_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) (T : ℝ≥0) (f : ℂ → ℝ) (hf : Measurable f) : + Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) T] + (levelApproximationRealTest κ B γ G u T f) := by + have hd := levelApproximationDensity_past_measurable hm hc κ u hu γ G F hw T + let : MeasurableSpace Ω := Filtration.natural B (fun t => (hm t).stronglyMeasurable) T + have he (ω : Ω) : levelApproximationRealTest κ B γ G u T f ω = + ∫ z, (levelApproximationDensity κ B γ G u T ω z).toReal * f z := by + apply integral_withDensity_eq_integral_toReal_smul hd.of_uncurry_left + filter_upwards [] with z + unfold levelApproximationDensity + split_ifs <;> simp + simp_rw [show levelApproximationRealTest κ B γ G u T f = + (fun ω => ∫ z, (levelApproximationDensity κ B γ G u T ω z).toReal * f z) + from funext he] + exact (hd.ennreal_toReal.mul (hf.comp measurable_snd)).stronglyMeasurable.integral_prod_right'.measurable + +lemma convexLevelMeasure_test_past_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (n : ℕ) (T : ℝ≥0) + (f : C_c(upperHalfPlane,ℝ)) : + Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) T] + (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n T ω) := by + have he (ω : Ω) := integral_convexLevelMeasure hκ hκ8 γ G F ω (hc ω) (hw ω) + u s w n T (hw₀ n) f + simp_rw [he] + apply Finset.measurable_sum + intro i hi + exact measurable_const.mul (levelApproximationRealTest_past_measurable hm hc κ + (u (n+i)) (hu (n+i)) γ G F hw T (extendHalfPlaneTest f) + (extendHalfPlaneTest_measurable f f.continuous.measurable)) + +/-- Every actual vague limit at a fixed time has a version measurable in +that time's uncompleted natural filtration. The convex sequence is unchanged. -/ +theorem level_radon_limit_has_past_measurable_version + {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) + (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (T : ℝ≥0) + (ν : Ω → Measure upperHalfPlane) + (hν : ∀ᵐ ω ∂P, (ν ω).Regular) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n T ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) : + ∃ η : Ω → Measure upperHalfPlane, + Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) T] η ∧ + (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ η =ᵐ[P] ν := by + let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + exact exists_subsigma_radon_version + (Filtration.natural B (fun t => (hm t).stronglyMeasurable) T) P _ ν + (fun n ω => convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hc ω) (hw ω) + u s w n T) + (fun n f => convexLevelMeasure_test_past_measurable hm hc hκ hκ8 γ G F hw + u hu s w hw₀ n T f) hν hlim + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelApproximationKernel.lean b/lean/OAI/Probability/SLE/Mass/LevelApproximationKernel.lean index ae889c711298f87970c9377e88da97af8f412409..fd6bad3955af2d7d08f83b9a0bcfb0b9b6b2aa31 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelApproximationKernel.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelApproximationKernel.lean @@ -1,241 +1,241 @@ -import OAI.Probability.SLE.Mass.LevelApproximants -import OAI.Probability.SLE.Loewner.ActualRadiusStops - -namespace OAI - -/-! The conditional Green kernel of the part of a level measure still to -come. The level hit is represented by an actual Brownian stopping time; -no measurability of a geometrically defined filtration is assumed. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -def levelRemainderGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ := - if (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u then - stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u else 0 - -def levelCurrentGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ := - if (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u then - greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z else 0 - -lemma levelRemainderGreen_nonneg - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {γ : Ω → ℝ≥0 → ℂ} - {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (t : ℝ≥0) (z : ℂ) : 0 ≤ levelRemainderGreen κ B γ G u t z ω := by - unfold levelRemainderGreen - split_ifs - · exact stoppedGreen_nonneg hw κ z u - · exact le_rfl - -lemma levelCurrentGreen_nonneg - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {γ : Ω → ℝ≥0 → ℂ} - {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (t : ℝ≥0) (z : ℂ) : 0 ≤ levelCurrentGreen κ B γ G u t z ω := by - unfold levelCurrentGreen - split_ifs - · exact greenDensity_nonneg hw κ t z - · exact le_rfl - -lemma actual_greenDensity_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : Measurable (fun q : Ω × ℂ => - greenDensity κ (fun s => Real.sqrt κ * B s q.1) (γ q.1) (G q.1) t q.2) := by - let U (s : ℝ≥0) (ω : Ω) := Real.sqrt κ * B s ω - let c := spatialClippedFamily U (hBm.const_mul _) (fun ω => (hBc ω).const_mul _) - have hj := c.jointGreen_measurable κ B hBm - have hp : Measurable (fun q : Ω × ℂ => (t, q.2, q.1)) := - measurable_const.prodMk (measurable_snd.prodMk measurable_fst) - have hh := hj.comp hp - have he (q : Ω × ℂ) : c.jointGreen κ B (t,q.2,q.1) = - greenDensity κ (fun s => Real.sqrt κ * B s q.1) (γ q.1) (G q.1) t q.2 := - c.jointGreen_eq κ B q.1 ((hBc q.1).const_mul _) (fun _ => rfl) (hw q.1) t q.2 - simpa only [Function.comp_def, he] using hh - -lemma levelRemainderGreen_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : Measurable (fun q : Ω × ℂ => levelRemainderGreen κ B γ G u t q.2 q.1) := by - exact Measurable.ite - (measurableSet_lt measurable_const (actual_radiusLevel_joint_measurable hm hBc κ u hu γ G F hw)) - (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw) measurable_const - -lemma levelCurrentGreen_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : Measurable (fun q : Ω × ℂ => levelCurrentGreen κ B γ G u t q.2 q.1) := by - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - exact Measurable.ite - (measurableSet_lt measurable_const (actual_radiusLevel_joint_measurable hm hBc κ u hu γ G F hw)) - (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t) measurable_const - -theorem levelRemainderGreen_condExp - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) : - let hm (s : ℝ≥0) : Measurable (B s) := - hBm.comp (measurable_const.prodMk measurable_id) - let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable) - Integrable (levelRemainderGreen κ B γ G u t z) P ∧ - Integrable (levelCurrentGreen κ B γ G u t z) P ∧ - P[levelRemainderGreen κ B γ G u t z | ℱ t] =ᵐ[P] - levelCurrentGreen κ B γ G u t z := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let hm (s : ℝ≥0) : Measurable (B s) := - hBm.comp (measurable_const.prodMk measurable_id) - let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable) - obtain ⟨M, Z, _, hZ, _, _, hClosed, _, heq, _⟩ := - exists_physical_green_stopping_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu - obtain ⟨τ, hτ, _, hτeq⟩ := exists_monotone_radiusLevel_stops hB hm hBm hBc hB0 - κ hκ hκ8 γ G F hw z hz (fun _ => u) (fun _ => hu) (fun _ _ _ => le_rfl) - let A : Set Ω := {ω | ¬ τ 0 ω ≤ (t : WithTop ℝ≥0)} - have hA : MeasurableSet[ℱ t] A := (hτ 0 t).compl - have hAg : MeasurableSet A := ℱ.le t _ hA - have hR : levelRemainderGreen κ B γ G u t z =ᵐ[P] A.indicator Z := by - filter_upwards [heq, hτeq] with ω he hte - have htest : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ω ∈ A := by - change (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ¬ τ 0 ω ≤ (t : ℝ≥0∞) - rw [hte 0] - exact lt_iff_not_ge - by_cases ha : ω ∈ A - · rw [levelRemainderGreen, ite_eq_left (htest.mpr ha), indicator_of_mem ha, he.2] - · rw [levelRemainderGreen, ite_eq_right (mt htest.mp ha), indicator_of_notMem ha] - have hC : levelCurrentGreen κ B γ G u t z =ᵐ[P] A.indicator (M t) := by - filter_upwards [heq, hτeq] with ω he hte - have htest : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ω ∈ A := by - change (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ¬ τ 0 ω ≤ (t : ℝ≥0∞) - rw [hte 0] - exact lt_iff_not_ge - by_cases ha : ω ∈ A - · rw [levelCurrentGreen, ite_eq_left (htest.mpr ha), indicator_of_mem ha, - he.1 t, physicalStoppedGreen, ite_eq_right (not_le.mpr (htest.mpr ha))] - · rw [levelCurrentGreen, ite_eq_right (mt htest.mp ha), indicator_of_notMem ha] - refine ⟨(hZ.indicator hAg).congr hR.symm, - ((integrable_condExp.congr (hClosed t).symm).indicator hAg).congr hC.symm, ?_⟩ - apply (condExp_congr_ae hR).trans - apply (condExp_indicator hZ hA).trans - apply EventuallyEq.trans _ hC.symm - filter_upwards [hClosed t] with ω he - by_cases ha : ω ∈ A - · simpa only [indicator_of_mem ha,ℱ,hm] using he.symm - · simp only [indicator_of_notMem ha] - -theorem levelRemainderGreen_setIntegral - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) - (A : Set Ω) - (hA : MeasurableSet[Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] A) : - (∫ ω in A, levelRemainderGreen κ B γ G u t z ω ∂P) = - ∫ ω in A, levelCurrentGreen κ B γ G u t z ω ∂P := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let ℱ := Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) - obtain ⟨hR, _, he⟩ := levelRemainderGreen_condExp hB hBm hBc hB0 hκ hκ8 - γ G F hw z hz u hu t - have : IsFiniteMeasure (P.trim (ℱ.le t)) := isFiniteMeasure_trim (ℱ.le t) - exact (setIntegral_condExp (ℱ.le t) hR hA).symm.trans - (integral_congr_ae (ae_restrict_of_ae he)) - -theorem levelRemainderGreen_setLIntegral - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) - (A : Set Ω) - (hA : MeasurableSet[Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] A) : - (∫⁻ ω in A, ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) ∂P) = - ∫⁻ ω in A, ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂P := by - obtain ⟨hR,hC,_⟩ := levelRemainderGreen_condExp hB hBm hBc hB0 hκ hκ8 γ G F hw z hz u hu t - have hRn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelRemainderGreen κ B γ G u t z ω := - ae_restrict_of_ae (hw.mono (fun ω hω => levelRemainderGreen_nonneg ω hω u t z)) - have hCn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelCurrentGreen κ B γ G u t z ω := - ae_restrict_of_ae (hw.mono (fun ω hω => levelCurrentGreen_nonneg ω hω u t z)) - rw [← ofReal_integral_eq_lintegral_ofReal hR.integrableOn hRn, - ← ofReal_integral_eq_lintegral_ofReal hC.integrableOn hCn] - rw [levelRemainderGreen_setIntegral hB hBm hBc hB0 hκ hκ8 γ G F hw z hz u hu t A hA] - -/-- The finite-level kernel identity against every nonnegative spatial -test, on every event known at time `t`. -/ -theorem levelRemainderGreen_spatial_kernel - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (K : Set ℂ) - (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (f : ℂ → ℝ≥0∞) (hf : Measurable f) (A : Set Ω) - (hA : MeasurableSet[Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] A) : - (∫⁻ ω in A, (∫⁻ z in K, f z * ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) ∂volume) ∂P) = - ∫⁻ ω in A, (∫⁻ z in K, f z * ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂volume) ∂P := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - have hmR := levelRemainderGreen_joint_measurable hm hBc κ u hu γ G F hw t - have hmC := levelCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw t - have hR := (hf.comp measurable_snd).mul hmR.ennreal_ofReal - have hC := (hf.comp measurable_snd).mul hmC.ennreal_ofReal - conv_lhs => rw [lintegral_lintegral_swap hR.aemeasurable] - conv_rhs => rw [lintegral_lintegral_swap hC.aemeasurable] - apply setLIntegral_congr_fun hK - intro z hz - dsimp only - have hpz : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const - have hmRz : Measurable (fun ω => ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω)) := - by - have hh := hmR.ennreal_ofReal.comp hpz - simpa only [Function.comp_def] using hh - have hmCz : Measurable (fun ω => ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω)) := - by - have hh := hmC.ennreal_ofReal.comp hpz - simpa only [Function.comp_def] using hh - rw [lintegral_const_mul _ hmRz, lintegral_const_mul _ hmCz] - rw [levelRemainderGreen_setLIntegral hB hBm hBc hB0 hκ hκ8 γ G F (ae_of_all _ hw) - z (hKH hz) u hu t A hA] - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximants +import OAI.Probability.SLE.Loewner.ActualRadiusStops + +namespace OAI + +/-! The conditional Green kernel of the part of a level measure still to +come. The level hit is represented by an actual Brownian stopping time; +no measurability of a geometrically defined filtration is assumed. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +def levelRemainderGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ := + if (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u then + stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u else 0 + +def levelCurrentGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0) (z : ℂ) (ω : Ω) : ℝ := + if (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u then + greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z else 0 + +lemma levelRemainderGreen_nonneg + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {γ : Ω → ℝ≥0 → ℂ} + {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (t : ℝ≥0) (z : ℂ) : 0 ≤ levelRemainderGreen κ B γ G u t z ω := by + unfold levelRemainderGreen + split_ifs + · exact stoppedGreen_nonneg hw κ z u + · exact le_rfl + +lemma levelCurrentGreen_nonneg + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {γ : Ω → ℝ≥0 → ℂ} + {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (t : ℝ≥0) (z : ℂ) : 0 ≤ levelCurrentGreen κ B γ G u t z ω := by + unfold levelCurrentGreen + split_ifs + · exact greenDensity_nonneg hw κ t z + · exact le_rfl + +lemma actual_greenDensity_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : Measurable (fun q : Ω × ℂ => + greenDensity κ (fun s => Real.sqrt κ * B s q.1) (γ q.1) (G q.1) t q.2) := by + let U (s : ℝ≥0) (ω : Ω) := Real.sqrt κ * B s ω + let c := spatialClippedFamily U (hBm.const_mul _) (fun ω => (hBc ω).const_mul _) + have hj := c.jointGreen_measurable κ B hBm + have hp : Measurable (fun q : Ω × ℂ => (t, q.2, q.1)) := + measurable_const.prodMk (measurable_snd.prodMk measurable_fst) + have hh := hj.comp hp + have he (q : Ω × ℂ) : c.jointGreen κ B (t,q.2,q.1) = + greenDensity κ (fun s => Real.sqrt κ * B s q.1) (γ q.1) (G q.1) t q.2 := + c.jointGreen_eq κ B q.1 ((hBc q.1).const_mul _) (fun _ => rfl) (hw q.1) t q.2 + simpa only [Function.comp_def, he] using hh + +lemma levelRemainderGreen_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : Measurable (fun q : Ω × ℂ => levelRemainderGreen κ B γ G u t q.2 q.1) := by + exact Measurable.ite + (measurableSet_lt measurable_const (actual_radiusLevel_joint_measurable hm hBc κ u hu γ G F hw)) + (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw) measurable_const + +lemma levelCurrentGreen_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : Measurable (fun q : Ω × ℂ => levelCurrentGreen κ B γ G u t q.2 q.1) := by + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + exact Measurable.ite + (measurableSet_lt measurable_const (actual_radiusLevel_joint_measurable hm hBc κ u hu γ G F hw)) + (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t) measurable_const + +theorem levelRemainderGreen_condExp + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) : + let hm (s : ℝ≥0) : Measurable (B s) := + hBm.comp (measurable_const.prodMk measurable_id) + let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable) + Integrable (levelRemainderGreen κ B γ G u t z) P ∧ + Integrable (levelCurrentGreen κ B γ G u t z) P ∧ + P[levelRemainderGreen κ B γ G u t z | ℱ t] =ᵐ[P] + levelCurrentGreen κ B γ G u t z := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let hm (s : ℝ≥0) : Measurable (B s) := + hBm.comp (measurable_const.prodMk measurable_id) + let ℱ := Filtration.natural B (fun s => (hm s).stronglyMeasurable) + obtain ⟨M, Z, _, hZ, _, _, hClosed, _, heq, _⟩ := + exists_physical_green_stopping_regular hB hBm hBc hB0 κ hκ hκ8 γ G F hw z hz u hu + obtain ⟨τ, hτ, _, hτeq⟩ := exists_monotone_radiusLevel_stops hB hm hBm hBc hB0 + κ hκ hκ8 γ G F hw z hz (fun _ => u) (fun _ => hu) (fun _ _ _ => le_rfl) + let A : Set Ω := {ω | ¬ τ 0 ω ≤ (t : WithTop ℝ≥0)} + have hA : MeasurableSet[ℱ t] A := (hτ 0 t).compl + have hAg : MeasurableSet A := ℱ.le t _ hA + have hR : levelRemainderGreen κ B γ G u t z =ᵐ[P] A.indicator Z := by + filter_upwards [heq, hτeq] with ω he hte + have htest : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ω ∈ A := by + change (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ¬ τ 0 ω ≤ (t : ℝ≥0∞) + rw [hte 0] + exact lt_iff_not_ge + by_cases ha : ω ∈ A + · rw [levelRemainderGreen, ite_eq_left (htest.mpr ha), indicator_of_mem ha, he.2] + · rw [levelRemainderGreen, ite_eq_right (mt htest.mp ha), indicator_of_notMem ha] + have hC : levelCurrentGreen κ B γ G u t z =ᵐ[P] A.indicator (M t) := by + filter_upwards [heq, hτeq] with ω he hte + have htest : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ω ∈ A := by + change (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ ¬ τ 0 ω ≤ (t : ℝ≥0∞) + rw [hte 0] + exact lt_iff_not_ge + by_cases ha : ω ∈ A + · rw [levelCurrentGreen, ite_eq_left (htest.mpr ha), indicator_of_mem ha, + he.1 t, physicalStoppedGreen, ite_eq_right (not_le.mpr (htest.mpr ha))] + · rw [levelCurrentGreen, ite_eq_right (mt htest.mp ha), indicator_of_notMem ha] + refine ⟨(hZ.indicator hAg).congr hR.symm, + ((integrable_condExp.congr (hClosed t).symm).indicator hAg).congr hC.symm, ?_⟩ + apply (condExp_congr_ae hR).trans + apply (condExp_indicator hZ hA).trans + apply EventuallyEq.trans _ hC.symm + filter_upwards [hClosed t] with ω he + by_cases ha : ω ∈ A + · simpa only [indicator_of_mem ha,ℱ,hm] using he.symm + · simp only [indicator_of_notMem ha] + +theorem levelRemainderGreen_setIntegral + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) + (A : Set Ω) + (hA : MeasurableSet[Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] A) : + (∫ ω in A, levelRemainderGreen κ B γ G u t z ω ∂P) = + ∫ ω in A, levelCurrentGreen κ B γ G u t z ω ∂P := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let ℱ := Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) + obtain ⟨hR, _, he⟩ := levelRemainderGreen_condExp hB hBm hBc hB0 hκ hκ8 + γ G F hw z hz u hu t + have : IsFiniteMeasure (P.trim (ℱ.le t)) := isFiniteMeasure_trim (ℱ.le t) + exact (setIntegral_condExp (ℱ.le t) hR hA).symm.trans + (integral_congr_ae (ae_restrict_of_ae he)) + +theorem levelRemainderGreen_setLIntegral + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ᵐ ω ∂P, IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) + (A : Set Ω) + (hA : MeasurableSet[Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] A) : + (∫⁻ ω in A, ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) ∂P) = + ∫⁻ ω in A, ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂P := by + obtain ⟨hR,hC,_⟩ := levelRemainderGreen_condExp hB hBm hBc hB0 hκ hκ8 γ G F hw z hz u hu t + have hRn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelRemainderGreen κ B γ G u t z ω := + ae_restrict_of_ae (hw.mono (fun ω hω => levelRemainderGreen_nonneg ω hω u t z)) + have hCn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelCurrentGreen κ B γ G u t z ω := + ae_restrict_of_ae (hw.mono (fun ω hω => levelCurrentGreen_nonneg ω hω u t z)) + rw [← ofReal_integral_eq_lintegral_ofReal hR.integrableOn hRn, + ← ofReal_integral_eq_lintegral_ofReal hC.integrableOn hCn] + rw [levelRemainderGreen_setIntegral hB hBm hBc hB0 hκ hκ8 γ G F hw z hz u hu t A hA] + +/-- The finite-level kernel identity against every nonnegative spatial +test, on every event known at time `t`. -/ +theorem levelRemainderGreen_spatial_kernel + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (K : Set ℂ) + (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (f : ℂ → ℝ≥0∞) (hf : Measurable f) (A : Set Ω) + (hA : MeasurableSet[Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] A) : + (∫⁻ ω in A, (∫⁻ z in K, f z * ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) ∂volume) ∂P) = + ∫⁻ ω in A, (∫⁻ z in K, f z * ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂volume) ∂P := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + have hmR := levelRemainderGreen_joint_measurable hm hBc κ u hu γ G F hw t + have hmC := levelCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw t + have hR := (hf.comp measurable_snd).mul hmR.ennreal_ofReal + have hC := (hf.comp measurable_snd).mul hmC.ennreal_ofReal + conv_lhs => rw [lintegral_lintegral_swap hR.aemeasurable] + conv_rhs => rw [lintegral_lintegral_swap hC.aemeasurable] + apply setLIntegral_congr_fun hK + intro z hz + dsimp only + have hpz : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const + have hmRz : Measurable (fun ω => ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω)) := + by + have hh := hmR.ennreal_ofReal.comp hpz + simpa only [Function.comp_def] using hh + have hmCz : Measurable (fun ω => ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω)) := + by + have hh := hmC.ennreal_ofReal.comp hpz + simpa only [Function.comp_def] using hh + rw [lintegral_const_mul _ hmRz, lintegral_const_mul _ hmCz] + rw [levelRemainderGreen_setLIntegral hB hBm hBc hB0 hκ hκ8 γ G F (ae_of_all _ hw) + z (hKH hz) u hu t A hA] + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelApproximationL2.lean b/lean/OAI/Probability/SLE/Mass/LevelApproximationL2.lean index 4bdf549c8ac2ea4c7c8f1251892a1ce235f43d4c..8602ab67683a5dfa4c54e30d1ff110690f5fe556 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelApproximationL2.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelApproximationL2.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Mass.LevelApproximationTests -import OAI.Probability.SLE.Mass.L2ConvexTails - -namespace OAI - -/-! Uniform Hilbert bounds and one common deterministic convex extraction -for the actual finite-level measure tests. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -lemma eLpNorm_toReal_two_le - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (A : Ω → ℝ≥0∞) (hA : Measurable A) (M : ℝ≥0∞) - (hM : (∫⁻ ω, A ω ^ 2 ∂P) ≤ M) : - eLpNorm (fun ω => (A ω).toReal) 2 P ≤ M ^ (1/2 : ℝ) := by - have hn (ω : Ω) : ‖(A ω).toReal‖ₑ ≤ A ω := by - rw [← ofReal_norm,Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg] - exact ENNReal.ofReal_toReal_le - have hi : (∫⁻ ω, ‖(A ω).toReal‖ₑ ^ 2 ∂P) ≤ M := - (lintegral_mono (fun ω => pow_le_pow_left' (hn ω) 2)).trans hM - rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num : (2 : ℝ≥0∞) ≠ 0) - ENNReal.ofNat_ne_top hA.ennreal_toReal.aestronglyMeasurable] - simpa only [ENNReal.toReal_ofNat,ENNReal.rpow_two] using - ENNReal.rpow_le_rpow hi (by norm_num : (0 : ℝ) ≤ 1/2) - -lemma memLp_toReal_two - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (A : Ω → ℝ≥0∞) (hA : Measurable A) (M : ℝ≥0∞) (hM : M < ⊤) - (hAM : (∫⁻ ω, A ω ^ 2 ∂P) ≤ M) : - MemLp (fun ω => (A ω).toReal) 2 P := by - apply lt_of_le_of_lt (eLpNorm_toReal_two_le P A hA M hAM) - exact ENNReal.rpow_lt_top_of_nonneg (by norm_num) hM.ne - -lemma norm_toLp_toReal_two_le - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (A : Ω → ℝ≥0∞) (hA : Measurable A) (M : ℝ≥0∞) (hM : M < ⊤) - (hAM : (∫⁻ ω, A ω ^ 2 ∂P) ≤ M) - (hLp : MemLp (fun ω => (A ω).toReal) 2 P) : - ‖hLp.toLp (fun ω => (A ω).toReal)‖ ≤ (M ^ (1/2 : ℝ)).toReal := by - rw [Lp.norm_toLp] - exact ENNReal.toReal_mono (ENNReal.rpow_ne_top_of_nonneg (by norm_num) hM.ne) - (eLpNorm_toReal_two_le P A hA M hAM) - -theorem exists_common_levelApproximation_test_limits - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (K : ℕ → Set ℂ) (hK : ∀ j, IsCompact (K j)) (hKH : ∀ j, K j ⊆ upperHalfPlane) - (f : ℕ → ℂ → ℝ≥0∞) (hfm : ∀ j, Measurable (f j)) - (C : ℕ → ℝ≥0∞) (hC : ∀ j, C j < ⊤) - (hf : ∀ j z, z ∈ K j → f j z ≤ C j) (hz : ∀ j z, z ∉ K j → f j z = 0) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : - ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ - (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ᵐ ω ∂P, ∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i * - (levelApproximationTest κ B γ G (u (n+i)) (t j) (f j) ω).toReal) - atTop (𝓝 (x j ω))) := by - let A (n j : ℕ) (ω : Ω) := levelApproximationTest κ B γ G (u n) (t j) (f j) ω - have hAm (n j : ℕ) : Measurable (A n j) := - levelApproximationTest_measurable hm hBc κ (u n) (hu n) γ G F hw (t j) (f j) (hfm j) - choose M hM hbound using fun j => uniform_levelApproximationTest_moments hB hm hBm hBc hB0 - hκ hκ8 γ G F hw hgrowth (K j) (hK j) (hKH j) (f j) (C j) (hC j) (hf j) (hz j) 2 - have hLp (n j : ℕ) : MemLp (fun ω => (A n j ω).toReal) 2 P := - memLp_toReal_two P (A n j) (hAm n j) (M j) (hM j) (hbound j (u n) (hu n) (t j)) - have hnorm (j : ℕ) : ∃ D : ℝ, ∀ n, ‖(hLp n j).toLp (fun ω => (A n j ω).toReal)‖ ≤ D := by - refine ⟨(M j ^ (1/2 : ℝ)).toReal,fun n => ?_⟩ - exact norm_toLp_toReal_two_le P (A n j) (hAm n j) (M j) (hM j) - (hbound j (u n) (hu n) (t j)) (hLp n j) - obtain ⟨x,s,w,hw₀,hw₁,_,hconv⟩ := exists_randomVariable_convex_tail_limit - (fun n j ω => (A n j ω).toReal) hLp hnorm - exact ⟨x,s,w,hw₀,hw₁,hconv⟩ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximationTests +import OAI.Probability.SLE.Mass.L2ConvexTails + +namespace OAI + +/-! Uniform Hilbert bounds and one common deterministic convex extraction +for the actual finite-level measure tests. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +lemma eLpNorm_toReal_two_le + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (A : Ω → ℝ≥0∞) (hA : Measurable A) (M : ℝ≥0∞) + (hM : (∫⁻ ω, A ω ^ 2 ∂P) ≤ M) : + eLpNorm (fun ω => (A ω).toReal) 2 P ≤ M ^ (1/2 : ℝ) := by + have hn (ω : Ω) : ‖(A ω).toReal‖ₑ ≤ A ω := by + rw [← ofReal_norm,Real.norm_eq_abs,abs_of_nonneg ENNReal.toReal_nonneg] + exact ENNReal.ofReal_toReal_le + have hi : (∫⁻ ω, ‖(A ω).toReal‖ₑ ^ 2 ∂P) ≤ M := + (lintegral_mono (fun ω => pow_le_pow_left' (hn ω) 2)).trans hM + rw [eLpNorm_eq_lintegral_rpow_enorm_toReal (by norm_num : (2 : ℝ≥0∞) ≠ 0) + ENNReal.ofNat_ne_top hA.ennreal_toReal.aestronglyMeasurable] + simpa only [ENNReal.toReal_ofNat,ENNReal.rpow_two] using + ENNReal.rpow_le_rpow hi (by norm_num : (0 : ℝ) ≤ 1/2) + +lemma memLp_toReal_two + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (A : Ω → ℝ≥0∞) (hA : Measurable A) (M : ℝ≥0∞) (hM : M < ⊤) + (hAM : (∫⁻ ω, A ω ^ 2 ∂P) ≤ M) : + MemLp (fun ω => (A ω).toReal) 2 P := by + apply lt_of_le_of_lt (eLpNorm_toReal_two_le P A hA M hAM) + exact ENNReal.rpow_lt_top_of_nonneg (by norm_num) hM.ne + +lemma norm_toLp_toReal_two_le + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (A : Ω → ℝ≥0∞) (hA : Measurable A) (M : ℝ≥0∞) (hM : M < ⊤) + (hAM : (∫⁻ ω, A ω ^ 2 ∂P) ≤ M) + (hLp : MemLp (fun ω => (A ω).toReal) 2 P) : + ‖hLp.toLp (fun ω => (A ω).toReal)‖ ≤ (M ^ (1/2 : ℝ)).toReal := by + rw [Lp.norm_toLp] + exact ENNReal.toReal_mono (ENNReal.rpow_ne_top_of_nonneg (by norm_num) hM.ne) + (eLpNorm_toReal_two_le P A hA M hAM) + +theorem exists_common_levelApproximation_test_limits + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (K : ℕ → Set ℂ) (hK : ∀ j, IsCompact (K j)) (hKH : ∀ j, K j ⊆ upperHalfPlane) + (f : ℕ → ℂ → ℝ≥0∞) (hfm : ∀ j, Measurable (f j)) + (C : ℕ → ℝ≥0∞) (hC : ∀ j, C j < ⊤) + (hf : ∀ j z, z ∈ K j → f j z ≤ C j) (hz : ∀ j z, z ∉ K j → f j z = 0) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : + ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ + (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ᵐ ω ∂P, ∀ j, Tendsto (fun n => ∑ i ∈ s n, w n i * + (levelApproximationTest κ B γ G (u (n+i)) (t j) (f j) ω).toReal) + atTop (𝓝 (x j ω))) := by + let A (n j : ℕ) (ω : Ω) := levelApproximationTest κ B γ G (u n) (t j) (f j) ω + have hAm (n j : ℕ) : Measurable (A n j) := + levelApproximationTest_measurable hm hBc κ (u n) (hu n) γ G F hw (t j) (f j) (hfm j) + choose M hM hbound using fun j => uniform_levelApproximationTest_moments hB hm hBm hBc hB0 + hκ hκ8 γ G F hw hgrowth (K j) (hK j) (hKH j) (f j) (C j) (hC j) (hf j) (hz j) 2 + have hLp (n j : ℕ) : MemLp (fun ω => (A n j ω).toReal) 2 P := + memLp_toReal_two P (A n j) (hAm n j) (M j) (hM j) (hbound j (u n) (hu n) (t j)) + have hnorm (j : ℕ) : ∃ D : ℝ, ∀ n, ‖(hLp n j).toLp (fun ω => (A n j ω).toReal)‖ ≤ D := by + refine ⟨(M j ^ (1/2 : ℝ)).toReal,fun n => ?_⟩ + exact norm_toLp_toReal_two_le P (A n j) (hAm n j) (M j) (hM j) + (hbound j (u n) (hu n) (t j)) (hLp n j) + obtain ⟨x,s,w,hw₀,hw₁,_,hconv⟩ := exists_randomVariable_convex_tail_limit + (fun n j ω => (A n j ω).toReal) hLp hnorm + exact ⟨x,s,w,hw₀,hw₁,hconv⟩ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelApproximationMean.lean b/lean/OAI/Probability/SLE/Mass/LevelApproximationMean.lean index 05cc1c935d15e2f2a124a2fff88239643ce8174f..c22130b351240123e951573324c441b3212c33a7 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelApproximationMean.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelApproximationMean.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Mass.LevelApproximants - -namespace OAI - -/-! Mean measure of the actual finite-level approximants. This identity -uses the proved one-point stopped law and needs no univalence input. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -theorem levelMass_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (u : ℝ) (hu : 0 ≤ u) : - (∫⁻ ω, levelMass κ B γ G u K ω ∂P) = ∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - unfold levelMass - rw [lintegral_lintegral_swap - (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal.aemeasurable] - apply setLIntegral_congr_fun hK - intro z hz - obtain ⟨hi,he⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F - (ae_of_all _ hw) z (hKH hz) u hu - dsimp only - rw [← ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun ω => - stoppedGreen_nonneg (hw ω) κ z u)),he] - -theorem levelApproximation_terminal_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (u : ℝ) (hu : 0 ≤ u) : - (∫⁻ ω, levelApproximation κ B γ G u ∞ ω K ∂P) = - ∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) := by - simp only [levelApproximation_apply_top κ B γ G u _ hK] - exact levelMass_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw K hK hKH u hu - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximants + +namespace OAI + +/-! Mean measure of the actual finite-level approximants. This identity +uses the proved one-point stopped law and needs no univalence input. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +theorem levelMass_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (u : ℝ) (hu : 0 ≤ u) : + (∫⁻ ω, levelMass κ B γ G u K ω ∂P) = ∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + unfold levelMass + rw [lintegral_lintegral_swap + (actual_stoppedGreen_joint_measurable hm hBc κ u hu γ G F hw).ennreal_ofReal.aemeasurable] + apply setLIntegral_congr_fun hK + intro z hz + obtain ⟨hi,he⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F + (ae_of_all _ hw) z (hKH hz) u hu + dsimp only + rw [← ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun ω => + stoppedGreen_nonneg (hw ω) κ z u)),he] + +theorem levelApproximation_terminal_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (u : ℝ) (hu : 0 ≤ u) : + (∫⁻ ω, levelApproximation κ B γ G u ∞ ω K ∂P) = + ∫⁻ z in K, ENNReal.ofReal (initialGreen κ z) := by + simp only [levelApproximation_apply_top κ B γ G u _ hK] + exact levelMass_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw K hK hKH u hu + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelApproximationSupport.lean b/lean/OAI/Probability/SLE/Mass/LevelApproximationSupport.lean index 9b401888e8c5458544fba5cc6c63c0036ac06a55..8fefd22a6ae0992451018e4d020828ce66412112 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelApproximationSupport.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelApproximationSupport.lean @@ -1,40 +1,40 @@ -import OAI.Probability.SLE.Mass.LevelApproximants - -namespace OAI - -/-! Finite-level mass is carried near the actual past trace. On an -interior compact, the neighborhood shrinks as the level tends to infinity. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -lemma levelApproximationDensity_eq_zero_of_avoidance - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hU : Continuous (fun s => Real.sqrt κ*B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im) - (u : ℝ) (hlevel : (2*z.im)*Real.exp (-u) < r) (t : ℝ≥0) - (havoid : ∀ s ∈ Icc 0 t, r ≤ dist (γ ω s) z) : - levelApproximationDensity κ B γ G u t ω z = 0 := by - by_contra hn - have hτ : radiusLevelTime (γ ω) (G ω) z u ≤ (t : ℝ≥0∞) := by - by_contra h - exact hn (by simp only [levelApproximationDensity,ite_eq_right h]) - obtain ⟨a,ha,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top (γ ω) (G ω) z u - (hτ.trans_lt ENNReal.coe_lt_top) - have hat : a ≤ t := by - rw [radiusLevelTime_eq_of_live_level hU hw z u a ha hR] at hτ - exact ENNReal.coe_le_coe.mp hτ - have hbound := hw.radius_ge_of_no_trace_approach hU a z ha hr hy - (fun s hs => havoid s ⟨hs.1,hs.2.trans hat⟩) - exact (not_lt_of_ge hbound) (hR.trans_lt hlevel) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximants + +namespace OAI + +/-! Finite-level mass is carried near the actual past trace. On an +interior compact, the neighborhood shrinks as the level tends to infinity. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +lemma levelApproximationDensity_eq_zero_of_avoidance + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hU : Continuous (fun s => Real.sqrt κ*B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ*B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (r : ℝ) (hr : 0 < r) (hy : r ≤ z.im) + (u : ℝ) (hlevel : (2*z.im)*Real.exp (-u) < r) (t : ℝ≥0) + (havoid : ∀ s ∈ Icc 0 t, r ≤ dist (γ ω s) z) : + levelApproximationDensity κ B γ G u t ω z = 0 := by + by_contra hn + have hτ : radiusLevelTime (γ ω) (G ω) z u ≤ (t : ℝ≥0∞) := by + by_contra h + exact hn (by simp only [levelApproximationDensity,ite_eq_right h]) + obtain ⟨a,ha,hR⟩ := exists_live_level_of_radiusLevelTime_lt_top (γ ω) (G ω) z u + (hτ.trans_lt ENNReal.coe_lt_top) + have hat : a ≤ t := by + rw [radiusLevelTime_eq_of_live_level hU hw z u a ha hR] at hτ + exact ENNReal.coe_le_coe.mp hτ + have hbound := hw.radius_ge_of_no_trace_approach hU a z ha hr hy + (fun s hs => havoid s ⟨hs.1,hs.2.trans hat⟩) + exact (not_lt_of_ge hbound) (hR.trans_lt hlevel) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelApproximationTests.lean b/lean/OAI/Probability/SLE/Mass/LevelApproximationTests.lean index 4bbdd7507866252f39de4e2e58a683641a668e24..0f19b66e5e71982b6631f46aeb1ba8b48878e9a4 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelApproximationTests.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelApproximationTests.lean @@ -1,88 +1,88 @@ -import OAI.Probability.SLE.Mass.CompactLevelMoments - -namespace OAI - -/-! Bounded nonnegative tests of the actual finite-level measures inherit -the compact moment bounds used for simultaneous convex extraction. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -def levelApproximationTest {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ≥0∞) (ω : Ω) : ℝ≥0∞ := - ∫⁻ z, f z ∂levelApproximation κ B γ G u t ω - -lemma lintegral_le_const_mul_measure_of_support - (μ : Measure ℂ) (K : Set ℂ) (hK : MeasurableSet K) - (f : ℂ → ℝ≥0∞) (C : ℝ≥0∞) (hf : ∀ z ∈ K, f z ≤ C) - (hz : ∀ z ∉ K, f z = 0) : (∫⁻ z, f z ∂μ) ≤ C * μ K := by - classical - apply (lintegral_mono (fun z => ?_)).trans (lintegral_indicator_const hK C).le - by_cases h : z ∈ K - · simpa only [indicator_of_mem h] using hf z h - · simp only [hz z h,indicator_of_notMem h,le_refl] - -lemma levelApproximationTest_le - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ≥0∞) (ω : Ω) - (K : Set ℂ) (hK : MeasurableSet K) (C : ℝ≥0∞) - (hf : ∀ z ∈ K, f z ≤ C) (hz : ∀ z ∉ K, f z = 0) : - levelApproximationTest κ B γ G u t f ω ≤ C * levelMass κ B γ G u K ω := by - apply (lintegral_le_const_mul_measure_of_support - (levelApproximation κ B γ G u t ω) K hK f C hf hz).trans - gcongr - exact levelApproximation_apply_le κ B γ G u t ω hK - -lemma levelApproximationTest_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0∞) (f : ℂ → ℝ≥0∞) (hf : Measurable f) : - Measurable (levelApproximationTest κ B γ G u t f) := by - have hd := levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t - have he (ω : Ω) : levelApproximationTest κ B γ G u t f ω = - ∫⁻ z, levelApproximationDensity κ B γ G u t ω z * f z := by - exact lintegral_withDensity_eq_lintegral_mul volume hd.of_uncurry_left hf - simp_rw [show levelApproximationTest κ B γ G u t f = - (fun ω => ∫⁻ z, levelApproximationDensity κ B γ G u t ω z * f z) from funext he] - exact (hd.mul (hf.comp measurable_snd)).lintegral_prod_right' - -theorem uniform_levelApproximationTest_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) - (f : ℂ → ℝ≥0∞) (C : ℝ≥0∞) (hC : C < ⊤) - (hf : ∀ z ∈ K, f z ≤ C) (hz : ∀ z ∉ K, f z = 0) (m : ℕ) : - ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, - (∫⁻ ω, levelApproximationTest κ B γ G u t f ω ^ m ∂P) ≤ M := by - obtain ⟨M,hM,hbound⟩ := uniform_levelMass_compact_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth K hK hKH m - refine ⟨C^m*M,ENNReal.mul_lt_top (ENNReal.pow_lt_top hC) hM,?_⟩ - intro u hu t - calc - _ ≤ ∫⁻ ω, (C*levelMass κ B γ G u K ω)^m ∂P := - lintegral_mono (fun ω => pow_le_pow_left' - (levelApproximationTest_le κ B γ G u t f ω K hK.measurableSet C hf hz) m) - _ = C^m*(∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) := by - simp_rw [mul_pow] - exact lintegral_const_mul' _ _ (ENNReal.pow_ne_top hC.ne) - _ ≤ _ := by gcongr; exact hbound u hu - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CompactLevelMoments + +namespace OAI + +/-! Bounded nonnegative tests of the actual finite-level measures inherit +the compact moment bounds used for simultaneous convex extraction. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +def levelApproximationTest {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ≥0∞) (ω : Ω) : ℝ≥0∞ := + ∫⁻ z, f z ∂levelApproximation κ B γ G u t ω + +lemma lintegral_le_const_mul_measure_of_support + (μ : Measure ℂ) (K : Set ℂ) (hK : MeasurableSet K) + (f : ℂ → ℝ≥0∞) (C : ℝ≥0∞) (hf : ∀ z ∈ K, f z ≤ C) + (hz : ∀ z ∉ K, f z = 0) : (∫⁻ z, f z ∂μ) ≤ C * μ K := by + classical + apply (lintegral_mono (fun z => ?_)).trans (lintegral_indicator_const hK C).le + by_cases h : z ∈ K + · simpa only [indicator_of_mem h] using hf z h + · simp only [hz z h,indicator_of_notMem h,le_refl] + +lemma levelApproximationTest_le + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0∞) (f : ℂ → ℝ≥0∞) (ω : Ω) + (K : Set ℂ) (hK : MeasurableSet K) (C : ℝ≥0∞) + (hf : ∀ z ∈ K, f z ≤ C) (hz : ∀ z ∉ K, f z = 0) : + levelApproximationTest κ B γ G u t f ω ≤ C * levelMass κ B γ G u K ω := by + apply (lintegral_le_const_mul_measure_of_support + (levelApproximation κ B γ G u t ω) K hK f C hf hz).trans + gcongr + exact levelApproximation_apply_le κ B γ G u t ω hK + +lemma levelApproximationTest_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0∞) (f : ℂ → ℝ≥0∞) (hf : Measurable f) : + Measurable (levelApproximationTest κ B γ G u t f) := by + have hd := levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t + have he (ω : Ω) : levelApproximationTest κ B γ G u t f ω = + ∫⁻ z, levelApproximationDensity κ B γ G u t ω z * f z := by + exact lintegral_withDensity_eq_lintegral_mul volume hd.of_uncurry_left hf + simp_rw [show levelApproximationTest κ B γ G u t f = + (fun ω => ∫⁻ z, levelApproximationDensity κ B γ G u t ω z * f z) from funext he] + exact (hd.mul (hf.comp measurable_snd)).lintegral_prod_right' + +theorem uniform_levelApproximationTest_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) + (f : ℂ → ℝ≥0∞) (C : ℝ≥0∞) (hC : C < ⊤) + (hf : ∀ z ∈ K, f z ≤ C) (hz : ∀ z ∉ K, f z = 0) (m : ℕ) : + ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, + (∫⁻ ω, levelApproximationTest κ B γ G u t f ω ^ m ∂P) ≤ M := by + obtain ⟨M,hM,hbound⟩ := uniform_levelMass_compact_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth K hK hKH m + refine ⟨C^m*M,ENNReal.mul_lt_top (ENNReal.pow_lt_top hC) hM,?_⟩ + intro u hu t + calc + _ ≤ ∫⁻ ω, (C*levelMass κ B γ G u K ω)^m ∂P := + lintegral_mono (fun ω => pow_le_pow_left' + (levelApproximationTest_le κ B γ G u t f ω K hK.measurableSet C hf hz) m) + _ = C^m*(∫⁻ ω, levelMass κ B γ G u K ω ^ m ∂P) := by + simp_rw [mul_pow] + exact lintegral_const_mul' _ _ (ENNReal.pow_ne_top hC.ne) + _ ≤ _ := by gcongr; exact hbound u hu + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelConditionalMean.lean b/lean/OAI/Probability/SLE/Mass/LevelConditionalMean.lean index 431dde4eda947ddbd976423efa58356f75fda6d4..6d9424f77963db77586559180b670875596ff651 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelConditionalMean.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelConditionalMean.lean @@ -1,146 +1,146 @@ -import OAI.Probability.SLE.Mass.LevelCurrentTests -import OAI.Probability.SLE.Mass.ConvexMeanMeasure -import OAI.Probability.SLE.Mass.LevelRandomRadon - -namespace OAI - -/-! The exact conditional decomposition is preserved by the common -finite convex combinations. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge - -lemma bind_mono_of_measurable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] - (P : Measure Ω) {μ ν : Ω → Measure X} (hμ : Measurable μ) (hν : Measurable ν) - (h : ∀ ω, μ ω ≤ ν ω) : P.bind μ ≤ P.bind ν := by - apply Measure.le_iff.mpr - intro A hA - rw [Measure.bind_apply hA hμ.aemeasurable,Measure.bind_apply hA hν.aemeasurable] - exact lintegral_mono (fun ω => h ω A) - -lemma bind_finset_smul {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] - (P : Measure Ω) (s : Finset ℕ) (w : ℕ → ℝ≥0∞) (hw : ∀ i, w i ≠ ⊤) - (μ : ℕ → Ω → Measure X) (hμ : ∀ i, Measurable (μ i)) : - P.bind (fun ω => ∑ i ∈ s, w i • μ i ω) = ∑ i ∈ s, w i • P.bind (μ i) := by - have hm : Measurable (fun ω => ∑ i ∈ s, w i • μ i ω) := by - apply Measure.measurable_of_measurable_coe - intro A hA - simp only [Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - exact Finset.measurable_sum _ (fun i _ => measurable_const.mul ((Measure.measurable_coe hA).comp (hμ i))) - apply Measure.ext - intro A hA - rw [Measure.bind_apply hA hm.aemeasurable] - simp only [Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - rw [lintegral_finsetSum s (f := fun i ω => w i * μ i ω A) - (fun i _ => measurable_const.mul ((Measure.measurable_coe hA).comp (hμ i)))] - apply Finset.sum_congr rfl - intro i _ - rw [lintegral_const_mul' _ _ (hw i),Measure.bind_apply hA (hμ i).aemeasurable] - -namespace Multipoint - -lemma halfPlaneLevelMeasure_conditional_split - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : - (P.restrict E).bind (halfPlaneLevelMeasure κ B γ G u ∞) = - (P.restrict E).bind (halfPlaneLevelMeasure κ B γ G u t) + - (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G u t) := by - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - rw [← halfPlaneRemainderMeasure_conditional_mean hB hBm hBc hB0 hκ hκ8 γ G F hw u hu t E hE] - apply Measure.ext - intro A hA - rw [Measure.add_apply, - Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw ∞).aemeasurable, - Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw t).aemeasurable, - Measure.bind_apply hA (halfPlaneRemainderMeasure_measurable hm hBc κ u hu γ G F hw t).aemeasurable] - simp_rw [halfPlaneLevelMeasure_split hm hBc κ u hu γ G F hw t,Measure.add_apply] - exact lintegral_add_left (((Measure.measurable_coe hA).comp - (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw t))) _ - -def convexCurrentMean {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0) : - Measure upperHalfPlane := - ∑ i ∈ s n, ENNReal.ofReal (w n i) • P.bind (halfPlaneCurrentMeasure κ B γ G (u (n+i)) t) - -lemma convexLevelMeasure_conditional_split - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (n : ℕ) (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : - (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n ∞) = - (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n t) + - convexCurrentMean (P.restrict E) κ B γ G u s w n t := by - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - unfold convexLevelMeasure convexCurrentMean - rw [bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ - (fun i => halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw ∞), - bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ - (fun i => halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t)] - simp_rw [halfPlaneLevelMeasure_conditional_split hB hBm hBc hB0 hκ hκ8 γ G F hw - _ (hu _) t E hE,smul_add,Finset.sum_add_distrib] - - -lemma convexLevelMeasure_mean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (n : ℕ) (t : ℝ≥0∞) (E : Set Ω) : - (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n t) ≤ initialGreenMeasure κ := by - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - rw [← convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu s w hw₀ hw₁ n] - apply Measure.le_iff.mpr - intro A hA - rw [Measure.bind_apply hA (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n t).aemeasurable, - Measure.bind_apply hA (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞).aemeasurable] - exact lintegral_mono' Measure.restrict_le_self (fun ω => - convexLevelMeasure_mono κ B γ G u s w n ω le_top A) - -lemma convexCurrentMean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (n : ℕ) (t : ℝ≥0) (E : Set Ω) : - convexCurrentMean (P.restrict E) κ B γ G u s w n t ≤ initialGreenMeasure κ := by - apply Measure.le_iff.mpr - intro A _ - simp only [convexCurrentMean,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - calc - _ ≤ ∑ i ∈ s n, ENNReal.ofReal (w n i) * initialGreenMeasure κ A := - Finset.sum_le_sum (fun i _ => mul_le_mul' le_rfl - (halfPlaneCurrentMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw (u (n+i)) (hu _) t E A)) - _ = _ := by - rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg (hw₀ n),hw₁ n,ENNReal.ofReal_one,one_mul] - -end Multipoint -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelCurrentTests +import OAI.Probability.SLE.Mass.ConvexMeanMeasure +import OAI.Probability.SLE.Mass.LevelRandomRadon + +namespace OAI + +/-! The exact conditional decomposition is preserved by the common +finite convex combinations. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge + +lemma bind_mono_of_measurable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] + (P : Measure Ω) {μ ν : Ω → Measure X} (hμ : Measurable μ) (hν : Measurable ν) + (h : ∀ ω, μ ω ≤ ν ω) : P.bind μ ≤ P.bind ν := by + apply Measure.le_iff.mpr + intro A hA + rw [Measure.bind_apply hA hμ.aemeasurable,Measure.bind_apply hA hν.aemeasurable] + exact lintegral_mono (fun ω => h ω A) + +lemma bind_finset_smul {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] + (P : Measure Ω) (s : Finset ℕ) (w : ℕ → ℝ≥0∞) (hw : ∀ i, w i ≠ ⊤) + (μ : ℕ → Ω → Measure X) (hμ : ∀ i, Measurable (μ i)) : + P.bind (fun ω => ∑ i ∈ s, w i • μ i ω) = ∑ i ∈ s, w i • P.bind (μ i) := by + have hm : Measurable (fun ω => ∑ i ∈ s, w i • μ i ω) := by + apply Measure.measurable_of_measurable_coe + intro A hA + simp only [Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + exact Finset.measurable_sum _ (fun i _ => measurable_const.mul ((Measure.measurable_coe hA).comp (hμ i))) + apply Measure.ext + intro A hA + rw [Measure.bind_apply hA hm.aemeasurable] + simp only [Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + rw [lintegral_finsetSum s (f := fun i ω => w i * μ i ω A) + (fun i _ => measurable_const.mul ((Measure.measurable_coe hA).comp (hμ i)))] + apply Finset.sum_congr rfl + intro i _ + rw [lintegral_const_mul' _ _ (hw i),Measure.bind_apply hA (hμ i).aemeasurable] + +namespace Multipoint + +lemma halfPlaneLevelMeasure_conditional_split + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : + (P.restrict E).bind (halfPlaneLevelMeasure κ B γ G u ∞) = + (P.restrict E).bind (halfPlaneLevelMeasure κ B γ G u t) + + (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G u t) := by + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + rw [← halfPlaneRemainderMeasure_conditional_mean hB hBm hBc hB0 hκ hκ8 γ G F hw u hu t E hE] + apply Measure.ext + intro A hA + rw [Measure.add_apply, + Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw ∞).aemeasurable, + Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw t).aemeasurable, + Measure.bind_apply hA (halfPlaneRemainderMeasure_measurable hm hBc κ u hu γ G F hw t).aemeasurable] + simp_rw [halfPlaneLevelMeasure_split hm hBc κ u hu γ G F hw t,Measure.add_apply] + exact lintegral_add_left (((Measure.measurable_coe hA).comp + (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw t))) _ + +def convexCurrentMean {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0) : + Measure upperHalfPlane := + ∑ i ∈ s n, ENNReal.ofReal (w n i) • P.bind (halfPlaneCurrentMeasure κ B γ G (u (n+i)) t) + +lemma convexLevelMeasure_conditional_split + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (n : ℕ) (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : + (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n ∞) = + (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n t) + + convexCurrentMean (P.restrict E) κ B γ G u s w n t := by + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + unfold convexLevelMeasure convexCurrentMean + rw [bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ + (fun i => halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw ∞), + bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ + (fun i => halfPlaneLevelMeasure_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t)] + simp_rw [halfPlaneLevelMeasure_conditional_split hB hBm hBc hB0 hκ hκ8 γ G F hw + _ (hu _) t E hE,smul_add,Finset.sum_add_distrib] + + +lemma convexLevelMeasure_mean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (n : ℕ) (t : ℝ≥0∞) (E : Set Ω) : + (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n t) ≤ initialGreenMeasure κ := by + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + rw [← convexLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu s w hw₀ hw₁ n] + apply Measure.le_iff.mpr + intro A hA + rw [Measure.bind_apply hA (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n t).aemeasurable, + Measure.bind_apply hA (convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n ∞).aemeasurable] + exact lintegral_mono' Measure.restrict_le_self (fun ω => + convexLevelMeasure_mono κ B γ G u s w n ω le_top A) + +lemma convexCurrentMean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (n : ℕ) (t : ℝ≥0) (E : Set Ω) : + convexCurrentMean (P.restrict E) κ B γ G u s w n t ≤ initialGreenMeasure κ := by + apply Measure.le_iff.mpr + intro A _ + simp only [convexCurrentMean,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + calc + _ ≤ ∑ i ∈ s n, ENNReal.ofReal (w n i) * initialGreenMeasure κ A := + Finset.sum_le_sum (fun i _ => mul_le_mul' le_rfl + (halfPlaneCurrentMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw (u (n+i)) (hu _) t E A)) + _ = _ := by + rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg (hw₀ n),hw₁ n,ENNReal.ofReal_one,one_mul] + +end Multipoint +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelCurrentLimit.lean b/lean/OAI/Probability/SLE/Mass/LevelCurrentLimit.lean index a659b6f8b1e81b651228bc8077382fd09f9bcbe1..9178eb2ad8c01ca7ab16c8633ab302d3f52a821d 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelCurrentLimit.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelCurrentLimit.lean @@ -1,146 +1,146 @@ -import OAI.Probability.SLE.Mass.LevelRemainderMeasure -import OAI.Probability.SLE.Mass.LevelGreenLimit - -namespace OAI - -/-! Increasing finite-level remaining kernels converge to the actual killed -Green kernel, also after averaging on an arbitrary event. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -def halfPlaneGreenMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal - (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z))).comap Subtype.val - -lemma halfPlaneGreenMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : Measurable (halfPlaneGreenMeasure κ B γ G t) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t).ennreal_ofReal.lintegral_prod_right' - -lemma levelCurrentGreen_le_green - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hw : IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (t : ℝ≥0) (z : ℂ) : - levelCurrentGreen κ B γ G u t z ω ≤ - greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z := by - unfold levelCurrentGreen - split_ifs - · exact le_rfl - · exact greenDensity_nonneg hw κ t z - -lemma halfPlaneCurrentMeasure_le_green - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hw : IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) (u : ℝ) (t : ℝ≥0) : - halfPlaneCurrentMeasure κ B γ G u t ω ≤ halfPlaneGreenMeasure κ B γ G t ω := by - apply Measure.le_iff.mpr - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneCurrentMeasure,halfPlaneGreenMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - exact lintegral_mono (fun z => ENNReal.ofReal_le_ofReal (levelCurrentGreen_le_green ω hw u t z)) - -theorem currentGreen_spatial_mean_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (K : Set ℂ) - (f : ℂ → ℝ≥0∞) (hf : Measurable f) (E : Set Ω) : - Tendsto (fun n => ∫⁻ ω in E, (∫⁻ z in K, - f z * ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂volume) ∂P) - atTop (𝓝 (∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal - (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂volume) ∂P)) := by - let a (n : ℕ) (q : Ω × ℂ) := - f q.2 * ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t q.2 q.1) - have ha (n : ℕ) : Measurable (a n) := - (hf.comp measurable_snd).mul - (levelCurrentGreen_joint_measurable hBm hBc κ (u n) (hu n) γ G F hw t).ennreal_ofReal - have ham (ω : Ω) : Monotone (fun n => fun z => a n (ω,z)) := by - intro n m hnm z - exact mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal - (levelCurrentGreen_mono ω (hBc ω) (hw ω) z t (hu n) (hmono hnm))) - have hal (ω : Ω) : Tendsto (fun n => ∫⁻ z in K, a n (ω,z)) atTop - (𝓝 (∫⁻ z in K, f z * ENNReal.ofReal - (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z))) := by - apply lintegral_tendsto_of_tendsto_of_monotone (μ := volume.restrict K) - (f := fun n z => a n (ω,z)) - (fun n => ((ha n).comp (measurable_const.prodMk measurable_id)).aemeasurable) - (Eventually.of_forall (fun z n m hnm => ham ω hnm z)) - filter_upwards [] with z - apply tendsto_const_nhds.congr' - exact (levelCurrentGreen_eventually_eq ω (hBc ω) (hw ω) z t u hu hlim).mono - (fun n hn => congrArg (fun x : ℝ => f z * ENNReal.ofReal x) hn.symm) - exact lintegral_tendsto_of_tendsto_of_monotone - (fun n => (ha n).lintegral_prod_right'.aemeasurable) - (Eventually.of_forall (fun ω n m hnm => lintegral_mono (ham ω hnm))) - (Eventually.of_forall hal) - -theorem halfPlaneCurrentMeasure_mean_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) - (A : Set upperHalfPlane) (hA : MeasurableSet A) : - Tendsto (fun n => (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u n) t) A) - atTop (𝓝 ((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) A)) := by - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [Measure.bind_apply hA - (halfPlaneCurrentMeasure_measurable hBm hBc κ (u _) (hu _) γ G F hw t).aemeasurable, - Measure.bind_apply hA (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable, - halfPlaneCurrentMeasure,halfPlaneGreenMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - simpa only [one_mul] using currentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim t _ (fun _ => 1) measurable_const E - -theorem halfPlaneGreenMeasure_mean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (E : Set Ω) : - (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) ≤ initialGreenMeasure κ := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - apply Measure.le_iff.mpr - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by - rintro _ ⟨x,_,rfl⟩; exact x.property - rw [Measure.bind_apply hA (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable] - simp only [halfPlaneGreenMeasure,initialGreenMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - apply (lintegral_mono' Measure.restrict_le_self (le_refl _)).trans - rw [lintegral_lintegral_swap - (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t).ennreal_ofReal.aemeasurable] - exact setLIntegral_mono' hAI (fun z hz => - greenDensity_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw z (hAH hz) t) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelRemainderMeasure +import OAI.Probability.SLE.Mass.LevelGreenLimit + +namespace OAI + +/-! Increasing finite-level remaining kernels converge to the actual killed +Green kernel, also after averaging on an arbitrary event. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +def halfPlaneGreenMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal + (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z))).comap Subtype.val + +lemma halfPlaneGreenMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : Measurable (halfPlaneGreenMeasure κ B γ G t) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t).ennreal_ofReal.lintegral_prod_right' + +lemma levelCurrentGreen_le_green + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hw : IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (t : ℝ≥0) (z : ℂ) : + levelCurrentGreen κ B γ G u t z ω ≤ + greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z := by + unfold levelCurrentGreen + split_ifs + · exact le_rfl + · exact greenDensity_nonneg hw κ t z + +lemma halfPlaneCurrentMeasure_le_green + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hw : IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) (u : ℝ) (t : ℝ≥0) : + halfPlaneCurrentMeasure κ B γ G u t ω ≤ halfPlaneGreenMeasure κ B γ G t ω := by + apply Measure.le_iff.mpr + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneCurrentMeasure,halfPlaneGreenMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + exact lintegral_mono (fun z => ENNReal.ofReal_le_ofReal (levelCurrentGreen_le_green ω hw u t z)) + +theorem currentGreen_spatial_mean_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (K : Set ℂ) + (f : ℂ → ℝ≥0∞) (hf : Measurable f) (E : Set Ω) : + Tendsto (fun n => ∫⁻ ω in E, (∫⁻ z in K, + f z * ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t z ω) ∂volume) ∂P) + atTop (𝓝 (∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal + (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂volume) ∂P)) := by + let a (n : ℕ) (q : Ω × ℂ) := + f q.2 * ENNReal.ofReal (levelCurrentGreen κ B γ G (u n) t q.2 q.1) + have ha (n : ℕ) : Measurable (a n) := + (hf.comp measurable_snd).mul + (levelCurrentGreen_joint_measurable hBm hBc κ (u n) (hu n) γ G F hw t).ennreal_ofReal + have ham (ω : Ω) : Monotone (fun n => fun z => a n (ω,z)) := by + intro n m hnm z + exact mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal + (levelCurrentGreen_mono ω (hBc ω) (hw ω) z t (hu n) (hmono hnm))) + have hal (ω : Ω) : Tendsto (fun n => ∫⁻ z in K, a n (ω,z)) atTop + (𝓝 (∫⁻ z in K, f z * ENNReal.ofReal + (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z))) := by + apply lintegral_tendsto_of_tendsto_of_monotone (μ := volume.restrict K) + (f := fun n z => a n (ω,z)) + (fun n => ((ha n).comp (measurable_const.prodMk measurable_id)).aemeasurable) + (Eventually.of_forall (fun z n m hnm => ham ω hnm z)) + filter_upwards [] with z + apply tendsto_const_nhds.congr' + exact (levelCurrentGreen_eventually_eq ω (hBc ω) (hw ω) z t u hu hlim).mono + (fun n hn => congrArg (fun x : ℝ => f z * ENNReal.ofReal x) hn.symm) + exact lintegral_tendsto_of_tendsto_of_monotone + (fun n => (ha n).lintegral_prod_right'.aemeasurable) + (Eventually.of_forall (fun ω n m hnm => lintegral_mono (ham ω hnm))) + (Eventually.of_forall hal) + +theorem halfPlaneCurrentMeasure_mean_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) + (A : Set upperHalfPlane) (hA : MeasurableSet A) : + Tendsto (fun n => (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u n) t) A) + atTop (𝓝 ((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) A)) := by + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [Measure.bind_apply hA + (halfPlaneCurrentMeasure_measurable hBm hBc κ (u _) (hu _) γ G F hw t).aemeasurable, + Measure.bind_apply hA (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable, + halfPlaneCurrentMeasure,halfPlaneGreenMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + simpa only [one_mul] using currentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim t _ (fun _ => 1) measurable_const E + +theorem halfPlaneGreenMeasure_mean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (E : Set Ω) : + (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) ≤ initialGreenMeasure κ := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + apply Measure.le_iff.mpr + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by + rintro _ ⟨x,_,rfl⟩; exact x.property + rw [Measure.bind_apply hA (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable] + simp only [halfPlaneGreenMeasure,initialGreenMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + apply (lintegral_mono' Measure.restrict_le_self (le_refl _)).trans + rw [lintegral_lintegral_swap + (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t).ennreal_ofReal.aemeasurable] + exact setLIntegral_mono' hAI (fun z hz => + greenDensity_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw z (hAH hz) t) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelCurrentTests.lean b/lean/OAI/Probability/SLE/Mass/LevelCurrentTests.lean index 89896d4e6d0200ac70974c2fad2525eb8c96f05c..a037c3ab3fa2b5a167099d045e7b1b2de30adc8e 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelCurrentTests.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelCurrentTests.lean @@ -1,151 +1,151 @@ -import OAI.Probability.SLE.Mass.LevelCurrentLimit -import OAI.Probability.SLE.Mass.SignedMeasureConvergence - -namespace OAI - -/-! Signed compact tests of the current Green mean pass through the -increasing radius levels. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma lintegral_halfPlane_density_test (ρ : ℂ → ℝ≥0∞) (hρ : Measurable ρ) - (f : upperHalfPlane → ℝ) (hf : Measurable f) : - (∫⁻ z : upperHalfPlane, ENNReal.ofReal (f z) - ∂(volume.withDensity ρ).comap Subtype.val) = - ∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ρ z ∂volume := by - calc - _ = ∫⁻ z : upperHalfPlane, ENNReal.ofReal (extendHalfPlaneTest f z) - ∂(volume.withDensity ρ).comap Subtype.val := by - congr 1 - funext z - rw [extendHalfPlaneTest_apply] - _ = ∫⁻ z in upperHalfPlane, ENNReal.ofReal (extendHalfPlaneTest f z) - ∂volume.withDensity ρ := - lintegral_subtype_comap (μ := volume.withDensity ρ) isOpen_upperHalfPlane.measurableSet - (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) - _ = ∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) ∂volume.withDensity ρ := by - apply setLIntegral_eq_of_support_subset - intro z hz - by_contra hn - exact hz (by - dsimp only - rw [extendHalfPlaneTest_of_not_mem f z hn,ENNReal.ofReal_zero]) - _ = _ := by - rw [lintegral_withDensity_eq_lintegral_mul _ hρ - (extendHalfPlaneTest_measurable f hf).ennreal_ofReal] - apply lintegral_congr - intro z - exact mul_comm _ _ - -lemma lintegral_density_kernel_test - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) - (ρ : Ω × ℂ → ℝ≥0∞) (hρ : Measurable ρ) - (he : ∀ ω, μ ω = (volume.withDensity (fun z => ρ (ω,z))).comap Subtype.val) - (f : upperHalfPlane → ℝ) (hf : Measurable f) : - (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind μ) = - ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ρ (ω,z) ∂volume) ∂P := by - rw [Measure.lintegral_bind hμ.aemeasurable hf.ennreal_ofReal.aemeasurable] - apply lintegral_congr - intro ω - rw [he ω] - exact lintegral_halfPlane_density_test _ - (hρ.comp (measurable_const.prodMk measurable_id)) f hf - -lemma halfPlaneCurrentMeasure_mean_lintegral_test - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (f : upperHalfPlane → ℝ) (hf : Measurable f) : - (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneCurrentMeasure κ B γ G u t)) = - ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * - ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂volume) ∂P := - lintegral_density_kernel_test P _ - (halfPlaneCurrentMeasure_measurable hBm hBc κ u hu γ G F hw t) _ - (levelCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw t).ennreal_ofReal - (fun _ => rfl) f hf - -lemma halfPlaneGreenMeasure_mean_lintegral_test - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (f : upperHalfPlane → ℝ) (hf : Measurable f) : - (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneGreenMeasure κ B γ G t)) = - ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ENNReal.ofReal - (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂volume) ∂P := - lintegral_density_kernel_test P _ (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t) _ - (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t).ennreal_ofReal - (fun _ => rfl) f hf - -lemma halfPlaneCurrentMeasure_mean_lintegral_test_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) - (f : upperHalfPlane → ℝ) (hf : Measurable f) : - Tendsto (fun n => ∫⁻ z, ENNReal.ofReal (f z) - ∂(P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u n) t)) atTop - (𝓝 (∫⁻ z, ENNReal.ofReal (f z) ∂(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) := by - simp_rw [halfPlaneCurrentMeasure_mean_lintegral_test _ hBm hBc κ _ (hu _) γ G F hw t f hf, - halfPlaneGreenMeasure_mean_lintegral_test _ hBm hBc κ γ G F hw t f hf] - simpa only [Measure.restrict_univ] using currentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim t univ (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) - (extendHalfPlaneTest_measurable f hf).ennreal_ofReal E - -lemma halfPlaneCurrentMeasure_mean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) : - (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G u t) ≤ initialGreenMeasure κ := by - apply le_trans ?_ (halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E) - apply Measure.le_iff.mpr - intro A hA - rw [Measure.bind_apply hA (halfPlaneCurrentMeasure_measurable hBm hBc κ u hu γ G F hw t).aemeasurable, - Measure.bind_apply hA (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable] - exact lintegral_mono (fun ω => halfPlaneCurrentMeasure_le_green ω (hw ω) u t A) - -theorem halfPlaneCurrentMeasure_mean_test_tendsto - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) - (f : C_c(upperHalfPlane,ℝ)) : - Tendsto (fun n => ∫ z, f z ∂(P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u n) t)) - atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) := by - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hfi : Integrable f (initialGreenMeasure κ) := f.integrable - apply integral_tendsto_of_lintegral_pos_neg - · exact fun n => hfi.mono_measure (halfPlaneCurrentMeasure_mean_le hB hBm hBc hB0 hκ hκ8 - γ G F hw (u n) (hu n) t E) - · exact hfi.mono_measure (halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E) - · exact halfPlaneCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim t E f f.continuous.measurable - · exact halfPlaneCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim t E (fun z => -f z) f.continuous.measurable.neg - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelCurrentLimit +import OAI.Probability.SLE.Mass.SignedMeasureConvergence + +namespace OAI + +/-! Signed compact tests of the current Green mean pass through the +increasing radius levels. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma lintegral_halfPlane_density_test (ρ : ℂ → ℝ≥0∞) (hρ : Measurable ρ) + (f : upperHalfPlane → ℝ) (hf : Measurable f) : + (∫⁻ z : upperHalfPlane, ENNReal.ofReal (f z) + ∂(volume.withDensity ρ).comap Subtype.val) = + ∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ρ z ∂volume := by + calc + _ = ∫⁻ z : upperHalfPlane, ENNReal.ofReal (extendHalfPlaneTest f z) + ∂(volume.withDensity ρ).comap Subtype.val := by + congr 1 + funext z + rw [extendHalfPlaneTest_apply] + _ = ∫⁻ z in upperHalfPlane, ENNReal.ofReal (extendHalfPlaneTest f z) + ∂volume.withDensity ρ := + lintegral_subtype_comap (μ := volume.withDensity ρ) isOpen_upperHalfPlane.measurableSet + (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) + _ = ∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) ∂volume.withDensity ρ := by + apply setLIntegral_eq_of_support_subset + intro z hz + by_contra hn + exact hz (by + dsimp only + rw [extendHalfPlaneTest_of_not_mem f z hn,ENNReal.ofReal_zero]) + _ = _ := by + rw [lintegral_withDensity_eq_lintegral_mul _ hρ + (extendHalfPlaneTest_measurable f hf).ennreal_ofReal] + apply lintegral_congr + intro z + exact mul_comm _ _ + +lemma lintegral_density_kernel_test + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) + (ρ : Ω × ℂ → ℝ≥0∞) (hρ : Measurable ρ) + (he : ∀ ω, μ ω = (volume.withDensity (fun z => ρ (ω,z))).comap Subtype.val) + (f : upperHalfPlane → ℝ) (hf : Measurable f) : + (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind μ) = + ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ρ (ω,z) ∂volume) ∂P := by + rw [Measure.lintegral_bind hμ.aemeasurable hf.ennreal_ofReal.aemeasurable] + apply lintegral_congr + intro ω + rw [he ω] + exact lintegral_halfPlane_density_test _ + (hρ.comp (measurable_const.prodMk measurable_id)) f hf + +lemma halfPlaneCurrentMeasure_mean_lintegral_test + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (f : upperHalfPlane → ℝ) (hf : Measurable f) : + (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneCurrentMeasure κ B γ G u t)) = + ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * + ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂volume) ∂P := + lintegral_density_kernel_test P _ + (halfPlaneCurrentMeasure_measurable hBm hBc κ u hu γ G F hw t) _ + (levelCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw t).ennreal_ofReal + (fun _ => rfl) f hf + +lemma halfPlaneGreenMeasure_mean_lintegral_test + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (f : upperHalfPlane → ℝ) (hf : Measurable f) : + (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneGreenMeasure κ B γ G t)) = + ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ENNReal.ofReal + (greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z) ∂volume) ∂P := + lintegral_density_kernel_test P _ (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t) _ + (actual_greenDensity_joint_measurable hBm hBc κ γ G F hw t).ennreal_ofReal + (fun _ => rfl) f hf + +lemma halfPlaneCurrentMeasure_mean_lintegral_test_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) + (f : upperHalfPlane → ℝ) (hf : Measurable f) : + Tendsto (fun n => ∫⁻ z, ENNReal.ofReal (f z) + ∂(P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u n) t)) atTop + (𝓝 (∫⁻ z, ENNReal.ofReal (f z) ∂(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) := by + simp_rw [halfPlaneCurrentMeasure_mean_lintegral_test _ hBm hBc κ _ (hu _) γ G F hw t f hf, + halfPlaneGreenMeasure_mean_lintegral_test _ hBm hBc κ γ G F hw t f hf] + simpa only [Measure.restrict_univ] using currentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim t univ (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) + (extendHalfPlaneTest_measurable f hf).ennreal_ofReal E + +lemma halfPlaneCurrentMeasure_mean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) : + (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G u t) ≤ initialGreenMeasure κ := by + apply le_trans ?_ (halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E) + apply Measure.le_iff.mpr + intro A hA + rw [Measure.bind_apply hA (halfPlaneCurrentMeasure_measurable hBm hBc κ u hu γ G F hw t).aemeasurable, + Measure.bind_apply hA (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t).aemeasurable] + exact lintegral_mono (fun ω => halfPlaneCurrentMeasure_le_green ω (hw ω) u t A) + +theorem halfPlaneCurrentMeasure_mean_test_tendsto + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (t : ℝ≥0) (E : Set Ω) + (f : C_c(upperHalfPlane,ℝ)) : + Tendsto (fun n => ∫ z, f z ∂(P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G (u n) t)) + atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t))) := by + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hfi : Integrable f (initialGreenMeasure κ) := f.integrable + apply integral_tendsto_of_lintegral_pos_neg + · exact fun n => hfi.mono_measure (halfPlaneCurrentMeasure_mean_le hB hBm hBc hB0 hκ hκ8 + γ G F hw (u n) (hu n) t E) + · exact hfi.mono_measure (halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E) + · exact halfPlaneCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim t E f f.continuous.measurable + · exact halfPlaneCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim t E (fun z => -f z) f.continuous.measurable.neg + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelGreenLimit.lean b/lean/OAI/Probability/SLE/Mass/LevelGreenLimit.lean index bbe3ec72a67dce92d5a8caf38e4fdab69ef80b85..40b3748c4da108955317b20419648b3a02fd0d04 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelGreenLimit.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelGreenLimit.lean @@ -1,120 +1,120 @@ -import OAI.Probability.SLE.Mass.LevelApproximationKernel -import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus - -namespace OAI - -/-! As the radius level tends to infinity, the remaining finite-level -kernel increases to the killed Green potential. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -lemma levelCurrentGreen_mono - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hBc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (t : ℝ≥0) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v) : - levelCurrentGreen κ B γ G u t z ω ≤ levelCurrentGreen κ B γ G v t z ω := by - have hτ := radiusLevelTime_mono (hBc.const_mul _) hw z hu huv - unfold levelCurrentGreen - split_ifs with h₁ h₂ - · exact le_rfl - · exact False.elim (h₂ (h₁.trans_le hτ)) - · exact greenDensity_nonneg hw κ t z - · exact le_rfl - -lemma levelCurrentGreen_eventually_eq - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hBc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (t : ℝ≥0) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) - (hlim : Tendsto u atTop atTop) : - ∀ᶠ n in atTop, levelCurrentGreen κ B γ G (u n) t z ω = - greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z := by - by_cases hz : z ∈ survivingDomain (γ ω) t - · filter_upwards [hlim.eventually (eventually_gt_atTop - (logRadiusClock (Real.log z.im) (logLoewnerCoordinates (G ω) z t)))] with n hn - have hτ : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z (u n) := by - apply lt_of_not_ge - rw [radiusLevelTime_le_iff_clock (hBc.const_mul _) hw z (u n) (hu n) t hz] - exact not_le.mpr hn - exact ite_eq_left hτ - · filter_upwards [] with n - unfold levelCurrentGreen greenDensity - simp only [ite_eq_right hz,ite_self] - -theorem levelCurrentGreen_mean_le - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun s => B s ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) : - (∫⁻ ω, ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂P) ≤ - ENNReal.ofReal (initialGreen κ z) := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - have he := levelRemainderGreen_setLIntegral hB hBm hBc hB0 hκ hκ8 γ G F - (ae_of_all _ hw) z hz u hu t univ MeasurableSet.univ - simp only [Measure.restrict_univ] at he - rw [← he] - calc - _ ≤ ∫⁻ ω, ENNReal.ofReal (stoppedGreen κ (fun s => Real.sqrt κ * B s ω) - (γ ω) (G ω) z u) ∂P := by - apply lintegral_mono - intro ω - unfold levelRemainderGreen - dsimp only - split_ifs - · exact le_rfl - · rw [ENNReal.ofReal_zero] - positivity - _ = _ := by - obtain ⟨hi,he⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F - (ae_of_all _ hw) z hz u hu - rw [← ofReal_integral_eq_lintegral_ofReal hi - (ae_of_all _ (fun ω => stoppedGreen_nonneg (hw ω) κ z u)),he] - -theorem greenDensity_mean_le - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun s => B s ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (t : ℝ≥0) : - (∫⁻ ω, ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) - (γ ω) (G ω) t z) ∂P) ≤ ENNReal.ofReal (initialGreen κ z) := by - have hm (n : ℕ) : Measurable (fun ω => ENNReal.ofReal (levelCurrentGreen κ B γ G n t z ω)) := by - have hj := (levelCurrentGreen_joint_measurable hBm hBc κ n (Nat.cast_nonneg n) γ G F hw t).ennreal_ofReal - have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const - have hh := hj.comp hp - simpa only [Function.comp_def] using hh - have hmono : ∀ᵐ ω ∂P, Monotone (fun n : ℕ => ENNReal.ofReal (levelCurrentGreen κ B γ G n t z ω)) := by - filter_upwards [] with ω - intro m n hmn - exact ENNReal.ofReal_le_ofReal (levelCurrentGreen_mono ω (hBc ω) (hw ω) z t - (Nat.cast_nonneg m) (Nat.cast_le.mpr hmn)) - have hlim : ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => ENNReal.ofReal (levelCurrentGreen κ B γ G n t z ω)) - atTop (𝓝 (ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) - (γ ω) (G ω) t z))) := by - filter_upwards [] with ω - apply tendsto_const_nhds.congr' - exact (levelCurrentGreen_eventually_eq ω (hBc ω) (hw ω) z t (fun n : ℕ => (n : ℝ)) - (fun n => Nat.cast_nonneg n) tendsto_natCast_atTop_atTop).mono (fun _ h => congrArg ENNReal.ofReal h.symm) - exact le_of_tendsto (lintegral_tendsto_of_tendsto_of_monotone (fun n => (hm n).aemeasurable) hmono hlim) - (Eventually.of_forall (fun n => levelCurrentGreen_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw z hz - n (Nat.cast_nonneg n) t)) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximationKernel +import OAI.Probability.SLE.Multipoint.MultipointRadiusStatus + +namespace OAI + +/-! As the radius level tends to infinity, the remaining finite-level +kernel increases to the killed Green potential. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +lemma levelCurrentGreen_mono + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hBc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (t : ℝ≥0) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v) : + levelCurrentGreen κ B γ G u t z ω ≤ levelCurrentGreen κ B γ G v t z ω := by + have hτ := radiusLevelTime_mono (hBc.const_mul _) hw z hu huv + unfold levelCurrentGreen + split_ifs with h₁ h₂ + · exact le_rfl + · exact False.elim (h₂ (h₁.trans_le hτ)) + · exact greenDensity_nonneg hw κ t z + · exact le_rfl + +lemma levelCurrentGreen_eventually_eq + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hBc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (t : ℝ≥0) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) + (hlim : Tendsto u atTop atTop) : + ∀ᶠ n in atTop, levelCurrentGreen κ B γ G (u n) t z ω = + greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) t z := by + by_cases hz : z ∈ survivingDomain (γ ω) t + · filter_upwards [hlim.eventually (eventually_gt_atTop + (logRadiusClock (Real.log z.im) (logLoewnerCoordinates (G ω) z t)))] with n hn + have hτ : (t : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z (u n) := by + apply lt_of_not_ge + rw [radiusLevelTime_le_iff_clock (hBc.const_mul _) hw z (u n) (hu n) t hz] + exact not_le.mpr hn + exact ite_eq_left hτ + · filter_upwards [] with n + unfold levelCurrentGreen greenDensity + simp only [ite_eq_right hz,ite_self] + +theorem levelCurrentGreen_mean_le + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun s => B s ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) : + (∫⁻ ω, ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω) ∂P) ≤ + ENNReal.ofReal (initialGreen κ z) := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + have he := levelRemainderGreen_setLIntegral hB hBm hBc hB0 hκ hκ8 γ G F + (ae_of_all _ hw) z hz u hu t univ MeasurableSet.univ + simp only [Measure.restrict_univ] at he + rw [← he] + calc + _ ≤ ∫⁻ ω, ENNReal.ofReal (stoppedGreen κ (fun s => Real.sqrt κ * B s ω) + (γ ω) (G ω) z u) ∂P := by + apply lintegral_mono + intro ω + unfold levelRemainderGreen + dsimp only + split_ifs + · exact le_rfl + · rw [ENNReal.ofReal_zero] + positivity + _ = _ := by + obtain ⟨hi,he⟩ := finite_stopped_green_regular hB hBm hBc hB0 κ hκ hκ8 γ G F + (ae_of_all _ hw) z hz u hu + rw [← ofReal_integral_eq_lintegral_ofReal hi + (ae_of_all _ (fun ω => stoppedGreen_nonneg (hw ω) κ z u)),he] + +theorem greenDensity_mean_le + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun s => B s ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (t : ℝ≥0) : + (∫⁻ ω, ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) + (γ ω) (G ω) t z) ∂P) ≤ ENNReal.ofReal (initialGreen κ z) := by + have hm (n : ℕ) : Measurable (fun ω => ENNReal.ofReal (levelCurrentGreen κ B γ G n t z ω)) := by + have hj := (levelCurrentGreen_joint_measurable hBm hBc κ n (Nat.cast_nonneg n) γ G F hw t).ennreal_ofReal + have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const + have hh := hj.comp hp + simpa only [Function.comp_def] using hh + have hmono : ∀ᵐ ω ∂P, Monotone (fun n : ℕ => ENNReal.ofReal (levelCurrentGreen κ B γ G n t z ω)) := by + filter_upwards [] with ω + intro m n hmn + exact ENNReal.ofReal_le_ofReal (levelCurrentGreen_mono ω (hBc ω) (hw ω) z t + (Nat.cast_nonneg m) (Nat.cast_le.mpr hmn)) + have hlim : ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => ENNReal.ofReal (levelCurrentGreen κ B γ G n t z ω)) + atTop (𝓝 (ENNReal.ofReal (greenDensity κ (fun s => Real.sqrt κ * B s ω) + (γ ω) (G ω) t z))) := by + filter_upwards [] with ω + apply tendsto_const_nhds.congr' + exact (levelCurrentGreen_eventually_eq ω (hBc ω) (hw ω) z t (fun n : ℕ => (n : ℝ)) + (fun n => Nat.cast_nonneg n) tendsto_natCast_atTop_atTop).mono (fun _ h => congrArg ENNReal.ofReal h.symm) + exact le_of_tendsto (lintegral_tendsto_of_tendsto_of_monotone (fun n => (hm n).aemeasurable) hmono hlim) + (Eventually.of_forall (fun n => levelCurrentGreen_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw z hz + n (Nat.cast_nonneg n) t)) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelMeanMeasure.lean b/lean/OAI/Probability/SLE/Mass/LevelMeanMeasure.lean index b8e23585d020f67994337172667867ce68db91b9..16abc371ce2896ee9436ae8a3717e0178a4f2a93 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelMeanMeasure.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelMeanMeasure.lean @@ -1,94 +1,94 @@ -import OAI.Probability.SLE.Mass.HalfPlaneLevelMeasures -import OAI.Probability.SLE.Mass.LevelApproximationMean -import OAI.Probability.SLE.Mass.RadonKernelIntegral - -namespace OAI - -/-! The exact mean measure of every terminal level approximation. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -def initialGreenMeasure (κ : ℝ) : Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal (initialGreen κ z))).comap Subtype.val - -lemma initialGreenMeasure_finite_on_compacts (κ : ℝ) : - IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := by - constructor - intro K hK - let L := (Subtype.val : upperHalfPlane → ℂ) '' K - have hL : IsCompact L := hK.image continuous_subtype_val - have hLH : L ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property - obtain ⟨C,hC⟩ := hL.bddAbove_image ((initialGreen_continuousOn κ).mono hLH) - rw [initialGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hL.measurableSet] - apply lt_of_le_of_lt (setLIntegral_mono' hL.measurableSet (fun z hz => - ENNReal.ofReal_le_ofReal (hC ⟨z,hz,rfl⟩))) - rw [setLIntegral_const] - exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top hL.measure_lt_top - -lemma halfPlaneLevelMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0∞) : Measurable (halfPlaneLevelMeasure κ B γ G u t) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet] - exact levelApproximation_measurable_apply hm hBc κ u hu γ G F hw t hAI - -theorem halfPlaneLevelMeasure_terminal_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) : - P.bind (halfPlaneLevelMeasure κ B γ G u ∞) = initialGreenMeasure κ := by - apply Measure.ext - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by - rintro _ ⟨x,_,rfl⟩; exact x.property - rw [Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw ∞).aemeasurable] - simp only [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - levelApproximation_apply_top κ B γ G u _ hAI] - rw [initialGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact levelMass_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw _ hAI hAH u hu - -theorem halfPlaneLevelMeasure_terminal_test_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (f : C_c(upperHalfPlane,ℝ)) : - (∫ ω, (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G u ∞ ω) ∂P) = - ∫ z, f z ∂initialGreenMeasure κ := by - have he := halfPlaneLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu - rw [← he] - apply integral_bind_of_integrable_sections P _ - (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw ∞) f f.continuous.measurable - · intro ω - let := halfPlaneLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u ∞ - exact f.integrable - · rw [he] - let := initialGreenMeasure_finite_on_compacts κ - exact f.integrable - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.HalfPlaneLevelMeasures +import OAI.Probability.SLE.Mass.LevelApproximationMean +import OAI.Probability.SLE.Mass.RadonKernelIntegral + +namespace OAI + +/-! The exact mean measure of every terminal level approximation. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +def initialGreenMeasure (κ : ℝ) : Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal (initialGreen κ z))).comap Subtype.val + +lemma initialGreenMeasure_finite_on_compacts (κ : ℝ) : + IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := by + constructor + intro K hK + let L := (Subtype.val : upperHalfPlane → ℂ) '' K + have hL : IsCompact L := hK.image continuous_subtype_val + have hLH : L ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property + obtain ⟨C,hC⟩ := hL.bddAbove_image ((initialGreen_continuousOn κ).mono hLH) + rw [initialGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hL.measurableSet] + apply lt_of_le_of_lt (setLIntegral_mono' hL.measurableSet (fun z hz => + ENNReal.ofReal_le_ofReal (hC ⟨z,hz,rfl⟩))) + rw [setLIntegral_const] + exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top hL.measure_lt_top + +lemma halfPlaneLevelMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0∞) : Measurable (halfPlaneLevelMeasure κ B γ G u t) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet] + exact levelApproximation_measurable_apply hm hBc κ u hu γ G F hw t hAI + +theorem halfPlaneLevelMeasure_terminal_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) : + P.bind (halfPlaneLevelMeasure κ B γ G u ∞) = initialGreenMeasure κ := by + apply Measure.ext + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by + rintro _ ⟨x,_,rfl⟩; exact x.property + rw [Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw ∞).aemeasurable] + simp only [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + levelApproximation_apply_top κ B γ G u _ hAI] + rw [initialGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact levelMass_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw _ hAI hAH u hu + +theorem halfPlaneLevelMeasure_terminal_test_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (f : C_c(upperHalfPlane,ℝ)) : + (∫ ω, (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G u ∞ ω) ∂P) = + ∫ z, f z ∂initialGreenMeasure κ := by + have he := halfPlaneLevelMeasure_terminal_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu + rw [← he] + apply integral_bind_of_integrable_sections P _ + (halfPlaneLevelMeasure_measurable hm hBc κ u hu γ G F hw ∞) f f.continuous.measurable + · intro ω + let := halfPlaneLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u ∞ + exact f.integrable + · rw [he] + let := initialGreenMeasure_finite_on_compacts κ + exact f.integrable + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonExtraction.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonExtraction.lean index a5af8df967af92a4ad7163d658d002d6f09f68ce..6deeff9125423e2d47006033e72e918e42b107e7 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonExtraction.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonExtraction.lean @@ -1,188 +1,188 @@ -import OAI.Probability.SLE.Mass.HalfPlaneLevelMeasures -import OAI.Probability.SLE.Mass.RadonTestFamily -import OAI.Probability.SLE.Mass.L2MeanConvergence - -namespace OAI - -/-! One sequence of convex level measures converges vaguely at every -prescribed countable time, on one event of probability one. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -def convexLevelMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (u : ℕ → ℝ) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0∞) (ω : Ω) : - Measure upperHalfPlane := - ∑ i ∈ s n, ENNReal.ofReal (w n i) • halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω - -lemma convexLevelMeasure_finite_on_compacts - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (ω : Ω) (hBc : Continuous (fun t => B t ω)) - (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0∞) : - IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w n t ω) := by - constructor - intro K hK - rw [convexLevelMeasure,Measure.finsetSum_apply] - apply ENNReal.sum_lt_top.mpr - intro i hi - rw [Measure.smul_apply,smul_eq_mul] - let := halfPlaneLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω hBc hw (u (n+i)) t - exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top hK.measure_lt_top - -lemma integral_convexLevelMeasure - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (ω : Ω) (hBc : Continuous (fun t => B t ω)) - (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0∞) - (hw₀ : ∀ i ∈ s n, 0 ≤ w n i) (f : C_c(upperHalfPlane,ℝ)) : - (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) = - ∑ i ∈ s n, w n i * levelApproximationRealTest κ B γ G (u (n+i)) t (extendHalfPlaneTest f) ω := by - have hi (i : ℕ) : Integrable f - (ENNReal.ofReal (w n i) • halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) := by - let := halfPlaneLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω hBc hw (u (n+i)) t - let : IsFiniteMeasureOnCompacts - (ENNReal.ofReal (w n i) • halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) := - IsFiniteMeasureOnCompacts.smul - (halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) ENNReal.ofReal_ne_top - exact f.integrable - rw [convexLevelMeasure,integral_finsetSum_measure (fun i _ => hi i)] - apply Finset.sum_congr rfl - intro i his - rw [integral_smul_measure,ENNReal.toReal_ofReal (hw₀ i his),smul_eq_mul, - integral_halfPlaneLevelMeasure] - -theorem exists_common_halfPlane_test_limits_with_pairings - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (f : ℕ → C_c(upperHalfPlane,ℝ)) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) - (t : ℕ → ℝ≥0∞) : - ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Tendsto (fun n => ∫ ω, (∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) - atTop (𝓝 (∫ ω, x j ω ∂P))) ∧ - (∀ j (a : Ω → ℝ), MemLp a 2 P → - Tendsto (fun n => ∫ ω, a ω * (∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) - atTop (𝓝 (∫ ω, a ω * x j ω ∂P))) ∧ - (∀ᵐ ω ∂P, ∀ j, Tendsto (fun n => ∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (x j ω))) := by - let φ (j : ℕ) := extendHalfPlaneTest (f j) - let K (j : ℕ) := (Subtype.val : upperHalfPlane → ℂ) '' tsupport (f j) - have hK (j : ℕ) : IsCompact (K j) := (f j).hasCompactSupport.image continuous_subtype_val - have hKH (j : ℕ) : K j ⊆ upperHalfPlane := by - rintro _ ⟨x,_,rfl⟩ - exact x.property - have hφ (j : ℕ) : Measurable (φ j) := extendHalfPlaneTest_measurable _ (f j).continuous.measurable - have info (j : ℕ) := uniform_signed_levelTest_L2 hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - (K j) (hK j) (hKH j) (φ j) (hφ j) ‖(f j).toBoundedContinuousFunction‖ - (fun z _ => extendHalfPlaneTest_norm_le _ z) (extendHalfPlaneTest_support (f j)) - let A (n j : ℕ) := levelApproximationRealTest κ B γ G (u n) (t j) (φ j) - have hLp (n j : ℕ) : MemLp (A n j) 2 P := (info j).1 (u n) (hu n) (t j) - have hbound (j : ℕ) : ∃ D : ℝ, ∀ n, ‖(hLp n j).toLp (A n j)‖ ≤ D := by - obtain ⟨D,hD⟩ := (info j).2 - exact ⟨D,fun n => hD (u n) (hu n) (t j) (hLp n j)⟩ - obtain ⟨x,s,w,hw₀,hw₁,hL2,hc⟩ := exists_randomVariable_convex_tail_limit A hLp hbound - refine ⟨x,s,w,hw₀,hw₁,?_,?_,?_⟩ - · intro j - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - have hmean := weighted_randomVariable_integral_tendsto P (fun n => A n j) - (fun n => hLp n j) s w (x j) (hL2 j) - convert hmean using 1 - funext n - apply integral_congr_ae - filter_upwards [] with ω - exact integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) - (hw₀ n) (f j) - · intro j a ha - have hmean := weighted_randomVariable_pairing_tendsto P (fun n => A n j) - (fun n => hLp n j) s w (x j) (hL2 j) a ha - convert hmean using 1 - funext n - apply integral_congr_ae - filter_upwards [] with ω - apply congrArg (fun v : ℝ => a ω * v) - exact integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) - (hw₀ n) (f j) - · filter_upwards [hc] with ω hω - intro j - simpa only [integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w _ _ - (hw₀ _) (f j),A,φ] using hω j - -theorem exists_common_halfPlane_test_limits - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (f : ℕ → C_c(upperHalfPlane,ℝ)) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) - (t : ℕ → ℝ≥0∞) : - ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Tendsto (fun n => ∫ ω, (∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) - atTop (𝓝 (∫ ω, x j ω ∂P))) ∧ - (∀ᵐ ω ∂P, ∀ j, Tendsto (fun n => ∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (x j ω))) := by - obtain ⟨x,s,w,hw₀,hw₁,hm,_,ha⟩ := exists_common_halfPlane_test_limits_with_pairings - hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth f u hu t - exact ⟨x,s,w,hw₀,hw₁,hm,ha⟩ - -/-- The same convex level measures have one Radon limit at all times in -an arbitrary countable list. The spatial determining family is constructed, -and is not an additional input. -/ -theorem exists_common_level_radon_limits - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : - ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ᵐ ω ∂P, ∃ ν : ℕ → Measure upperHalfPlane, - (∀ j, (ν j).Regular ∧ IsFiniteMeasureOnCompacts (ν j)) ∧ - ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν j))) := by - let D := localTestFamily upperHalfPlane - obtain ⟨x,s,w,hw₀,hw₁,_,hconv⟩ := exists_common_halfPlane_test_limits hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth (fun j => D.test (Nat.unpair j).2) u hu (fun j => t (Nat.unpair j).1) - refine ⟨s,w,hw₀,hw₁,?_⟩ - filter_upwards [hconv] with ω hω - have hex (j : ℕ) : ∃ ν : Measure upperHalfPlane, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ - ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν)) := by - let (n : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w n (t j) ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) - apply D.exists_radon_limit - intro k - refine ⟨x (Nat.pair j k) ω,?_⟩ - simpa only [Nat.unpair_pair] using hω (Nat.pair j k) - choose ν hν using hex - exact ⟨ν,fun j => ⟨(hν j).1,(hν j).2.1⟩,fun j => (hν j).2.2⟩ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.HalfPlaneLevelMeasures +import OAI.Probability.SLE.Mass.RadonTestFamily +import OAI.Probability.SLE.Mass.L2MeanConvergence + +namespace OAI + +/-! One sequence of convex level measures converges vaguely at every +prescribed countable time, on one event of probability one. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +def convexLevelMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (u : ℕ → ℝ) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0∞) (ω : Ω) : + Measure upperHalfPlane := + ∑ i ∈ s n, ENNReal.ofReal (w n i) • halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω + +lemma convexLevelMeasure_finite_on_compacts + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (ω : Ω) (hBc : Continuous (fun t => B t ω)) + (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0∞) : + IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w n t ω) := by + constructor + intro K hK + rw [convexLevelMeasure,Measure.finsetSum_apply] + apply ENNReal.sum_lt_top.mpr + intro i hi + rw [Measure.smul_apply,smul_eq_mul] + let := halfPlaneLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω hBc hw (u (n+i)) t + exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top hK.measure_lt_top + +lemma integral_convexLevelMeasure + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (ω : Ω) (hBc : Continuous (fun t => B t ω)) + (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (t : ℝ≥0∞) + (hw₀ : ∀ i ∈ s n, 0 ≤ w n i) (f : C_c(upperHalfPlane,ℝ)) : + (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) = + ∑ i ∈ s n, w n i * levelApproximationRealTest κ B γ G (u (n+i)) t (extendHalfPlaneTest f) ω := by + have hi (i : ℕ) : Integrable f + (ENNReal.ofReal (w n i) • halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) := by + let := halfPlaneLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω hBc hw (u (n+i)) t + let : IsFiniteMeasureOnCompacts + (ENNReal.ofReal (w n i) • halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) := + IsFiniteMeasureOnCompacts.smul + (halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) ENNReal.ofReal_ne_top + exact f.integrable + rw [convexLevelMeasure,integral_finsetSum_measure (fun i _ => hi i)] + apply Finset.sum_congr rfl + intro i his + rw [integral_smul_measure,ENNReal.toReal_ofReal (hw₀ i his),smul_eq_mul, + integral_halfPlaneLevelMeasure] + +theorem exists_common_halfPlane_test_limits_with_pairings + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (f : ℕ → C_c(upperHalfPlane,ℝ)) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) + (t : ℕ → ℝ≥0∞) : + ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Tendsto (fun n => ∫ ω, (∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) + atTop (𝓝 (∫ ω, x j ω ∂P))) ∧ + (∀ j (a : Ω → ℝ), MemLp a 2 P → + Tendsto (fun n => ∫ ω, a ω * (∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) + atTop (𝓝 (∫ ω, a ω * x j ω ∂P))) ∧ + (∀ᵐ ω ∂P, ∀ j, Tendsto (fun n => ∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (x j ω))) := by + let φ (j : ℕ) := extendHalfPlaneTest (f j) + let K (j : ℕ) := (Subtype.val : upperHalfPlane → ℂ) '' tsupport (f j) + have hK (j : ℕ) : IsCompact (K j) := (f j).hasCompactSupport.image continuous_subtype_val + have hKH (j : ℕ) : K j ⊆ upperHalfPlane := by + rintro _ ⟨x,_,rfl⟩ + exact x.property + have hφ (j : ℕ) : Measurable (φ j) := extendHalfPlaneTest_measurable _ (f j).continuous.measurable + have info (j : ℕ) := uniform_signed_levelTest_L2 hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + (K j) (hK j) (hKH j) (φ j) (hφ j) ‖(f j).toBoundedContinuousFunction‖ + (fun z _ => extendHalfPlaneTest_norm_le _ z) (extendHalfPlaneTest_support (f j)) + let A (n j : ℕ) := levelApproximationRealTest κ B γ G (u n) (t j) (φ j) + have hLp (n j : ℕ) : MemLp (A n j) 2 P := (info j).1 (u n) (hu n) (t j) + have hbound (j : ℕ) : ∃ D : ℝ, ∀ n, ‖(hLp n j).toLp (A n j)‖ ≤ D := by + obtain ⟨D,hD⟩ := (info j).2 + exact ⟨D,fun n => hD (u n) (hu n) (t j) (hLp n j)⟩ + obtain ⟨x,s,w,hw₀,hw₁,hL2,hc⟩ := exists_randomVariable_convex_tail_limit A hLp hbound + refine ⟨x,s,w,hw₀,hw₁,?_,?_,?_⟩ + · intro j + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + have hmean := weighted_randomVariable_integral_tendsto P (fun n => A n j) + (fun n => hLp n j) s w (x j) (hL2 j) + convert hmean using 1 + funext n + apply integral_congr_ae + filter_upwards [] with ω + exact integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) + (hw₀ n) (f j) + · intro j a ha + have hmean := weighted_randomVariable_pairing_tendsto P (fun n => A n j) + (fun n => hLp n j) s w (x j) (hL2 j) a ha + convert hmean using 1 + funext n + apply integral_congr_ae + filter_upwards [] with ω + apply congrArg (fun v : ℝ => a ω * v) + exact integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) + (hw₀ n) (f j) + · filter_upwards [hc] with ω hω + intro j + simpa only [integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w _ _ + (hw₀ _) (f j),A,φ] using hω j + +theorem exists_common_halfPlane_test_limits + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (f : ℕ → C_c(upperHalfPlane,ℝ)) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) + (t : ℕ → ℝ≥0∞) : + ∃ x : ℕ → Lp ℝ 2 P, ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Tendsto (fun n => ∫ ω, (∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) + atTop (𝓝 (∫ ω, x j ω ∂P))) ∧ + (∀ᵐ ω ∂P, ∀ j, Tendsto (fun n => ∫ z, f j z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (x j ω))) := by + obtain ⟨x,s,w,hw₀,hw₁,hm,_,ha⟩ := exists_common_halfPlane_test_limits_with_pairings + hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth f u hu t + exact ⟨x,s,w,hw₀,hw₁,hm,ha⟩ + +/-- The same convex level measures have one Radon limit at all times in +an arbitrary countable list. The spatial determining family is constructed, +and is not an additional input. -/ +theorem exists_common_level_radon_limits + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : + ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ᵐ ω ∂P, ∃ ν : ℕ → Measure upperHalfPlane, + (∀ j, (ν j).Regular ∧ IsFiniteMeasureOnCompacts (ν j)) ∧ + ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν j))) := by + let D := localTestFamily upperHalfPlane + obtain ⟨x,s,w,hw₀,hw₁,_,hconv⟩ := exists_common_halfPlane_test_limits hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth (fun j => D.test (Nat.unpair j).2) u hu (fun j => t (Nat.unpair j).1) + refine ⟨s,w,hw₀,hw₁,?_⟩ + filter_upwards [hconv] with ω hω + have hex (j : ℕ) : ∃ ν : Measure upperHalfPlane, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ + ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν)) := by + let (n : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w n (t j) ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) + apply D.exists_radon_limit + intro k + refine ⟨x (Nat.pair j k) ω,?_⟩ + simpa only [Nat.unpair_pair] using hω (Nat.pair j k) + choose ν hν using hex + exact ⟨ν,fun j => ⟨(hν j).1,(hν j).2.1⟩,fun j => (hν j).2.2⟩ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonKernel.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonKernel.lean index 601fb8b18cf22a8567d336a2596165c954f038cc..b348aebfda0cb6dfd4d684b007fb178018c7546c 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonKernel.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonKernel.lean @@ -1,129 +1,129 @@ -import OAI.Probability.SLE.Mass.ConvexCurrentMean -import OAI.Probability.SLE.Mass.LevelRadonMoments -import OAI.Probability.SLE.Mass.RadonTestDetermines - -namespace OAI - -/-! Passage of the conditional Green kernel to the actual common Radon -limits. The hypotheses on the limits are exactly those furnished by the -common convex extraction, including its retained L² pairings. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma level_radon_mean_finite_on_compacts - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) - (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) (E : Set Ω) : - IsFiniteMeasureOnCompacts ((P.restrict E).bind ν) := by - constructor - intro K hK - rw [Measure.bind_apply hK.measurableSet hνm.aemeasurable] - apply lt_of_le_of_lt (lintegral_mono' Measure.restrict_le_self (le_refl _)) - simpa only [pow_one] using level_radon_limit_compact_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth u hu s w hw₀ hw₁ t ν hν hlim K hK 1 - -theorem level_radon_conditional_kernel - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) (huLim : Tendsto u atTop atTop) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0) (ν : Fin 2 → Ω → Measure upperHalfPlane) - (hνm : ∀ j, Measurable (ν j)) - (hν : ∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) - (hlim : ∀ j, ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n - (if j = 0 then (t : ℝ≥0∞) else ∞) ω) atTop (𝓝 (∫ z, f z ∂ν j ω))) - (hpair : ∀ j k (a : Ω → ℝ), MemLp a 2 P → - Tendsto (fun n => ∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z - ∂convexLevelMeasure κ B γ G u s w n (if j = 0 then (t : ℝ≥0∞) else ∞) ω) ∂P) - atTop (𝓝 (∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z ∂ν j ω) ∂P))) - (E : Set Ω) - (hE : MeasurableSet[Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : - (P.restrict E).bind (ν 1) = (P.restrict E).bind (ν 0) + - (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let τ (j : Fin 2) : ℝ≥0∞ := if j = 0 then t else ∞ - let (j : Fin 2) (ω : Ω) : IsFiniteMeasureOnCompacts (ν j ω) := (hν j ω).2 - let (j : Fin 2) : IsFiniteMeasureOnCompacts ((P.restrict E).bind (ν j)) := - level_radon_mean_finite_on_compacts hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - u hu s w hw₀ hw₁ (τ j) (ν j) (hνm j) (fun _ => inferInstance) (hlim j) E - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hgreen := halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) := - ⟨fun K hK => lt_of_le_of_lt (hgreen K) hK.measure_lt_top⟩ - let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (ν 0) + - (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) := by - constructor - intro K hK - rw [Measure.add_apply] - exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩ - have hEg : MeasurableSet E := (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)).le t E hE - apply (localTestFamily upperHalfPlane).measure_eq_of_test_integrals - intro k - let f := (localTestFamily upperHalfPlane).test k - have hc (j : Fin 2) : Tendsto (fun n => ∫ z, f z - ∂(P.restrict E).bind (convexLevelMeasure κ B γ G u s w n (τ j))) atTop - (𝓝 (∫ z, f z ∂(P.restrict E).bind (ν j))) := by - apply mean_test_tendsto_of_L2_pairings P _ (ν j) - (fun n => convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n (τ j)) - (hνm j) f f.continuous.measurable - · intro n ω - let := convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (τ j) - exact f.integrable - · exact fun _ => f.integrable - · exact hEg - · intro n - exact f.integrable.mono_measure (convexLevelMeasure_mean_le hB hBm hBc hB0 hκ hκ8 - γ G F hw u hu s w hw₀ hw₁ n (τ j) E) - · exact f.integrable - · exact hpair j k - have ht := hc 0 - have hi := hc 1 - have hg := convexCurrentMean_test_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw - u hu hmono huLim s w hw₀ hw₁ t E f - simp [τ] at ht hi - have he (n : ℕ) : (∫ z, f z ∂(P.restrict E).bind (convexLevelMeasure κ B γ G u s w n ∞)) = - (∫ z, f z ∂(P.restrict E).bind (convexLevelMeasure κ B γ G u s w n t)) + - ∫ z, f z ∂convexCurrentMean (P.restrict E) κ B γ G u s w n t := by - rw [convexLevelMeasure_conditional_split hB hBm hBc hB0 hκ hκ8 γ G F hw u hu s w n t E hE] - exact integral_add_measure - (f.integrable.mono_measure (convexLevelMeasure_mean_le hB hBm hBc hB0 hκ hκ8 - γ G F hw u hu s w hw₀ hw₁ n t E)) - (f.integrable.mono_measure (convexCurrentMean_le hB hBm hBc hB0 hκ hκ8 - γ G F hw u hu s w hw₀ hw₁ n t E)) - rw [integral_add_measure f.integrable f.integrable] - apply tendsto_nhds_unique hi - exact (ht.add hg).congr' (Eventually.of_forall (fun n => (he n).symm)) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.ConvexCurrentMean +import OAI.Probability.SLE.Mass.LevelRadonMoments +import OAI.Probability.SLE.Mass.RadonTestDetermines + +namespace OAI + +/-! Passage of the conditional Green kernel to the actual common Radon +limits. The hypotheses on the limits are exactly those furnished by the +common convex extraction, including its retained L² pairings. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma level_radon_mean_finite_on_compacts + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) + (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) (E : Set Ω) : + IsFiniteMeasureOnCompacts ((P.restrict E).bind ν) := by + constructor + intro K hK + rw [Measure.bind_apply hK.measurableSet hνm.aemeasurable] + apply lt_of_le_of_lt (lintegral_mono' Measure.restrict_le_self (le_refl _)) + simpa only [pow_one] using level_radon_limit_compact_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth u hu s w hw₀ hw₁ t ν hν hlim K hK 1 + +theorem level_radon_conditional_kernel + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) (huLim : Tendsto u atTop atTop) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0) (ν : Fin 2 → Ω → Measure upperHalfPlane) + (hνm : ∀ j, Measurable (ν j)) + (hν : ∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) + (hlim : ∀ j, ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n + (if j = 0 then (t : ℝ≥0∞) else ∞) ω) atTop (𝓝 (∫ z, f z ∂ν j ω))) + (hpair : ∀ j k (a : Ω → ℝ), MemLp a 2 P → + Tendsto (fun n => ∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z + ∂convexLevelMeasure κ B γ G u s w n (if j = 0 then (t : ℝ≥0∞) else ∞) ω) ∂P) + atTop (𝓝 (∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z ∂ν j ω) ∂P))) + (E : Set Ω) + (hE : MeasurableSet[Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : + (P.restrict E).bind (ν 1) = (P.restrict E).bind (ν 0) + + (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t) := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let τ (j : Fin 2) : ℝ≥0∞ := if j = 0 then t else ∞ + let (j : Fin 2) (ω : Ω) : IsFiniteMeasureOnCompacts (ν j ω) := (hν j ω).2 + let (j : Fin 2) : IsFiniteMeasureOnCompacts ((P.restrict E).bind (ν j)) := + level_radon_mean_finite_on_compacts hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + u hu s w hw₀ hw₁ (τ j) (ν j) (hνm j) (fun _ => inferInstance) (hlim j) E + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hgreen := halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t E + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) := + ⟨fun K hK => lt_of_le_of_lt (hgreen K) hK.measure_lt_top⟩ + let : IsFiniteMeasureOnCompacts ((P.restrict E).bind (ν 0) + + (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) := by + constructor + intro K hK + rw [Measure.add_apply] + exact ENNReal.add_lt_top.mpr ⟨hK.measure_lt_top,hK.measure_lt_top⟩ + have hEg : MeasurableSet E := (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)).le t E hE + apply (localTestFamily upperHalfPlane).measure_eq_of_test_integrals + intro k + let f := (localTestFamily upperHalfPlane).test k + have hc (j : Fin 2) : Tendsto (fun n => ∫ z, f z + ∂(P.restrict E).bind (convexLevelMeasure κ B γ G u s w n (τ j))) atTop + (𝓝 (∫ z, f z ∂(P.restrict E).bind (ν j))) := by + apply mean_test_tendsto_of_L2_pairings P _ (ν j) + (fun n => convexLevelMeasure_measurable hm hBc κ γ G F hw u hu s w n (τ j)) + (hνm j) f f.continuous.measurable + · intro n ω + let := convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (τ j) + exact f.integrable + · exact fun _ => f.integrable + · exact hEg + · intro n + exact f.integrable.mono_measure (convexLevelMeasure_mean_le hB hBm hBc hB0 hκ hκ8 + γ G F hw u hu s w hw₀ hw₁ n (τ j) E) + · exact f.integrable + · exact hpair j k + have ht := hc 0 + have hi := hc 1 + have hg := convexCurrentMean_test_tendsto hB hBm hBc hB0 hκ hκ8 γ G F hw + u hu hmono huLim s w hw₀ hw₁ t E f + simp [τ] at ht hi + have he (n : ℕ) : (∫ z, f z ∂(P.restrict E).bind (convexLevelMeasure κ B γ G u s w n ∞)) = + (∫ z, f z ∂(P.restrict E).bind (convexLevelMeasure κ B γ G u s w n t)) + + ∫ z, f z ∂convexCurrentMean (P.restrict E) κ B γ G u s w n t := by + rw [convexLevelMeasure_conditional_split hB hBm hBc hB0 hκ hκ8 γ G F hw u hu s w n t E hE] + exact integral_add_measure + (f.integrable.mono_measure (convexLevelMeasure_mean_le hB hBm hBc hB0 hκ hκ8 + γ G F hw u hu s w hw₀ hw₁ n t E)) + (f.integrable.mono_measure (convexCurrentMean_le hB hBm hBc hB0 hκ hκ8 + γ G F hw u hu s w hw₀ hw₁ n t E)) + rw [integral_add_measure f.integrable f.integrable] + apply tendsto_nhds_unique hi + exact (ht.add hg).congr' (Eventually.of_forall (fun n => (he n).symm)) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonMean.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonMean.lean index af299573c336a25210fa75e770a8a9f28f9274f8..8e84c49b823d7e2613666e80786479586905900c 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonMean.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonMean.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Mass.ConvexMeanMeasure -import OAI.Probability.SLE.Mass.LevelRadonMoments -import OAI.Probability.SLE.Mass.RadonTestDetermines - -namespace OAI - -/-! The actual terminal Radon limit has exactly the initial Green mean -measure. Local moment bounds justify signed compact tests of that mean. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem level_radon_terminal_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) - (hν : ∀ ω, (ν ω).Regular) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) - (hmeanlim : ∀ k, Tendsto (fun n => ∫ ω, (∫ z, (localTestFamily upperHalfPlane).test k z - ∂convexLevelMeasure κ B γ G u s w n ∞ ω) ∂P) - atTop (𝓝 (∫ ω, (∫ z, (localTestFamily upperHalfPlane).test k z ∂ν ω) ∂P))) : - P.bind ν = initialGreenMeasure κ := by - let (ω : Ω) : (ν ω).Regular := hν ω - let : IsFiniteMeasureOnCompacts (P.bind ν) := by - constructor - intro K hK - rw [Measure.bind_apply hK.measurableSet hνm.aemeasurable] - simpa only [pow_one] using level_radon_limit_compact_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth u hu s w hw₀ hw₁ ∞ ν (fun _ => inferInstance) hlim K hK 1 - let := initialGreenMeasure_finite_on_compacts κ - apply (localTestFamily upperHalfPlane).measure_eq_of_test_integrals - intro k - let f := (localTestFamily upperHalfPlane).test k - have hmconst (n : ℕ) : (∫ ω, (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) ∂P) = - ∫ z, f z ∂initialGreenMeasure κ := - (convexLevelMeasure_terminal_test_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu s w hw₀ hw₁ n f).2 - have he : (∫ ω, (∫ z, f z ∂ν ω) ∂P) = ∫ z, f z ∂initialGreenMeasure κ := by - apply tendsto_nhds_unique (hmeanlim k) - apply tendsto_const_nhds.congr' - exact Eventually.of_forall (fun n => (hmconst n).symm) - rw [← he] - exact (integral_bind_of_integrable_sections P ν hνm f f.continuous.measurable - (fun _ => f.integrable) f.integrable).symm - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.ConvexMeanMeasure +import OAI.Probability.SLE.Mass.LevelRadonMoments +import OAI.Probability.SLE.Mass.RadonTestDetermines + +namespace OAI + +/-! The actual terminal Radon limit has exactly the initial Green mean +measure. Local moment bounds justify signed compact tests of that mean. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem level_radon_terminal_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) + (hν : ∀ ω, (ν ω).Regular) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) + (hmeanlim : ∀ k, Tendsto (fun n => ∫ ω, (∫ z, (localTestFamily upperHalfPlane).test k z + ∂convexLevelMeasure κ B γ G u s w n ∞ ω) ∂P) + atTop (𝓝 (∫ ω, (∫ z, (localTestFamily upperHalfPlane).test k z ∂ν ω) ∂P))) : + P.bind ν = initialGreenMeasure κ := by + let (ω : Ω) : (ν ω).Regular := hν ω + let : IsFiniteMeasureOnCompacts (P.bind ν) := by + constructor + intro K hK + rw [Measure.bind_apply hK.measurableSet hνm.aemeasurable] + simpa only [pow_one] using level_radon_limit_compact_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth u hu s w hw₀ hw₁ ∞ ν (fun _ => inferInstance) hlim K hK 1 + let := initialGreenMeasure_finite_on_compacts κ + apply (localTestFamily upperHalfPlane).measure_eq_of_test_integrals + intro k + let f := (localTestFamily upperHalfPlane).test k + have hmconst (n : ℕ) : (∫ ω, (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n ∞ ω) ∂P) = + ∫ z, f z ∂initialGreenMeasure κ := + (convexLevelMeasure_terminal_test_mean hB hm hBm hBc hB0 hκ hκ8 γ G F hw u hu s w hw₀ hw₁ n f).2 + have he : (∫ ω, (∫ z, f z ∂ν ω) ∂P) = ∫ z, f z ∂initialGreenMeasure κ := by + apply tendsto_nhds_unique (hmeanlim k) + apply tendsto_const_nhds.congr' + exact Eventually.of_forall (fun n => (hmconst n).symm) + rw [← he] + exact (integral_bind_of_integrable_sections P ν hνm f f.continuous.measurable + (fun _ => f.integrable) f.integrable).symm + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonMoments.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonMoments.lean index eaa8298dc9c62afbccfb8ced406a8f5d16bd7e91..60ec5267e4990670b4156cd501cb843ea3e92b74 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonMoments.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonMoments.lean @@ -1,166 +1,166 @@ -import OAI.Probability.SLE.Mass.LevelRandomRadon -import OAI.Probability.SLE.Mass.ConvexMomentLimits - -namespace OAI - -/-! Every compact moment bound passes from the actual level measures to -any of their common convex Radon limits. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma extendHalfPlaneTest_nonneg (f : upperHalfPlane → ℝ) (hf : ∀ z, 0 ≤ f z) : - ∀ z, 0 ≤ extendHalfPlaneTest f z := by - intro z - by_cases hz : z ∈ upperHalfPlane - · exact (extendHalfPlaneTest_apply f ⟨z,hz⟩).symm ▸ hf ⟨z,hz⟩ - · rw [extendHalfPlaneTest_of_not_mem f z hz] - -lemma ofReal_integral_halfPlaneLevelMeasure - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (ω : Ω) (hBc : Continuous (fun t => B t ω)) - (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (t : ℝ≥0∞) (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : - ENNReal.ofReal (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G u t ω) = - levelApproximationTest κ B γ G u t (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) ω := by - let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f - have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val - have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property - have hfin := (levelApproximation_apply_le κ B γ G u t ω hK.measurableSet).trans_lt - (levelMass_lt_top_of_compact hκ hκ8 γ G F ω hBc hw K hK hKH u) - rw [integral_halfPlaneLevelMeasure] - exact ofReal_integral_eq_lintegral_ofReal - (integrable_of_bounded_support _ K hK.measurableSet hfin _ - (extendHalfPlaneTest_measurable f f.continuous.measurable) _ - (fun z _ => extendHalfPlaneTest_norm_le f z) (extendHalfPlaneTest_support f)) - (Eventually.of_forall (extendHalfPlaneTest_nonneg f hf)) - -theorem uniform_halfPlaneLevelMeasure_test_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) : - ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, - (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G u t ω)^m ∂P) ≤ M := by - let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f - have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val - have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property - obtain ⟨M,hM,hbound⟩ := uniform_levelApproximationTest_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth K hK hKH (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) - (ENNReal.ofReal ‖f.toBoundedContinuousFunction‖) ENNReal.ofReal_lt_top - (fun z _ => ENNReal.ofReal_le_ofReal ((le_abs_self _).trans (extendHalfPlaneTest_norm_le f z))) - (fun z hz => by rw [extendHalfPlaneTest_support f z hz,ENNReal.ofReal_zero]) m - refine ⟨M,hM,fun u hu t => ?_⟩ - simpa only [ofReal_integral_halfPlaneLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) u t f hf] - using hbound u hu t - -theorem uniform_convexLevelMeasure_test_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) : - ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ n (t : ℝ≥0∞), - (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω)^m ∂P) ≤ M := by - obtain ⟨M,hM,hbound⟩ := uniform_halfPlaneLevelMeasure_test_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth f hf m - refine ⟨M,hM,fun n t => ?_⟩ - have hb := lintegral_pow_convex_sum_le P (s n) (w n) - (fun i ω => ∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) - (hw₀ n) (hw₁ n) (fun i _ ω => integral_nonneg hf) - (fun i _ => by - simp_rw [integral_halfPlaneLevelMeasure] - exact levelApproximationRealTest_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t - _ (extendHalfPlaneTest_measurable f f.continuous.measurable)) - m M (fun i _ => hbound (u (n+i)) (hu _) t) - simpa only [integral_convexLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) u s w n t (hw₀ n) f, - integral_halfPlaneLevelMeasure] using hb - -theorem level_radon_limit_test_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) : - (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂ν ω)^m ∂P) < ⊤ := by - obtain ⟨M,hM,hbound⟩ := uniform_convexLevelMeasure_test_moments hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth u hu s w hw₀ hw₁ f hf m - apply lt_of_le_of_lt (lintegral_le_of_ae_tendsto_bound P _ _ (fun n => - ((convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n t f).ennreal_ofReal.pow_const m).aemeasurable) - ?_ M (fun n => hbound n t)) hM - filter_upwards [hlim] with ω hω - exact (ENNReal.continuous_pow m).continuousAt.tendsto.comp - ((ENNReal.continuous_ofReal.tendsto _).comp (hω f)) - -lemma measure_compact_le_nonneg_test (ν : Measure upperHalfPlane) - [IsFiniteMeasureOnCompacts ν] (K : Set upperHalfPlane) (hK : IsCompact K) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (hKf : ∀ z ∈ K, 1 ≤ f z) : - ν K ≤ ENNReal.ofReal (∫ z, f z ∂ν) := by - rw [ofReal_integral_eq_lintegral_ofReal f.integrable (Eventually.of_forall hf)] - calc - _ = ∫⁻ z, K.indicator (fun _ => (1 : ℝ≥0∞)) z ∂ν := by - rw [lintegral_indicator_const hK.measurableSet,one_mul] - _ ≤ _ := by - apply lintegral_mono - intro z - by_cases hz : z ∈ K - · simpa only [indicator_of_mem hz,ENNReal.one_le_ofReal] using hKf z hz - · simp only [indicator_of_notMem hz,zero_le] - -theorem level_radon_limit_compact_moments - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) - (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) - (K : Set upperHalfPlane) (hK : IsCompact K) (m : ℕ) : - (∫⁻ ω, (ν ω K)^m ∂P) < ⊤ := by - obtain ⟨f,hf,hKf⟩ := exists_compact_test_cutoff K hK - apply lt_of_le_of_lt (lintegral_mono (fun ω => ?_)) - (level_radon_limit_test_moments hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - u hu s w hw₀ hw₁ t ν hlim f hf m) - let := hν ω - exact pow_le_pow_left' (measure_compact_le_nonneg_test (ν ω) K hK f hf hKf) m - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelRandomRadon +import OAI.Probability.SLE.Mass.ConvexMomentLimits + +namespace OAI + +/-! Every compact moment bound passes from the actual level measures to +any of their common convex Radon limits. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma extendHalfPlaneTest_nonneg (f : upperHalfPlane → ℝ) (hf : ∀ z, 0 ≤ f z) : + ∀ z, 0 ≤ extendHalfPlaneTest f z := by + intro z + by_cases hz : z ∈ upperHalfPlane + · exact (extendHalfPlaneTest_apply f ⟨z,hz⟩).symm ▸ hf ⟨z,hz⟩ + · rw [extendHalfPlaneTest_of_not_mem f z hz] + +lemma ofReal_integral_halfPlaneLevelMeasure + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (ω : Ω) (hBc : Continuous (fun t => B t ω)) + (hw : IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (t : ℝ≥0∞) (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) : + ENNReal.ofReal (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G u t ω) = + levelApproximationTest κ B γ G u t (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) ω := by + let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f + have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val + have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property + have hfin := (levelApproximation_apply_le κ B γ G u t ω hK.measurableSet).trans_lt + (levelMass_lt_top_of_compact hκ hκ8 γ G F ω hBc hw K hK hKH u) + rw [integral_halfPlaneLevelMeasure] + exact ofReal_integral_eq_lintegral_ofReal + (integrable_of_bounded_support _ K hK.measurableSet hfin _ + (extendHalfPlaneTest_measurable f f.continuous.measurable) _ + (fun z _ => extendHalfPlaneTest_norm_le f z) (extendHalfPlaneTest_support f)) + (Eventually.of_forall (extendHalfPlaneTest_nonneg f hf)) + +theorem uniform_halfPlaneLevelMeasure_test_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) : + ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ u : ℝ, 0 ≤ u → ∀ t : ℝ≥0∞, + (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂halfPlaneLevelMeasure κ B γ G u t ω)^m ∂P) ≤ M := by + let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f + have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val + have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property + obtain ⟨M,hM,hbound⟩ := uniform_levelApproximationTest_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth K hK hKH (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) + (ENNReal.ofReal ‖f.toBoundedContinuousFunction‖) ENNReal.ofReal_lt_top + (fun z _ => ENNReal.ofReal_le_ofReal ((le_abs_self _).trans (extendHalfPlaneTest_norm_le f z))) + (fun z hz => by rw [extendHalfPlaneTest_support f z hz,ENNReal.ofReal_zero]) m + refine ⟨M,hM,fun u hu t => ?_⟩ + simpa only [ofReal_integral_halfPlaneLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) u t f hf] + using hbound u hu t + +theorem uniform_convexLevelMeasure_test_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) : + ∃ M : ℝ≥0∞, M < ⊤ ∧ ∀ n (t : ℝ≥0∞), + (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω)^m ∂P) ≤ M := by + obtain ⟨M,hM,hbound⟩ := uniform_halfPlaneLevelMeasure_test_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth f hf m + refine ⟨M,hM,fun n t => ?_⟩ + have hb := lintegral_pow_convex_sum_le P (s n) (w n) + (fun i ω => ∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω) + (hw₀ n) (hw₁ n) (fun i _ ω => integral_nonneg hf) + (fun i _ => by + simp_rw [integral_halfPlaneLevelMeasure] + exact levelApproximationRealTest_measurable hm hBc κ (u (n+i)) (hu _) γ G F hw t + _ (extendHalfPlaneTest_measurable f f.continuous.measurable)) + m M (fun i _ => hbound (u (n+i)) (hu _) t) + simpa only [integral_convexLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) u s w n t (hw₀ n) f, + integral_halfPlaneLevelMeasure] using hb + +theorem level_radon_limit_test_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (m : ℕ) : + (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂ν ω)^m ∂P) < ⊤ := by + obtain ⟨M,hM,hbound⟩ := uniform_convexLevelMeasure_test_moments hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth u hu s w hw₀ hw₁ f hf m + apply lt_of_le_of_lt (lintegral_le_of_ae_tendsto_bound P _ _ (fun n => + ((convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n t f).ennreal_ofReal.pow_const m).aemeasurable) + ?_ M (fun n => hbound n t)) hM + filter_upwards [hlim] with ω hω + exact (ENNReal.continuous_pow m).continuousAt.tendsto.comp + ((ENNReal.continuous_ofReal.tendsto _).comp (hω f)) + +lemma measure_compact_le_nonneg_test (ν : Measure upperHalfPlane) + [IsFiniteMeasureOnCompacts ν] (K : Set upperHalfPlane) (hK : IsCompact K) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) (hKf : ∀ z ∈ K, 1 ≤ f z) : + ν K ≤ ENNReal.ofReal (∫ z, f z ∂ν) := by + rw [ofReal_integral_eq_lintegral_ofReal f.integrable (Eventually.of_forall hf)] + calc + _ = ∫⁻ z, K.indicator (fun _ => (1 : ℝ≥0∞)) z ∂ν := by + rw [lintegral_indicator_const hK.measurableSet,one_mul] + _ ≤ _ := by + apply lintegral_mono + intro z + by_cases hz : z ∈ K + · simpa only [indicator_of_mem hz,ENNReal.one_le_ofReal] using hKf z hz + · simp only [indicator_of_notMem hz,zero_le] + +theorem level_radon_limit_compact_moments + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) + (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) + (K : Set upperHalfPlane) (hK : IsCompact K) (m : ℕ) : + (∫⁻ ω, (ν ω K)^m ∂P) < ⊤ := by + obtain ⟨f,hf,hKf⟩ := exists_compact_test_cutoff K hK + apply lt_of_le_of_lt (lintegral_mono (fun ω => ?_)) + (level_radon_limit_test_moments hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + u hu s w hw₀ hw₁ t ν hlim f hf m) + let := hν ω + exact pow_le_pow_left' (measure_compact_le_nonneg_test (ν ω) K hK f hf hKf) m + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonNonatomic.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonNonatomic.lean index 3a64fc5a80f94f258fca775a7dac7aeb19f0cd6d..419524b918cfba1b642951b65e7ccafefe188b84 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonNonatomic.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonNonatomic.lean @@ -1,124 +1,124 @@ -import OAI.Probability.SLE.Mass.LevelRadonTwoPoint -import OAI.Probability.SLE.Mass.MassGridCutoffs -import OAI.Probability.SLE.Gauge.ShrinkingCoverAtoms - -namespace OAI - -/-! Nonatomicity of every actual convex Radon limit, from the two-point -bound and one deterministic family of shrinking grid cutoffs. -/ - -noncomputable section -open Set Metric Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem level_radon_limit_compact_null_singletons - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) - (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) - (K : Set upperHalfPlane) (hK : IsCompact K) : - ∀ᵐ ω ∂P, ∀ x ∈ K, ν ω {x}=0 := by - obtain ⟨χ,hχ,hχK⟩ := exists_unit_compact_test_cutoff K hK - let L := (Subtype.val : upperHalfPlane → ℂ) '' tsupport χ - have hL : IsCompact L := χ.hasCompactSupport.image continuous_subtype_val - have hLH : L ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property - obtain ⟨b,hb,hbL⟩ := compact_positive_radius_cap hL hLH - have hKI := hK.image (continuous_subtype_val : Continuous (Subtype.val : upperHalfPlane → ℂ)) - obtain ⟨R,hR,hRK⟩ := hKI.isBounded.exists_pos_norm_le - let r : ℕ → ℝ := fun n => 1/((n:ℝ)+1) - have hr (n : ℕ) : 0 < r n := by dsimp [r]; positivity - let I (n : ℕ) := coverGridIndex R (r n) - let f (n : ℕ) (i : ℤ×ℤ) := massGridCutoff χ (r n) i - have hf (n : ℕ) (i : ℤ×ℤ) (z : upperHalfPlane) : 0 ≤ f n i z := - (massGridCutoff_bounds χ hχ (r n) i z).1 - let A (n : ℕ) (ω : Ω) := ∑ i ∈ I n, ENNReal.ofReal (∫ z, f n i z ∂ν ω)^2 - have hA (n : ℕ) : Measurable (A n) := by - apply Finset.measurable_sum - intro i hi - have he (ω : Ω) : ENNReal.ofReal (∫ z, f n i z ∂ν ω) = - ∫⁻ z, ENNReal.ofReal (f n i z) ∂ν ω := by - let := hν ω - exact ofReal_integral_eq_lintegral_ofReal (f n i).integrable (Eventually.of_forall (hf n i)) - simp_rw [he] - exact ((Measure.measurable_lintegral (f n i).continuous.measurable.ennreal_ofReal).comp hνm).pow_const 2 - apply ae_null_singletons_on_of_square_bounds P ν K A hA - · intro ω x hx n - have hxR : (x:ℂ) ∈ closedBall (0:ℂ) R := by - simpa only [mem_closedBall,dist_zero_right] using hRK x ⟨x,hx,rfl⟩ - obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (coverGrid_covers_disk hR.le (hr n) hxR) - have hfx : f n i x = 1 := massGridCutoff_eq_one χ (hr n) i x (hχK x hx) hxi - let := hν ω - have hμ : ν ω {x} ≤ ENNReal.ofReal (∫ z, f n i z ∂ν ω) := - measure_compact_le_nonneg_test (ν ω) {x} isCompact_singleton (f n i) (hf n i) - (by intro z hz; simpa only [mem_singleton_iff.mp hz,hfx] using (le_refl (1:ℝ))) - exact (pow_le_pow_left' hμ 2).trans - (Finset.single_le_sum (f := fun j => ENNReal.ofReal (∫ z, f n j z ∂ν ω)^2) - (fun _ _ => zero_le) hi) - · have hbound (n : ℕ) : (∫⁻ ω, A n ω ∂P) ≤ - ENNReal.ofReal (multipointConstant κ 2) * (81 * - ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel (codimension κ) b (6*r n) z - ∂Measure.pi (fun _ : Fin 2 => volume.restrict L)) := by - exact level_radon_cutoff_square_sum_le hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - u hu s w hw₀ hw₁ t ν hνm hν hlim L hL.measurableSet hLH b hb hbL (I n) (f n) - (hf n) (fun i z hz => massGridCutoff_common_support χ (r n) i z hz) - (6*r n) 81 (massGridCutoff_sum_bound χ hχ (hr n) (I n)) - have hr0 : Tendsto (fun n => 6*r n) atTop (𝓝 0) := by - simpa only [mul_zero] using - (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul 6 - have hnear := nearDiagonalKernel_integral_tendsto_zero (codimension_pos κ hκ8) - (by unfold codimension dimension; linarith) b L hL.isBounded (fun n => 6*r n) hr0 - have hupper : Tendsto (fun n => ENNReal.ofReal (multipointConstant κ 2) * (81 * - ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel (codimension κ) b (6*r n) z - ∂Measure.pi (fun _ : Fin 2 => volume.restrict L))) atTop (𝓝 0) := by - simpa only [mul_zero] using ENNReal.Tendsto.const_mul - (ENNReal.Tendsto.const_mul hnear (Or.inr (by norm_num : (81:ℝ≥0∞)≠⊤))) - (Or.inr ENNReal.ofReal_ne_top) - exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hupper - (fun _ => zero_le) hbound - -theorem level_radon_limit_nonatomic - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) - (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) : - ∀ᵐ ω ∂P, NullSingletonClass (ν ω) := by - let D := localTestFamily upperHalfPlane - apply ae_nullSingletonClass_of_exhaustion P ν D.compactSet - · intro x - obtain ⟨j,hj⟩ := D.exhaustive {x} isCompact_singleton - exact ⟨j,hj (mem_singleton x)⟩ - · intro j - exact level_radon_limit_compact_null_singletons hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - u hu s w hw₀ hw₁ t ν hνm hν hlim (D.compactSet j) (D.isCompact j) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelRadonTwoPoint +import OAI.Probability.SLE.Mass.MassGridCutoffs +import OAI.Probability.SLE.Gauge.ShrinkingCoverAtoms + +namespace OAI + +/-! Nonatomicity of every actual convex Radon limit, from the two-point +bound and one deterministic family of shrinking grid cutoffs. -/ + +noncomputable section +open Set Metric Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem level_radon_limit_compact_null_singletons + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) + (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) + (K : Set upperHalfPlane) (hK : IsCompact K) : + ∀ᵐ ω ∂P, ∀ x ∈ K, ν ω {x}=0 := by + obtain ⟨χ,hχ,hχK⟩ := exists_unit_compact_test_cutoff K hK + let L := (Subtype.val : upperHalfPlane → ℂ) '' tsupport χ + have hL : IsCompact L := χ.hasCompactSupport.image continuous_subtype_val + have hLH : L ⊆ upperHalfPlane := by rintro _ ⟨x,_,rfl⟩; exact x.property + obtain ⟨b,hb,hbL⟩ := compact_positive_radius_cap hL hLH + have hKI := hK.image (continuous_subtype_val : Continuous (Subtype.val : upperHalfPlane → ℂ)) + obtain ⟨R,hR,hRK⟩ := hKI.isBounded.exists_pos_norm_le + let r : ℕ → ℝ := fun n => 1/((n:ℝ)+1) + have hr (n : ℕ) : 0 < r n := by dsimp [r]; positivity + let I (n : ℕ) := coverGridIndex R (r n) + let f (n : ℕ) (i : ℤ×ℤ) := massGridCutoff χ (r n) i + have hf (n : ℕ) (i : ℤ×ℤ) (z : upperHalfPlane) : 0 ≤ f n i z := + (massGridCutoff_bounds χ hχ (r n) i z).1 + let A (n : ℕ) (ω : Ω) := ∑ i ∈ I n, ENNReal.ofReal (∫ z, f n i z ∂ν ω)^2 + have hA (n : ℕ) : Measurable (A n) := by + apply Finset.measurable_sum + intro i hi + have he (ω : Ω) : ENNReal.ofReal (∫ z, f n i z ∂ν ω) = + ∫⁻ z, ENNReal.ofReal (f n i z) ∂ν ω := by + let := hν ω + exact ofReal_integral_eq_lintegral_ofReal (f n i).integrable (Eventually.of_forall (hf n i)) + simp_rw [he] + exact ((Measure.measurable_lintegral (f n i).continuous.measurable.ennreal_ofReal).comp hνm).pow_const 2 + apply ae_null_singletons_on_of_square_bounds P ν K A hA + · intro ω x hx n + have hxR : (x:ℂ) ∈ closedBall (0:ℂ) R := by + simpa only [mem_closedBall,dist_zero_right] using hRK x ⟨x,hx,rfl⟩ + obtain ⟨i,hi,hxi⟩ := mem_iUnion₂.mp (coverGrid_covers_disk hR.le (hr n) hxR) + have hfx : f n i x = 1 := massGridCutoff_eq_one χ (hr n) i x (hχK x hx) hxi + let := hν ω + have hμ : ν ω {x} ≤ ENNReal.ofReal (∫ z, f n i z ∂ν ω) := + measure_compact_le_nonneg_test (ν ω) {x} isCompact_singleton (f n i) (hf n i) + (by intro z hz; simpa only [mem_singleton_iff.mp hz,hfx] using (le_refl (1:ℝ))) + exact (pow_le_pow_left' hμ 2).trans + (Finset.single_le_sum (f := fun j => ENNReal.ofReal (∫ z, f n j z ∂ν ω)^2) + (fun _ _ => zero_le) hi) + · have hbound (n : ℕ) : (∫⁻ ω, A n ω ∂P) ≤ + ENNReal.ofReal (multipointConstant κ 2) * (81 * + ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel (codimension κ) b (6*r n) z + ∂Measure.pi (fun _ : Fin 2 => volume.restrict L)) := by + exact level_radon_cutoff_square_sum_le hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + u hu s w hw₀ hw₁ t ν hνm hν hlim L hL.measurableSet hLH b hb hbL (I n) (f n) + (hf n) (fun i z hz => massGridCutoff_common_support χ (r n) i z hz) + (6*r n) 81 (massGridCutoff_sum_bound χ hχ (hr n) (I n)) + have hr0 : Tendsto (fun n => 6*r n) atTop (𝓝 0) := by + simpa only [mul_zero] using + (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul 6 + have hnear := nearDiagonalKernel_integral_tendsto_zero (codimension_pos κ hκ8) + (by unfold codimension dimension; linarith) b L hL.isBounded (fun n => 6*r n) hr0 + have hupper : Tendsto (fun n => ENNReal.ofReal (multipointConstant κ 2) * (81 * + ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel (codimension κ) b (6*r n) z + ∂Measure.pi (fun _ : Fin 2 => volume.restrict L))) atTop (𝓝 0) := by + simpa only [mul_zero] using ENNReal.Tendsto.const_mul + (ENNReal.Tendsto.const_mul hnear (Or.inr (by norm_num : (81:ℝ≥0∞)≠⊤))) + (Or.inr ENNReal.ofReal_ne_top) + exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds hupper + (fun _ => zero_le) hbound + +theorem level_radon_limit_nonatomic + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) + (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) : + ∀ᵐ ω ∂P, NullSingletonClass (ν ω) := by + let D := localTestFamily upperHalfPlane + apply ae_nullSingletonClass_of_exhaustion P ν D.compactSet + · intro x + obtain ⟨j,hj⟩ := D.exhaustive {x} isCompact_singleton + exact ⟨j,hj (mem_singleton x)⟩ + · intro j + exact level_radon_limit_compact_null_singletons hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + u hu s w hw₀ hw₁ t ν hνm hν hlim (D.compactSet j) (D.isCompact j) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonPairings.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonPairings.lean index ff2a2700224d20b66d710ce6e4a8cb003ddc46ce..745a78c39aaafc20d912ab71d8ab32090aaec4bf 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonPairings.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonPairings.lean @@ -1,81 +1,81 @@ -import OAI.Probability.SLE.Mass.LevelRandomRadon - -namespace OAI - -/-! The measurable common Radon versions retain the L² pairings. In -particular expectations on any past event pass through the same limits. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -theorem exists_measurable_level_radon_limits_with_pairings - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : - ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, ∃ ν : ℕ → Ω → Measure upperHalfPlane, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Measurable (ν j)) ∧ - (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ - (∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ - ∀ j k (a : Ω → ℝ), MemLp a 2 P → - Tendsto (fun n => ∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z - ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) - atTop (𝓝 (∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z ∂ν j ω) ∂P)) := by - let D := localTestFamily upperHalfPlane - obtain ⟨x,s,w,hw₀,hw₁,_,hp,hconv⟩ := exists_common_halfPlane_test_limits_with_pairings - hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth - (fun j => D.test (Nat.unpair j).2) u hu (fun j => t (Nat.unpair j).1) - have hex : ∀ᵐ ω ∂P, ∃ ν : ℕ → Measure upperHalfPlane, (∀ j, (ν j).Regular) ∧ - ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν j)) := by - filter_upwards [hconv] with ω hω - have hj (j : ℕ) : ∃ ν : Measure upperHalfPlane, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ - ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν)) := by - let (n : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w n (t j) ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) - apply D.exists_radon_limit - intro k - refine ⟨x (Nat.pair j k) ω,?_⟩ - simpa only [Nat.unpair_pair] using hω (Nat.pair j k) - choose ν hν using hj - exact ⟨ν,fun j => (hν j).1,fun j => (hν j).2.2⟩ - obtain ⟨ν,hνm,hνr,hlim⟩ := exists_measurable_countable_radon_limits P - (fun n j ω => convexLevelMeasure κ B γ G u s w n (t j) ω) - (fun n j f => convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n (t j) f) hex - refine ⟨s,w,ν,hw₀,hw₁,hνm,hνr,hlim,?_⟩ - intro j k a ha - have heq : (fun ω => x (Nat.pair j k) ω) =ᵐ[P] - (fun ω => ∫ z, D.test k z ∂ν j ω) := by - filter_upwards [hconv,hlim] with ω hc hl - have hc' := hc (Nat.pair j k) - simp only [Nat.unpair_pair] at hc' - exact tendsto_nhds_unique hc' (hl j (D.test k)) - have ht := hp (Nat.pair j k) a ha - simp only [Nat.unpair_pair] at ht - convert ht using 1 - apply congrArg (fun v : ℝ => 𝓝 v) - apply integral_congr_ae - filter_upwards [heq] with ω hω - exact congrArg (fun v : ℝ => a ω * v) hω.symm - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelRandomRadon + +namespace OAI + +/-! The measurable common Radon versions retain the L² pairings. In +particular expectations on any past event pass through the same limits. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +theorem exists_measurable_level_radon_limits_with_pairings + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : + ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, ∃ ν : ℕ → Ω → Measure upperHalfPlane, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Measurable (ν j)) ∧ + (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ + (∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ + ∀ j k (a : Ω → ℝ), MemLp a 2 P → + Tendsto (fun n => ∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z + ∂convexLevelMeasure κ B γ G u s w n (t j) ω) ∂P) + atTop (𝓝 (∫ ω, a ω * (∫ z, (localTestFamily upperHalfPlane).test k z ∂ν j ω) ∂P)) := by + let D := localTestFamily upperHalfPlane + obtain ⟨x,s,w,hw₀,hw₁,_,hp,hconv⟩ := exists_common_halfPlane_test_limits_with_pairings + hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth + (fun j => D.test (Nat.unpair j).2) u hu (fun j => t (Nat.unpair j).1) + have hex : ∀ᵐ ω ∂P, ∃ ν : ℕ → Measure upperHalfPlane, (∀ j, (ν j).Regular) ∧ + ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν j)) := by + filter_upwards [hconv] with ω hω + have hj (j : ℕ) : ∃ ν : Measure upperHalfPlane, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ + ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν)) := by + let (n : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w n (t j) ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w n (t j) + apply D.exists_radon_limit + intro k + refine ⟨x (Nat.pair j k) ω,?_⟩ + simpa only [Nat.unpair_pair] using hω (Nat.pair j k) + choose ν hν using hj + exact ⟨ν,fun j => (hν j).1,fun j => (hν j).2.2⟩ + obtain ⟨ν,hνm,hνr,hlim⟩ := exists_measurable_countable_radon_limits P + (fun n j ω => convexLevelMeasure κ B γ G u s w n (t j) ω) + (fun n j f => convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n (t j) f) hex + refine ⟨s,w,ν,hw₀,hw₁,hνm,hνr,hlim,?_⟩ + intro j k a ha + have heq : (fun ω => x (Nat.pair j k) ω) =ᵐ[P] + (fun ω => ∫ z, D.test k z ∂ν j ω) := by + filter_upwards [hconv,hlim] with ω hc hl + have hc' := hc (Nat.pair j k) + simp only [Nat.unpair_pair] at hc' + exact tendsto_nhds_unique hc' (hl j (D.test k)) + have ht := hp (Nat.pair j k) a ha + simp only [Nat.unpair_pair] at ht + convert ht using 1 + apply congrArg (fun v : ℝ => 𝓝 v) + apply integral_congr_ae + filter_upwards [heq] with ω hω + exact congrArg (fun v : ℝ => a ω * v) hω.symm + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonSupport.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonSupport.lean index 922ec052bfcdba92a361541d6909e16939b3e2f5..ecdccf8cdde2ad89fdaffbb6cd1b5d6f70eb1914 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonSupport.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonSupport.lean @@ -1,220 +1,220 @@ -import OAI.Probability.SLE.Mass.LevelApproximationSupport -import OAI.Probability.SLE.Mass.LevelRadonExtraction -import OAI.Probability.SLE.Mass.RadonTestLimitProperties - -namespace OAI - -/-! Actual past support of every convex-tail Radon limit. Compact tests -separated from the past trace vanish at all sufficiently high levels. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma level_density_eventually_zero_on_compact - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hBc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) - (hdis : Disjoint K (γ ω '' Icc 0 t)) - (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) : - ∀ᶠ n in atTop, ∀ z ∈ K, - levelApproximationDensity κ B γ G (u n) t ω z = 0 := by - let S := γ ω '' Icc 0 t - have hS : IsClosed S := (isCompact_Icc.image hw.1).isClosed - have hSne : S.Nonempty := ⟨γ ω 0,0,⟨le_rfl,bot_le⟩,rfl⟩ - obtain ⟨r,hr,hrK⟩ := hK.exists_forall_le' - (Complex.continuous_im.min (Metric.continuous_infDist_pt (s := S))).continuousOn - (fun z hz => lt_min (hKH hz) - ((hS.notMem_iff_infDist_pos hSne).mp (fun hs => Set.disjoint_left.mp hdis hz hs))) - obtain ⟨M,hM⟩ := hK.bddAbove_image Complex.continuous_im.continuousOn - have hlim : Tendsto (fun n => (2*M)*Real.exp (-u n)) atTop (𝓝 0) := by - simpa only [mul_zero,Function.comp_def] using - (Real.tendsto_exp_atBot.comp (tendsto_neg_atTop_atBot.comp hu)).const_mul (2*M) - filter_upwards [hlim.eventually (eventually_lt_nhds hr)] with n hn z hz - apply levelApproximationDensity_eq_zero_of_avoidance ω (hBc.const_mul _) hw z r hr - ((hrK z hz).trans (min_le_left _ _)) (u n) - ((mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hM ⟨z,hz,rfl⟩) - (by norm_num : (0:ℝ) ≤ 2)) (Real.exp_pos _).le).trans_lt hn) t - intro s hs - exact ((hrK z hz).trans (min_le_right _ _)).trans - ((Metric.infDist_le_dist_of_mem (show γ ω s ∈ S from ⟨s,hs,rfl⟩)).trans_eq (dist_comm _ _)) - -lemma halfPlane_level_test_eventually_zero_off_past - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hBc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)) - (hf : ∀ z ∈ tsupport f, (z : ℂ) ∉ γ ω '' Icc 0 t) - (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) : - ∀ᶠ n in atTop, ∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u n) t ω = 0 := by - let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f - have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val - have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨z,_,rfl⟩; exact z.property - have hdis : Disjoint K (γ ω '' Icc 0 t) := by - apply Set.disjoint_left.mpr - rintro _ ⟨z,hz,rfl⟩ hs - exact hf z hz hs - filter_upwards [level_density_eventually_zero_on_compact ω hBc hw t K hK hKH hdis u hu] - with n hn - have hzero : halfPlaneLevelMeasure κ B γ G (u n) t ω (tsupport f) = 0 := by - rw [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - levelApproximation,withDensity_apply _ hK.measurableSet] - apply le_antisymm ?_ bot_le - apply (setLIntegral_mono' hK.measurableSet (fun z hz => le_of_eq (hn z hz))).trans_eq - simp - apply integral_eq_zero_of_ae - filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with z hz - exact image_eq_zero_of_notMem_tsupport hz - -/-- Every compact test away from the actual past vanishes on every late -convex tail. The coefficients may be arbitrary: no normalization is used. -/ -lemma convex_level_test_eventually_zero_off_past - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hBc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)) - (hf : ∀ z ∈ tsupport f, (z : ℂ) ∉ γ ω '' Icc 0 t) - (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) : - ∀ᶠ n in atTop, ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω = 0 := by - let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f - have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val - have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨z,_,rfl⟩; exact z.property - have hdis : Disjoint K (γ ω '' Icc 0 t) := by - apply Set.disjoint_left.mpr - rintro _ ⟨z,hz,rfl⟩ hs - exact hf z hz hs - obtain ⟨N,hN⟩ := eventually_atTop.mp - (level_density_eventually_zero_on_compact ω hBc hw t K hK hKH hdis u hu) - filter_upwards [eventually_ge_atTop N] with n hn - have hzero : convexLevelMeasure κ B γ G u s w n t ω (tsupport f) = 0 := by - rw [convexLevelMeasure,Measure.finsetSum_apply] - apply Finset.sum_eq_zero - intro i _ - rw [Measure.smul_apply] - suffices he : halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω (tsupport f) = 0 by - simp only [he,smul_zero] - rw [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - levelApproximation,withDensity_apply _ hK.measurableSet] - apply le_antisymm ?_ bot_le - apply (setLIntegral_mono' hK.measurableSet - (fun z hz => le_of_eq (hN (n+i) (hn.trans (Nat.le_add_right n i)) z hz))).trans_eq - simp - apply integral_eq_zero_of_ae - filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with z hz - exact image_eq_zero_of_notMem_tsupport hz - -/-- Any vague limit of the actual convex level measures at a finite time -is carried by the actual trace up to that time. -/ -theorem convex_level_radon_limit_carried_by_past - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hBc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (ν : Measure upperHalfPlane) [ν.Regular] - (hlim : ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν))) : - ν {z : upperHalfPlane | (z : ℂ) ∉ γ ω '' Icc 0 t} = 0 := by - let S : Set upperHalfPlane := {z | (z : ℂ) ∈ γ ω '' Icc 0 t} - apply radon_limit_carried_by_closed _ ν hlim S - (((isCompact_Icc.image hw.1).isClosed).preimage continuous_subtype_val) - intro f hf - exact convex_level_test_eventually_zero_off_past ω hBc hw t f hf u hu s w - -theorem convex_level_radon_limit_zero_at_zero - {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) - (hBc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (ν : Measure upperHalfPlane) [ν.Regular] - (hlim : ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n 0 ω) - atTop (𝓝 (∫ z, f z ∂ν))) : ν = 0 := by - have h := convex_level_radon_limit_carried_by_past ω hBc hw 0 u hu s w ν hlim - have he : {z : upperHalfPlane | (z : ℂ) ∉ γ ω '' Icc 0 0} = univ := by - apply Set.eq_univ_of_forall - intro z hz - obtain ⟨a,ha,haz⟩ := hz - have ha0 : a = 0 := le_antisymm ha.2 ha.1 - subst a - have hz0 : (z : ℂ) = 0 := haz.symm.trans hw.2.1 - have hpos := z.property - change 0 < (z : ℂ).im at hpos - simp only [hz0,Complex.zero_im,lt_self_iff_false] at hpos - rw [he] at h - exact Measure.measure_univ_eq_zero.mp h - -/-- The support conclusion holds simultaneously for the prescribed -countable list of finite times on the same vague-convergence event. -/ -theorem ae_common_level_radon_limits_carried_by_past - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (hBc : ∀ ω, Continuous (fun s => B s ω)) - (hw : ∀ ω, IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (t : ℕ → ℝ≥0) (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (ν : ℕ → Ω → Measure upperHalfPlane) - (hν : ∀ j ω, (ν j ω).Regular) - (hlim : ∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν j ω))) : - ∀ᵐ ω ∂P, ∀ j, ν j ω {z : upperHalfPlane | (z : ℂ) ∉ γ ω '' Icc 0 (t j)} = 0 := by - filter_upwards [hlim] with ω hω - intro j - let := hν j ω - exact convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) (t j) u hu - s w (ν j ω) (hω j) - -lemma past_inter_surviving_subset_segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : - (γ '' Icc 0 t) ∩ survivingDomain γ s ⊆ γ '' Icc s t := by - rintro z ⟨⟨a,ha,rfl⟩,hlive⟩ - have hsa : s ≤ a := by - by_contra h - exact hlive.1.2 ⟨a,⟨ha.1,(lt_of_not_ge h).le⟩,rfl⟩ - exact ⟨a,⟨hsa,ha.2⟩,rfl⟩ - -/-- A positive increment dominated by the past mass and by the remaining -mass is carried by the new segment. This is the deterministic consequence -of past support and the conditional-kernel future support. -/ -theorem increment_le_restrict_trace_segment - (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (η μpast μfuture ν : Measure upperHalfPlane) - (hηpast : η ≤ μpast) (hηfuture : η ≤ μfuture) (hην : η ≤ ν) - (hpast : μpast {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 t} = 0) - (hfuture : μfuture {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ s} = 0) : - η ≤ ν.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := by - have hp : ∀ᵐ z : upperHalfPlane ∂η, (z : ℂ) ∈ γ '' Icc 0 t := by - have hz : η {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 t} = 0 := - le_zero_iff.mp ((hηpast {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 t}).trans_eq hpast) - filter_upwards [measure_eq_zero_iff_ae_notMem.mp hz] with z hz - exact not_not.mp hz - have hf : ∀ᵐ z : upperHalfPlane ∂η, (z : ℂ) ∈ survivingDomain γ s := by - have hz : η {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ s} = 0 := - le_zero_iff.mp ((hηfuture {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ s}).trans_eq hfuture) - filter_upwards [measure_eq_zero_iff_ae_notMem.mp hz] with z hz - exact not_not.mp hz - have hm : ∀ᵐ z ∂η, z ∈ {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := by - filter_upwards [hp,hf] with z hzp hzf - exact past_inter_surviving_subset_segment γ s t ⟨hzp,hzf⟩ - calc - η = η.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := - (Measure.restrict_eq_self_of_ae_mem hm).symm - _ ≤ ν.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := - Measure.restrict_mono le_rfl hην - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximationSupport +import OAI.Probability.SLE.Mass.LevelRadonExtraction +import OAI.Probability.SLE.Mass.RadonTestLimitProperties + +namespace OAI + +/-! Actual past support of every convex-tail Radon limit. Compact tests +separated from the past trace vanish at all sufficiently high levels. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma level_density_eventually_zero_on_compact + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hBc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (K : Set ℂ) (hK : IsCompact K) (hKH : K ⊆ upperHalfPlane) + (hdis : Disjoint K (γ ω '' Icc 0 t)) + (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) : + ∀ᶠ n in atTop, ∀ z ∈ K, + levelApproximationDensity κ B γ G (u n) t ω z = 0 := by + let S := γ ω '' Icc 0 t + have hS : IsClosed S := (isCompact_Icc.image hw.1).isClosed + have hSne : S.Nonempty := ⟨γ ω 0,0,⟨le_rfl,bot_le⟩,rfl⟩ + obtain ⟨r,hr,hrK⟩ := hK.exists_forall_le' + (Complex.continuous_im.min (Metric.continuous_infDist_pt (s := S))).continuousOn + (fun z hz => lt_min (hKH hz) + ((hS.notMem_iff_infDist_pos hSne).mp (fun hs => Set.disjoint_left.mp hdis hz hs))) + obtain ⟨M,hM⟩ := hK.bddAbove_image Complex.continuous_im.continuousOn + have hlim : Tendsto (fun n => (2*M)*Real.exp (-u n)) atTop (𝓝 0) := by + simpa only [mul_zero,Function.comp_def] using + (Real.tendsto_exp_atBot.comp (tendsto_neg_atTop_atBot.comp hu)).const_mul (2*M) + filter_upwards [hlim.eventually (eventually_lt_nhds hr)] with n hn z hz + apply levelApproximationDensity_eq_zero_of_avoidance ω (hBc.const_mul _) hw z r hr + ((hrK z hz).trans (min_le_left _ _)) (u n) + ((mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hM ⟨z,hz,rfl⟩) + (by norm_num : (0:ℝ) ≤ 2)) (Real.exp_pos _).le).trans_lt hn) t + intro s hs + exact ((hrK z hz).trans (min_le_right _ _)).trans + ((Metric.infDist_le_dist_of_mem (show γ ω s ∈ S from ⟨s,hs,rfl⟩)).trans_eq (dist_comm _ _)) + +lemma halfPlane_level_test_eventually_zero_off_past + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hBc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)) + (hf : ∀ z ∈ tsupport f, (z : ℂ) ∉ γ ω '' Icc 0 t) + (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) : + ∀ᶠ n in atTop, ∫ z, f z ∂halfPlaneLevelMeasure κ B γ G (u n) t ω = 0 := by + let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f + have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val + have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨z,_,rfl⟩; exact z.property + have hdis : Disjoint K (γ ω '' Icc 0 t) := by + apply Set.disjoint_left.mpr + rintro _ ⟨z,hz,rfl⟩ hs + exact hf z hz hs + filter_upwards [level_density_eventually_zero_on_compact ω hBc hw t K hK hKH hdis u hu] + with n hn + have hzero : halfPlaneLevelMeasure κ B γ G (u n) t ω (tsupport f) = 0 := by + rw [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + levelApproximation,withDensity_apply _ hK.measurableSet] + apply le_antisymm ?_ bot_le + apply (setLIntegral_mono' hK.measurableSet (fun z hz => le_of_eq (hn z hz))).trans_eq + simp + apply integral_eq_zero_of_ae + filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with z hz + exact image_eq_zero_of_notMem_tsupport hz + +/-- Every compact test away from the actual past vanishes on every late +convex tail. The coefficients may be arbitrary: no normalization is used. -/ +lemma convex_level_test_eventually_zero_off_past + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hBc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (f : C_c(upperHalfPlane,ℝ)) + (hf : ∀ z ∈ tsupport f, (z : ℂ) ∉ γ ω '' Icc 0 t) + (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) : + ∀ᶠ n in atTop, ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω = 0 := by + let K := (Subtype.val : upperHalfPlane → ℂ) '' tsupport f + have hK : IsCompact K := f.hasCompactSupport.image continuous_subtype_val + have hKH : K ⊆ upperHalfPlane := by rintro _ ⟨z,_,rfl⟩; exact z.property + have hdis : Disjoint K (γ ω '' Icc 0 t) := by + apply Set.disjoint_left.mpr + rintro _ ⟨z,hz,rfl⟩ hs + exact hf z hz hs + obtain ⟨N,hN⟩ := eventually_atTop.mp + (level_density_eventually_zero_on_compact ω hBc hw t K hK hKH hdis u hu) + filter_upwards [eventually_ge_atTop N] with n hn + have hzero : convexLevelMeasure κ B γ G u s w n t ω (tsupport f) = 0 := by + rw [convexLevelMeasure,Measure.finsetSum_apply] + apply Finset.sum_eq_zero + intro i _ + rw [Measure.smul_apply] + suffices he : halfPlaneLevelMeasure κ B γ G (u (n+i)) t ω (tsupport f) = 0 by + simp only [he,smul_zero] + rw [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + levelApproximation,withDensity_apply _ hK.measurableSet] + apply le_antisymm ?_ bot_le + apply (setLIntegral_mono' hK.measurableSet + (fun z hz => le_of_eq (hN (n+i) (hn.trans (Nat.le_add_right n i)) z hz))).trans_eq + simp + apply integral_eq_zero_of_ae + filter_upwards [measure_eq_zero_iff_ae_notMem.mp hzero] with z hz + exact image_eq_zero_of_notMem_tsupport hz + +/-- Any vague limit of the actual convex level measures at a finite time +is carried by the actual trace up to that time. -/ +theorem convex_level_radon_limit_carried_by_past + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hBc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (ν : Measure upperHalfPlane) [ν.Regular] + (hlim : ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν))) : + ν {z : upperHalfPlane | (z : ℂ) ∉ γ ω '' Icc 0 t} = 0 := by + let S : Set upperHalfPlane := {z | (z : ℂ) ∈ γ ω '' Icc 0 t} + apply radon_limit_carried_by_closed _ ν hlim S + (((isCompact_Icc.image hw.1).isClosed).preimage continuous_subtype_val) + intro f hf + exact convex_level_test_eventually_zero_off_past ω hBc hw t f hf u hu s w + +theorem convex_level_radon_limit_zero_at_zero + {Ω : Type*} {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} (ω : Ω) + (hBc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (ν : Measure upperHalfPlane) [ν.Regular] + (hlim : ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n 0 ω) + atTop (𝓝 (∫ z, f z ∂ν))) : ν = 0 := by + have h := convex_level_radon_limit_carried_by_past ω hBc hw 0 u hu s w ν hlim + have he : {z : upperHalfPlane | (z : ℂ) ∉ γ ω '' Icc 0 0} = univ := by + apply Set.eq_univ_of_forall + intro z hz + obtain ⟨a,ha,haz⟩ := hz + have ha0 : a = 0 := le_antisymm ha.2 ha.1 + subst a + have hz0 : (z : ℂ) = 0 := haz.symm.trans hw.2.1 + have hpos := z.property + change 0 < (z : ℂ).im at hpos + simp only [hz0,Complex.zero_im,lt_self_iff_false] at hpos + rw [he] at h + exact Measure.measure_univ_eq_zero.mp h + +/-- The support conclusion holds simultaneously for the prescribed +countable list of finite times on the same vague-convergence event. -/ +theorem ae_common_level_radon_limits_carried_by_past + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (hBc : ∀ ω, Continuous (fun s => B s ω)) + (hw : ∀ ω, IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (t : ℕ → ℝ≥0) (u : ℕ → ℝ) (hu : Tendsto u atTop atTop) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (ν : ℕ → Ω → Measure upperHalfPlane) + (hν : ∀ j ω, (ν j ω).Regular) + (hlim : ∀ᵐ ω ∂P, ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν j ω))) : + ∀ᵐ ω ∂P, ∀ j, ν j ω {z : upperHalfPlane | (z : ℂ) ∉ γ ω '' Icc 0 (t j)} = 0 := by + filter_upwards [hlim] with ω hω + intro j + let := hν j ω + exact convex_level_radon_limit_carried_by_past ω (hBc ω) (hw ω) (t j) u hu + s w (ν j ω) (hω j) + +lemma past_inter_surviving_subset_segment (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) : + (γ '' Icc 0 t) ∩ survivingDomain γ s ⊆ γ '' Icc s t := by + rintro z ⟨⟨a,ha,rfl⟩,hlive⟩ + have hsa : s ≤ a := by + by_contra h + exact hlive.1.2 ⟨a,⟨ha.1,(lt_of_not_ge h).le⟩,rfl⟩ + exact ⟨a,⟨hsa,ha.2⟩,rfl⟩ + +/-- A positive increment dominated by the past mass and by the remaining +mass is carried by the new segment. This is the deterministic consequence +of past support and the conditional-kernel future support. -/ +theorem increment_le_restrict_trace_segment + (γ : ℝ≥0 → ℂ) (s t : ℝ≥0) (η μpast μfuture ν : Measure upperHalfPlane) + (hηpast : η ≤ μpast) (hηfuture : η ≤ μfuture) (hην : η ≤ ν) + (hpast : μpast {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 t} = 0) + (hfuture : μfuture {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ s} = 0) : + η ≤ ν.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := by + have hp : ∀ᵐ z : upperHalfPlane ∂η, (z : ℂ) ∈ γ '' Icc 0 t := by + have hz : η {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 t} = 0 := + le_zero_iff.mp ((hηpast {z : upperHalfPlane | (z : ℂ) ∉ γ '' Icc 0 t}).trans_eq hpast) + filter_upwards [measure_eq_zero_iff_ae_notMem.mp hz] with z hz + exact not_not.mp hz + have hf : ∀ᵐ z : upperHalfPlane ∂η, (z : ℂ) ∈ survivingDomain γ s := by + have hz : η {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ s} = 0 := + le_zero_iff.mp ((hηfuture {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain γ s}).trans_eq hfuture) + filter_upwards [measure_eq_zero_iff_ae_notMem.mp hz] with z hz + exact not_not.mp hz + have hm : ∀ᵐ z ∂η, z ∈ {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := by + filter_upwards [hp,hf] with z hzp hzf + exact past_inter_surviving_subset_segment γ s t ⟨hzp,hzf⟩ + calc + η = η.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := + (Measure.restrict_eq_self_of_ae_mem hm).symm + _ ≤ ν.restrict {z : upperHalfPlane | (z : ℂ) ∈ γ '' Icc s t} := + Measure.restrict_mono le_rfl hην + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRadonTwoPoint.lean b/lean/OAI/Probability/SLE/Mass/LevelRadonTwoPoint.lean index af8cdcda1bc3134aa3dc8a0270a68ba582c29a21..aa397d88c8598fb0a06e3c8320421695af7db1b6 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRadonTwoPoint.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRadonTwoPoint.lean @@ -1,97 +1,97 @@ -import OAI.Probability.SLE.Mass.WeightedLevelMoments -import OAI.Probability.SLE.Multipoint.NearDiagonalKernel - -namespace OAI - -/-! Weighted two-point control of the actual limiting Radon measure. -Finite spatial cutoff sums are bounded by the near-diagonal kernel. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -theorem level_radon_limit_test_moment_le_cappedKernel - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) - (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) - (hz : ∀ z ∉ K, extendHalfPlaneTest f z = 0) (m : ℕ) : - (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂ν ω)^m ∂P) ≤ - ENNReal.ofReal (multipointConstant κ m) * - ∫⁻ z : Fin m → ℂ, (∏ i, ENNReal.ofReal (extendHalfPlaneTest f (z i))) * - cappedDistanceKernel (codimension κ) b z - ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by - apply level_radon_limit_test_moment_le hm hBc hκ hκ8 γ G F hw u hu s w hw₀ hw₁ - t ν hlim f hf m - intro n - simp_rw [ofReal_integral_halfPlaneLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) (u n) t f hf] - exact levelApproximationTest_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw - hgrowth K hK hKH (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) - (extendHalfPlaneTest_measurable f f.continuous.measurable).ennreal_ofReal - (fun z hzK => by rw [hz z hzK,ENNReal.ofReal_zero]) b hb hbR m (u n) (hu n) t - -theorem level_radon_cutoff_square_sum_le - {Ω ι : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) - (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) - (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) - atTop (𝓝 (∫ z, f z ∂ν ω))) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) - (I : Finset ι) (f : ι → C_c(upperHalfPlane,ℝ)) (hf : ∀ i z, 0 ≤ f i z) - (hz : ∀ i z, z ∉ K → extendHalfPlaneTest (f i) z = 0) - (r : ℝ) (N : ℝ≥0∞) - (hoverlap : ∀ x y, (∑ i ∈ I, ENNReal.ofReal (extendHalfPlaneTest (f i) x) * - ENNReal.ofReal (extendHalfPlaneTest (f i) y)) ≤ if dist x y ≤ r then N else 0) : - (∫⁻ ω, ∑ i ∈ I, ENNReal.ofReal (∫ z, f i z ∂ν ω)^2 ∂P) ≤ - ENNReal.ofReal (multipointConstant κ 2) * (N * - ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel (codimension κ) b r z - ∂Measure.pi (fun _ : Fin 2 => volume.restrict K)) := by - have hfm (i : ι) : Measurable (fun ω => ENNReal.ofReal (∫ z, f i z ∂ν ω)) := by - have he (ω : Ω) : ENNReal.ofReal (∫ z, f i z ∂ν ω) = - ∫⁻ z, ENNReal.ofReal (f i z) ∂ν ω := by - let := hν ω - exact ofReal_integral_eq_lintegral_ofReal (f i).integrable (Eventually.of_forall (hf i)) - simp_rw [he] - exact (Measure.measurable_lintegral (f i).continuous.measurable.ennreal_ofReal).comp hνm - rw [lintegral_finsetSum I (f := fun i ω => ENNReal.ofReal (∫ z, f i z ∂ν ω)^2) - (fun i _ => (hfm i).pow_const 2)] - apply (Finset.sum_le_sum (fun i _ => level_radon_limit_test_moment_le_cappedKernel - hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth u hu s w hw₀ hw₁ t ν hlim K hK hKH - b hb hbR (f i) (hf i) (hz i) 2)).trans - rw [← Finset.mul_sum] - exact mul_le_mul' le_rfl (sum_weighted_pair_kernel_le I - (fun i z => ENNReal.ofReal (extendHalfPlaneTest (f i) z)) - (fun i _ => (extendHalfPlaneTest_measurable (f i) (f i).continuous.measurable).ennreal_ofReal) - (codimension κ) b r N hoverlap K) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.WeightedLevelMoments +import OAI.Probability.SLE.Multipoint.NearDiagonalKernel + +namespace OAI + +/-! Weighted two-point control of the actual limiting Radon measure. +Finite spatial cutoff sums are bounded by the near-diagonal kernel. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +theorem level_radon_limit_test_moment_le_cappedKernel + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) + (f : C_c(upperHalfPlane,ℝ)) (hf : ∀ z, 0 ≤ f z) + (hz : ∀ z ∉ K, extendHalfPlaneTest f z = 0) (m : ℕ) : + (∫⁻ ω, ENNReal.ofReal (∫ z, f z ∂ν ω)^m ∂P) ≤ + ENNReal.ofReal (multipointConstant κ m) * + ∫⁻ z : Fin m → ℂ, (∏ i, ENNReal.ofReal (extendHalfPlaneTest f (z i))) * + cappedDistanceKernel (codimension κ) b z + ∂Measure.pi (fun _ : Fin m => volume.restrict K) := by + apply level_radon_limit_test_moment_le hm hBc hκ hκ8 γ G F hw u hu s w hw₀ hw₁ + t ν hlim f hf m + intro n + simp_rw [ofReal_integral_halfPlaneLevelMeasure hκ hκ8 γ G F _ (hBc _) (hw _) (u n) t f hf] + exact levelApproximationTest_moment_le_cappedKernel hB hm hBm hBc hB0 hκ hκ8 γ G F hw + hgrowth K hK hKH (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) + (extendHalfPlaneTest_measurable f f.continuous.measurable).ennreal_ofReal + (fun z hzK => by rw [hz z hzK,ENNReal.ofReal_zero]) b hb hbR m (u n) (hu n) t + +theorem level_radon_cutoff_square_sum_le + {Ω ι : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (t : ℝ≥0∞) (ν : Ω → Measure upperHalfPlane) (hνm : Measurable ν) + (hν : ∀ ω, IsFiniteMeasureOnCompacts (ν ω)) + (hlim : ∀ᵐ ω ∂P, ∀ f : C_c(upperHalfPlane,ℝ), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) + atTop (𝓝 (∫ z, f z ∂ν ω))) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (b : ℝ) (hb : 0 < b) (hbR : ∀ z ∈ K, b ≤ 2*z.im) + (I : Finset ι) (f : ι → C_c(upperHalfPlane,ℝ)) (hf : ∀ i z, 0 ≤ f i z) + (hz : ∀ i z, z ∉ K → extendHalfPlaneTest (f i) z = 0) + (r : ℝ) (N : ℝ≥0∞) + (hoverlap : ∀ x y, (∑ i ∈ I, ENNReal.ofReal (extendHalfPlaneTest (f i) x) * + ENNReal.ofReal (extendHalfPlaneTest (f i) y)) ≤ if dist x y ≤ r then N else 0) : + (∫⁻ ω, ∑ i ∈ I, ENNReal.ofReal (∫ z, f i z ∂ν ω)^2 ∂P) ≤ + ENNReal.ofReal (multipointConstant κ 2) * (N * + ∫⁻ z : Fin 2 → ℂ, nearDiagonalKernel (codimension κ) b r z + ∂Measure.pi (fun _ : Fin 2 => volume.restrict K)) := by + have hfm (i : ι) : Measurable (fun ω => ENNReal.ofReal (∫ z, f i z ∂ν ω)) := by + have he (ω : Ω) : ENNReal.ofReal (∫ z, f i z ∂ν ω) = + ∫⁻ z, ENNReal.ofReal (f i z) ∂ν ω := by + let := hν ω + exact ofReal_integral_eq_lintegral_ofReal (f i).integrable (Eventually.of_forall (hf i)) + simp_rw [he] + exact (Measure.measurable_lintegral (f i).continuous.measurable.ennreal_ofReal).comp hνm + rw [lintegral_finsetSum I (f := fun i ω => ENNReal.ofReal (∫ z, f i z ∂ν ω)^2) + (fun i _ => (hfm i).pow_const 2)] + apply (Finset.sum_le_sum (fun i _ => level_radon_limit_test_moment_le_cappedKernel + hB hm hBm hBc hB0 hκ hκ8 γ G F hw hgrowth u hu s w hw₀ hw₁ t ν hlim K hK hKH + b hb hbR (f i) (hf i) (hz i) 2)).trans + rw [← Finset.mul_sum] + exact mul_le_mul' le_rfl (sum_weighted_pair_kernel_le I + (fun i z => ENNReal.ofReal (extendHalfPlaneTest (f i) z)) + (fun i _ => (extendHalfPlaneTest_measurable (f i) (f i).continuous.measurable).ennreal_ofReal) + (codimension κ) b r N hoverlap K) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRandomRadon.lean b/lean/OAI/Probability/SLE/Mass/LevelRandomRadon.lean index 0127a893dddbfed864d3b8a259d45750849017f6..af252e81941340785363ad0f6bca93773868b0d6 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRandomRadon.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRandomRadon.lean @@ -1,107 +1,107 @@ -import OAI.Probability.SLE.Mass.LevelRadonExtraction -import OAI.Probability.SLE.Mass.RandomRadonLimit -import OAI.Probability.SLE.Mass.RadonTestLimitProperties - -namespace OAI - -/-! Measurable versions and simultaneous time ordering of the actual -common Radon limits. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma halfPlaneLevelMeasure_mono - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (u : ℝ) (ω : Ω) : - Monotone (fun t => halfPlaneLevelMeasure κ B γ G u t ω) := by - intro s t hst - apply Measure.le_iff.mpr - intro A hA - simp only [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet] - exact levelApproximation_mono κ B γ G u ω hst _ - -lemma convexLevelMeasure_mono - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (u : ℕ → ℝ) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (ω : Ω) : - Monotone (fun t => convexLevelMeasure κ B γ G u s w n t ω) := by - intro t₁ t₂ ht - apply Measure.le_iff.mpr - intro A hA - simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - exact Finset.sum_le_sum (fun i _ => mul_le_mul' le_rfl - (halfPlaneLevelMeasure_mono κ B γ G (u (n+i)) ω ht A)) - -lemma convexLevelMeasure_test_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (n : ℕ) (t : ℝ≥0∞) - (f : C_c(upperHalfPlane,ℝ)) : - Measurable (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) := by - have he (ω : Ω) := integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) - u s w n t (hw₀ n) f - simp_rw [he] - apply Finset.measurable_sum - intro i hi - exact measurable_const.mul (levelApproximationRealTest_measurable hm hBc κ (u (n+i)) - (hu (n+i)) γ G F hw t (extendHalfPlaneTest f) - (extendHalfPlaneTest_measurable f f.continuous.measurable)) - -theorem exists_measurable_level_radon_limits - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (hgrowth : PublishedDiskUnivalentGrowth) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : - ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, ∃ ν : ℕ → Ω → Measure upperHalfPlane, - (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ - (∀ j, Measurable (ν j)) ∧ - (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ - (∀ᵐ ω ∂P, - (∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ - ∀ i j, t i ≤ t j → ν i ω ≤ ν j ω) := by - obtain ⟨s,w,hw₀,hw₁,hex⟩ := exists_common_level_radon_limits hB hm hBm hBc hB0 hκ hκ8 - γ G F hw hgrowth u hu t - have hex' : ∀ᵐ ω ∂P, ∃ ν : ℕ → Measure upperHalfPlane, (∀ j, (ν j).Regular) ∧ - ∀ j (f : C_c(upperHalfPlane,ℝ)), - Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) - atTop (𝓝 (∫ z, f z ∂ν j)) := by - filter_upwards [hex] with ω hω - obtain ⟨ν,hν,hlim⟩ := hω - exact ⟨ν,fun j => (hν j).1,hlim⟩ - obtain ⟨ν,hνm,hνr,hlim⟩ := exists_measurable_countable_radon_limits P - (fun n j ω => convexLevelMeasure κ B γ G u s w n (t j) ω) - (fun n j f => convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n (t j) f) hex' - refine ⟨s,w,ν,hw₀,hw₁,hνm,hνr,?_⟩ - filter_upwards [hlim] with ω hω - refine ⟨hω,?_⟩ - intro i j hij - let (k : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w k (t i) ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w k (t i) - let (k : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w k (t j) ω) := - convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w k (t j) - let : (ν i ω).Regular := (hνr i ω).1 - let : (ν j ω).Regular := (hνr j ω).1 - exact radon_limits_mono _ _ (ν i ω) (ν j ω) (hω i) (hω j) - (fun k => convexLevelMeasure_mono κ B γ G u s w k ω hij) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelRadonExtraction +import OAI.Probability.SLE.Mass.RandomRadonLimit +import OAI.Probability.SLE.Mass.RadonTestLimitProperties + +namespace OAI + +/-! Measurable versions and simultaneous time ordering of the actual +common Radon limits. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma halfPlaneLevelMeasure_mono + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (u : ℝ) (ω : Ω) : + Monotone (fun t => halfPlaneLevelMeasure κ B γ G u t ω) := by + intro s t hst + apply Measure.le_iff.mpr + intro A hA + simp only [halfPlaneLevelMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet] + exact levelApproximation_mono κ B γ G u ω hst _ + +lemma convexLevelMeasure_mono + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) (u : ℕ → ℝ) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (ω : Ω) : + Monotone (fun t => convexLevelMeasure κ B γ G u s w n t ω) := by + intro t₁ t₂ ht + apply Measure.le_iff.mpr + intro A hA + simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + exact Finset.sum_le_sum (fun i _ => mul_le_mul' le_rfl + (halfPlaneLevelMeasure_mono κ B γ G (u (n+i)) ω ht A)) + +lemma convexLevelMeasure_test_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (n : ℕ) (t : ℝ≥0∞) + (f : C_c(upperHalfPlane,ℝ)) : + Measurable (fun ω => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n t ω) := by + have he (ω : Ω) := integral_convexLevelMeasure hκ hκ8 γ G F ω (hBc ω) (hw ω) + u s w n t (hw₀ n) f + simp_rw [he] + apply Finset.measurable_sum + intro i hi + exact measurable_const.mul (levelApproximationRealTest_measurable hm hBc κ (u (n+i)) + (hu (n+i)) γ G F hw t (extendHalfPlaneTest f) + (extendHalfPlaneTest_measurable f f.continuous.measurable)) + +theorem exists_measurable_level_radon_limits + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hm : ∀ t, Measurable (B t)) (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (hgrowth : PublishedDiskUnivalentGrowth) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (t : ℕ → ℝ≥0∞) : + ∃ s : ℕ → Finset ℕ, ∃ w : ℕ → ℕ → ℝ, ∃ ν : ℕ → Ω → Measure upperHalfPlane, + (∀ n i, i ∈ s n → 0 ≤ w n i) ∧ (∀ n, ∑ i ∈ s n, w n i = 1) ∧ + (∀ j, Measurable (ν j)) ∧ + (∀ j ω, (ν j ω).Regular ∧ IsFiniteMeasureOnCompacts (ν j ω)) ∧ + (∀ᵐ ω ∂P, + (∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν j ω))) ∧ + ∀ i j, t i ≤ t j → ν i ω ≤ ν j ω) := by + obtain ⟨s,w,hw₀,hw₁,hex⟩ := exists_common_level_radon_limits hB hm hBm hBc hB0 hκ hκ8 + γ G F hw hgrowth u hu t + have hex' : ∀ᵐ ω ∂P, ∃ ν : ℕ → Measure upperHalfPlane, (∀ j, (ν j).Regular) ∧ + ∀ j (f : C_c(upperHalfPlane,ℝ)), + Tendsto (fun n => ∫ z, f z ∂convexLevelMeasure κ B γ G u s w n (t j) ω) + atTop (𝓝 (∫ z, f z ∂ν j)) := by + filter_upwards [hex] with ω hω + obtain ⟨ν,hν,hlim⟩ := hω + exact ⟨ν,fun j => (hν j).1,hlim⟩ + obtain ⟨ν,hνm,hνr,hlim⟩ := exists_measurable_countable_radon_limits P + (fun n j ω => convexLevelMeasure κ B γ G u s w n (t j) ω) + (fun n j f => convexLevelMeasure_test_measurable hm hBc hκ hκ8 γ G F hw u hu s w hw₀ n (t j) f) hex' + refine ⟨s,w,ν,hw₀,hw₁,hνm,hνr,?_⟩ + filter_upwards [hlim] with ω hω + refine ⟨hω,?_⟩ + intro i j hij + let (k : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w k (t i) ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w k (t i) + let (k : ℕ) : IsFiniteMeasureOnCompacts (convexLevelMeasure κ B γ G u s w k (t j) ω) := + convexLevelMeasure_finite_on_compacts hκ hκ8 γ G F ω (hBc ω) (hw ω) u s w k (t j) + let : (ν i ω).Regular := (hνr i ω).1 + let : (ν j ω).Regular := (hνr j ω).1 + exact radon_limits_mono _ _ (ν i ω) (ν j ω) (hω i) (hω j) + (fun k => convexLevelMeasure_mono κ B γ G u s w k ω hij) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRemainderMeasure.lean b/lean/OAI/Probability/SLE/Mass/LevelRemainderMeasure.lean index 529f7d030e6e7662b0088e8e66d51f47665b634b..988cba3cdebaaded2777e15ab4000e648ae39579 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRemainderMeasure.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRemainderMeasure.lean @@ -1,113 +1,113 @@ -import OAI.Probability.SLE.Mass.LevelApproximationKernel -import OAI.Probability.SLE.Mass.LevelMeanMeasure - -namespace OAI - -/-! The unreached part of a finite-level approximation, as a positive -measure. The conditional Green identity holds as a measure identity. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -def halfPlaneRemainderMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω))).comap Subtype.val - -def halfPlaneCurrentMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω))).comap Subtype.val - -lemma levelDensity_split {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (t : ℝ≥0) (ω : Ω) (z : ℂ) : - levelApproximationDensity κ B γ G u ∞ ω z = - levelApproximationDensity κ B γ G u t ω z + - ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) := by - unfold levelApproximationDensity levelRemainderGreen - by_cases h : radiusLevelTime (γ ω) (G ω) z u ≤ t - · simp only [le_top,ite_true,ite_eq_left h,ite_eq_right (not_lt.mpr h), - ENNReal.ofReal_zero,add_zero] - · simp only [le_top,ite_true,ite_eq_right h,ite_eq_left (lt_of_not_ge h),zero_add] - -lemma halfPlaneRemainderMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : Measurable (halfPlaneRemainderMeasure κ B γ G u t) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneRemainderMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (levelRemainderGreen_joint_measurable hm hBc κ u hu γ G F hw t).ennreal_ofReal.lintegral_prod_right' - -lemma halfPlaneCurrentMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : Measurable (halfPlaneCurrentMeasure κ B γ G u t) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneCurrentMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (levelCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw t).ennreal_ofReal.lintegral_prod_right' - -lemma halfPlaneLevelMeasure_split - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (ω : Ω) : - halfPlaneLevelMeasure κ B γ G u ∞ ω = - halfPlaneLevelMeasure κ B γ G u t ω + halfPlaneRemainderMeasure κ B γ G u t ω := by - apply Measure.ext - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [Measure.add_apply,halfPlaneLevelMeasure,halfPlaneRemainderMeasure,levelApproximation, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - simp_rw [levelDensity_split κ B γ G u t ω] - exact lintegral_add_left - (levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t).of_uncurry_left _ - -theorem halfPlaneRemainderMeasure_conditional_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) - (hE : MeasurableSet[Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : - (P.restrict E).bind (halfPlaneRemainderMeasure κ B γ G u t) = - (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G u t) := by - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - apply Measure.ext - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by - rintro _ ⟨x,_,rfl⟩; exact x.property - rw [Measure.bind_apply hA (halfPlaneRemainderMeasure_measurable hm hBc κ u hu γ G F hw t).aemeasurable, - Measure.bind_apply hA (halfPlaneCurrentMeasure_measurable hBm hBc κ u hu γ G F hw t).aemeasurable] - simp only [halfPlaneRemainderMeasure,halfPlaneCurrentMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - simpa only [one_mul] using levelRemainderGreen_spatial_kernel hB hBm hBc hB0 hκ hκ8 γ G F hw - u hu t _ hAI hAH (fun _ => 1) measurable_const E hE - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximationKernel +import OAI.Probability.SLE.Mass.LevelMeanMeasure + +namespace OAI + +/-! The unreached part of a finite-level approximation, as a positive +measure. The conditional Green identity holds as a measure identity. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +def halfPlaneRemainderMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω))).comap Subtype.val + +def halfPlaneCurrentMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0) (ω : Ω) : Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal (levelCurrentGreen κ B γ G u t z ω))).comap Subtype.val + +lemma levelDensity_split {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (t : ℝ≥0) (ω : Ω) (z : ℂ) : + levelApproximationDensity κ B γ G u ∞ ω z = + levelApproximationDensity κ B γ G u t ω z + + ENNReal.ofReal (levelRemainderGreen κ B γ G u t z ω) := by + unfold levelApproximationDensity levelRemainderGreen + by_cases h : radiusLevelTime (γ ω) (G ω) z u ≤ t + · simp only [le_top,ite_true,ite_eq_left h,ite_eq_right (not_lt.mpr h), + ENNReal.ofReal_zero,add_zero] + · simp only [le_top,ite_true,ite_eq_right h,ite_eq_left (lt_of_not_ge h),zero_add] + +lemma halfPlaneRemainderMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : Measurable (halfPlaneRemainderMeasure κ B γ G u t) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneRemainderMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (levelRemainderGreen_joint_measurable hm hBc κ u hu γ G F hw t).ennreal_ofReal.lintegral_prod_right' + +lemma halfPlaneCurrentMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : Measurable (halfPlaneCurrentMeasure κ B γ G u t) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneCurrentMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (levelCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw t).ennreal_ofReal.lintegral_prod_right' + +lemma halfPlaneLevelMeasure_split + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (ω : Ω) : + halfPlaneLevelMeasure κ B γ G u ∞ ω = + halfPlaneLevelMeasure κ B γ G u t ω + halfPlaneRemainderMeasure κ B γ G u t ω := by + apply Measure.ext + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [Measure.add_apply,halfPlaneLevelMeasure,halfPlaneRemainderMeasure,levelApproximation, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + simp_rw [levelDensity_split κ B γ G u t ω] + exact lintegral_add_left + (levelApproximationDensity_joint_measurable hm hBc κ u hu γ G F hw t).of_uncurry_left _ + +theorem halfPlaneRemainderMeasure_conditional_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (t : ℝ≥0) (E : Set Ω) + (hE : MeasurableSet[Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) t] E) : + (P.restrict E).bind (halfPlaneRemainderMeasure κ B γ G u t) = + (P.restrict E).bind (halfPlaneCurrentMeasure κ B γ G u t) := by + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + apply Measure.ext + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by + rintro _ ⟨x,_,rfl⟩; exact x.property + rw [Measure.bind_apply hA (halfPlaneRemainderMeasure_measurable hm hBc κ u hu γ G F hw t).aemeasurable, + Measure.bind_apply hA (halfPlaneCurrentMeasure_measurable hBm hBc κ u hu γ G F hw t).aemeasurable] + simp only [halfPlaneRemainderMeasure,halfPlaneCurrentMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + simpa only [one_mul] using levelRemainderGreen_spatial_kernel hB hBm hBc hB0 hκ hκ8 γ G F hw + u hu t _ hAI hAH (fun _ => 1) measurable_const E hE + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelRemainderSupport.lean b/lean/OAI/Probability/SLE/Mass/LevelRemainderSupport.lean index 0363df1de48d89e7a3a86410a1d13a2c7478c6a0..e50ded051078517680a3a14ce5eb83b30306d109 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelRemainderSupport.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelRemainderSupport.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Geometry.RandomKernelSupport -import OAI.Probability.SLE.Mass.GreenMeasureSupport -import OAI.Probability.SLE.Geometry.CountableRemainderKernels - -namespace OAI - -/-! The positive remaining mass in the actual countable Radon construction -is carried by the surviving Loewner domain. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma surviving_complement_past_measurable - {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) : - @MeasurableSet (Ω × upperHalfPlane) - ((Filtration.natural B (fun r => (hm r).stronglyMeasurable) t).prod inferInstance) - {p | (p.2 : ℂ) ∉ survivingDomain (γ p.1) t} := by - have hlive := (actual_spatial_past_kernels hm hBc κ γ G F hw t).2.2 - let : MeasurableSpace Ω := Filtration.natural B (fun r => (hm r).stronglyMeasurable) t - let t0 : Iic t := ⟨t,by simp only [mem_Iic]; exact le_rfl⟩ - have hmap : Measurable (fun p : Ω × upperHalfPlane => - (t0,((p.2 : ℂ),p.1))) := by - apply Measurable.prodMk - · exact measurable_const - · exact ((measurable_subtype_coe : Measurable (fun z : upperHalfPlane => (z : ℂ))).comp - measurable_snd).prodMk measurable_fst - exact (hlive.preimage hmap).compl - -theorem ae_remainder_carried_by_surviving - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : ℝ≥0) (η : Ω → Measure upperHalfPlane) (hη : Measurable η) - (hηfin : ∀ ω, IsFiniteMeasureOnCompacts (η ω)) - (hevent : ∀ E : Set Ω, - MeasurableSet[Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) t] E → - (P.restrict E).bind η = (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) : - ∀ᵐ ω ∂P, η ω {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain (γ ω) t} = 0 := by - let D := localTestFamily upperHalfPlane - let ℱ := Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) - have hmean : P.bind η = P.bind (halfPlaneGreenMeasure κ B γ G t) := by - simpa only [Measure.restrict_univ] using hevent univ MeasurableSet.univ - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - refine ae_avoids_random_set_of_conditional_kernel (ℱ t) (ℱ.le t) P η - (halfPlaneGreenMeasure κ B γ G t) hη - (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t) - D.compactSet (fun n => (D.isCompact n).measurableSet) ?_ ?_ ?_ ?_ hevent - {p | (p.2 : ℂ) ∉ survivingDomain (γ p.1) t} ?_ ?_ - · intro z - obtain ⟨n,hn⟩ := D.exhaustive {z} isCompact_singleton - exact ⟨n,hn (mem_singleton z)⟩ - · intro n ω - let := hηfin ω - exact (D.isCompact n).measure_lt_top - · intro n - exact ae_halfPlaneGreenMeasure_compact_finite hB hBm hBc hB0 hκ hκ8 γ G F hw t - _ (D.isCompact n) - · intro n - rw [hmean] - have hb := halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t univ - (D.compactSet n) - rw [Measure.restrict_univ] at hb - exact lt_of_le_of_lt hb (D.isCompact n).measure_lt_top - · exact surviving_complement_past_measurable - (fun r => hBm.comp (measurable_const.prodMk measurable_id)) hBc κ γ G F hw t - · intro ω - exact halfPlaneGreenMeasure_off_surviving ω (hw ω) t - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Geometry.RandomKernelSupport +import OAI.Probability.SLE.Mass.GreenMeasureSupport +import OAI.Probability.SLE.Geometry.CountableRemainderKernels + +namespace OAI + +/-! The positive remaining mass in the actual countable Radon construction +is carried by the surviving Loewner domain. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma surviving_complement_past_measurable + {Ω : Type*} [mΩ : MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) : + @MeasurableSet (Ω × upperHalfPlane) + ((Filtration.natural B (fun r => (hm r).stronglyMeasurable) t).prod inferInstance) + {p | (p.2 : ℂ) ∉ survivingDomain (γ p.1) t} := by + have hlive := (actual_spatial_past_kernels hm hBc κ γ G F hw t).2.2 + let : MeasurableSpace Ω := Filtration.natural B (fun r => (hm r).stronglyMeasurable) t + let t0 : Iic t := ⟨t,by simp only [mem_Iic]; exact le_rfl⟩ + have hmap : Measurable (fun p : Ω × upperHalfPlane => + (t0,((p.2 : ℂ),p.1))) := by + apply Measurable.prodMk + · exact measurable_const + · exact ((measurable_subtype_coe : Measurable (fun z : upperHalfPlane => (z : ℂ))).comp + measurable_snd).prodMk measurable_fst + exact (hlive.preimage hmap).compl + +theorem ae_remainder_carried_by_surviving + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : ℝ≥0) (η : Ω → Measure upperHalfPlane) (hη : Measurable η) + (hηfin : ∀ ω, IsFiniteMeasureOnCompacts (η ω)) + (hevent : ∀ E : Set Ω, + MeasurableSet[Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) t] E → + (P.restrict E).bind η = (P.restrict E).bind (halfPlaneGreenMeasure κ B γ G t)) : + ∀ᵐ ω ∂P, η ω {z : upperHalfPlane | (z : ℂ) ∉ survivingDomain (γ ω) t} = 0 := by + let D := localTestFamily upperHalfPlane + let ℱ := Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable) + have hmean : P.bind η = P.bind (halfPlaneGreenMeasure κ B γ G t) := by + simpa only [Measure.restrict_univ] using hevent univ MeasurableSet.univ + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + refine ae_avoids_random_set_of_conditional_kernel (ℱ t) (ℱ.le t) P η + (halfPlaneGreenMeasure κ B γ G t) hη + (halfPlaneGreenMeasure_measurable hBm hBc κ γ G F hw t) + D.compactSet (fun n => (D.isCompact n).measurableSet) ?_ ?_ ?_ ?_ hevent + {p | (p.2 : ℂ) ∉ survivingDomain (γ p.1) t} ?_ ?_ + · intro z + obtain ⟨n,hn⟩ := D.exhaustive {z} isCompact_singleton + exact ⟨n,hn (mem_singleton z)⟩ + · intro n ω + let := hηfin ω + exact (D.isCompact n).measure_lt_top + · intro n + exact ae_halfPlaneGreenMeasure_compact_finite hB hBm hBc hB0 hκ hκ8 γ G F hw t + _ (D.isCompact n) + · intro n + rw [hmean] + have hb := halfPlaneGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw t univ + (D.compactSet n) + rw [Measure.restrict_univ] at hb + exact lt_of_le_of_lt hb (D.isCompact n).measure_lt_top + · exact surviving_complement_past_measurable + (fun r => hBm.comp (measurable_const.prodMk measurable_id)) hBc κ γ G F hw t + · intro ω + exact halfPlaneGreenMeasure_off_surviving ω (hw ω) t + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelStoppingConvex.lean b/lean/OAI/Probability/SLE/Mass/LevelStoppingConvex.lean index 33c2f9f38d23ce0dadc7bb0eb6b99db9c9e8ebcd..ed6b415d04ed7946ebe95589f99e34c5941cdf8e 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelStoppingConvex.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelStoppingConvex.lean @@ -1,128 +1,128 @@ -import OAI.Probability.SLE.Mass.LevelStoppingTests - -namespace OAI - -/-! The actual common convex tails preserve the conditional decomposition -at every Brownian stopping time, including infinite stops. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -def convexStoppingCurrentMean {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (S : Ω → ℝ≥0∞) : - Measure upperHalfPlane := - ∑ i ∈ s n, ENNReal.ofReal (w n i) • P.bind (halfPlaneStoppingCurrentMeasure κ B γ G (u (n+i)) S) - -lemma convexLevelMeasure_random_time_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (n : ℕ) (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun ω => convexLevelMeasure κ B γ G u s w n (S ω) ω) := by - apply Measure.measurable_of_measurable_coe - intro A hA - simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - apply Finset.measurable_sum - intro i _ - exact measurable_const.mul ((Measure.measurable_coe hA).comp - (halfPlaneLevelMeasure_random_time_measurable hm hc κ (u (n+i)) (hu _) γ G F hw S hS)) - -lemma convexLevelMeasure_stopping_conditional_split - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (n : ℕ) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) - (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : - (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n ∞) = - (P.restrict E).bind (fun ω => convexLevelMeasure κ B γ G u s w n (S ω) ω) + - convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S := by - let hm (r : ℝ≥0) : Measurable (B r) := hBm.comp (measurable_const.prodMk measurable_id) - unfold convexLevelMeasure convexStoppingCurrentMean - rw [bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ - (fun i => halfPlaneLevelMeasure_measurable hm hc κ (u (n+i)) (hu _) γ G F hw ∞), - bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ - (fun i => halfPlaneLevelMeasure_random_time_measurable hm hc κ (u (n+i)) (hu _) - γ G F hw S hS.measurable')] - simp_rw [halfPlaneLevelMeasure_stopping_conditional_split hB hBm hc hB0 hκ hκ8 γ G F hw - _ (hu _) S hS E hE,smul_add,Finset.sum_add_distrib] - -lemma convexStoppingCurrentMean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (n : ℕ) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) - (E : Set Ω) : - convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S ≤ initialGreenMeasure κ := by - apply Measure.le_iff.mpr - intro A _ - simp only [convexStoppingCurrentMean,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] - calc - _ ≤ ∑ i ∈ s n, ENNReal.ofReal (w n i) * initialGreenMeasure κ A := - Finset.sum_le_sum (fun i _ => mul_le_mul' le_rfl - (halfPlaneStoppingCurrentMeasure_mean_le hB hBm hc hB0 hκ hκ8 γ G F hw - (u (n+i)) (hu _) S hS E A)) - _ = _ := by - rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg (hw₀ n),hw₁ n,ENNReal.ofReal_one,one_mul] - -theorem convexStoppingCurrentMean_test_tendsto - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) (hlim : Tendsto u atTop atTop) - (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) - (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) - (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) - (E : Set Ω) (f : C_c(upperHalfPlane,ℝ)) : - Tendsto (fun n => ∫ z, f z ∂convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S) - atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S))) := by - let ρ := initialGreenMeasure κ - let : IsFiniteMeasureOnCompacts ρ := initialGreenMeasure_finite_on_compacts κ - let μ (j : ℕ) := (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u j) S) - have hμ (j : ℕ) : μ j ≤ ρ := - halfPlaneStoppingCurrentMeasure_mean_le hB hBm hc hB0 hκ hκ8 γ G F hw (u j) (hu j) S hS E - have hfi (j : ℕ) : Integrable f (μ j) := f.integrable.mono_measure (hμ j) - have he (n : ℕ) : (∫ z, f z ∂convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S) = - ∑ i ∈ s n, w n i * (∫ z, f z ∂μ (n+i)) := by - have hi (i : ℕ) : Integrable f (ENNReal.ofReal (w n i) • μ (n+i)) := - (hfi (n+i)).smul_measure ENNReal.ofReal_ne_top - change (∫ z, f z ∂∑ i ∈ s n, ENNReal.ofReal (w n i) • μ (n+i)) = _ - rw [integral_finsetSum_measure (fun i _ => hi i)] - apply Finset.sum_congr rfl - intro i his - rw [integral_smul_measure,ENNReal.toReal_ofReal (hw₀ n i his),smul_eq_mul] - simp only [he] - exact real_convex_tail_tendsto (fun j => ∫ z, f z ∂μ j) - (halfPlaneStoppingCurrentMeasure_mean_test_tendsto hB hBm hc hB0 hκ hκ8 γ G F hw - u hu hmono hlim S hS E f) s w hw₀ hw₁ - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelStoppingTests + +namespace OAI + +/-! The actual common convex tails preserve the conditional decomposition +at every Brownian stopping time, including infinite stops. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +def convexStoppingCurrentMean {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℕ → ℝ) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) (n : ℕ) (S : Ω → ℝ≥0∞) : + Measure upperHalfPlane := + ∑ i ∈ s n, ENNReal.ofReal (w n i) • P.bind (halfPlaneStoppingCurrentMeasure κ B γ G (u (n+i)) S) + +lemma convexLevelMeasure_random_time_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (n : ℕ) (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun ω => convexLevelMeasure κ B γ G u s w n (S ω) ω) := by + apply Measure.measurable_of_measurable_coe + intro A hA + simp only [convexLevelMeasure,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + apply Finset.measurable_sum + intro i _ + exact measurable_const.mul ((Measure.measurable_coe hA).comp + (halfPlaneLevelMeasure_random_time_measurable hm hc κ (u (n+i)) (hu _) γ G F hw S hS)) + +lemma convexLevelMeasure_stopping_conditional_split + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (n : ℕ) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) + (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : + (P.restrict E).bind (convexLevelMeasure κ B γ G u s w n ∞) = + (P.restrict E).bind (fun ω => convexLevelMeasure κ B γ G u s w n (S ω) ω) + + convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S := by + let hm (r : ℝ≥0) : Measurable (B r) := hBm.comp (measurable_const.prodMk measurable_id) + unfold convexLevelMeasure convexStoppingCurrentMean + rw [bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ + (fun i => halfPlaneLevelMeasure_measurable hm hc κ (u (n+i)) (hu _) γ G F hw ∞), + bind_finset_smul _ _ _ (fun _ => ENNReal.ofReal_ne_top) _ + (fun i => halfPlaneLevelMeasure_random_time_measurable hm hc κ (u (n+i)) (hu _) + γ G F hw S hS.measurable')] + simp_rw [halfPlaneLevelMeasure_stopping_conditional_split hB hBm hc hB0 hκ hκ8 γ G F hw + _ (hu _) S hS E hE,smul_add,Finset.sum_add_distrib] + +lemma convexStoppingCurrentMean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (n : ℕ) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) + (E : Set Ω) : + convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S ≤ initialGreenMeasure κ := by + apply Measure.le_iff.mpr + intro A _ + simp only [convexStoppingCurrentMean,Measure.finsetSum_apply,Measure.smul_apply,smul_eq_mul] + calc + _ ≤ ∑ i ∈ s n, ENNReal.ofReal (w n i) * initialGreenMeasure κ A := + Finset.sum_le_sum (fun i _ => mul_le_mul' le_rfl + (halfPlaneStoppingCurrentMeasure_mean_le hB hBm hc hB0 hκ hκ8 γ G F hw + (u (n+i)) (hu _) S hS E A)) + _ = _ := by + rw [← Finset.sum_mul,← ENNReal.ofReal_sum_of_nonneg (hw₀ n),hw₁ n,ENNReal.ofReal_one,one_mul] + +theorem convexStoppingCurrentMean_test_tendsto + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) (hlim : Tendsto u atTop atTop) + (s : ℕ → Finset ℕ) (w : ℕ → ℕ → ℝ) + (hw₀ : ∀ n i, i ∈ s n → 0 ≤ w n i) (hw₁ : ∀ n, ∑ i ∈ s n, w n i = 1) + (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) S) + (E : Set Ω) (f : C_c(upperHalfPlane,ℝ)) : + Tendsto (fun n => ∫ z, f z ∂convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S) + atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S))) := by + let ρ := initialGreenMeasure κ + let : IsFiniteMeasureOnCompacts ρ := initialGreenMeasure_finite_on_compacts κ + let μ (j : ℕ) := (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u j) S) + have hμ (j : ℕ) : μ j ≤ ρ := + halfPlaneStoppingCurrentMeasure_mean_le hB hBm hc hB0 hκ hκ8 γ G F hw (u j) (hu j) S hS E + have hfi (j : ℕ) : Integrable f (μ j) := f.integrable.mono_measure (hμ j) + have he (n : ℕ) : (∫ z, f z ∂convexStoppingCurrentMean (P.restrict E) κ B γ G u s w n S) = + ∑ i ∈ s n, w n i * (∫ z, f z ∂μ (n+i)) := by + have hi (i : ℕ) : Integrable f (ENNReal.ofReal (w n i) • μ (n+i)) := + (hfi (n+i)).smul_measure ENNReal.ofReal_ne_top + change (∫ z, f z ∂∑ i ∈ s n, ENNReal.ofReal (w n i) • μ (n+i)) = _ + rw [integral_finsetSum_measure (fun i _ => hi i)] + apply Finset.sum_congr rfl + intro i his + rw [integral_smul_measure,ENNReal.toReal_ofReal (hw₀ n i his),smul_eq_mul] + simp only [he] + exact real_convex_tail_tendsto (fun j => ∫ z, f z ∂μ j) + (halfPlaneStoppingCurrentMeasure_mean_test_tendsto hB hBm hc hB0 hκ hκ8 γ G F hw + u hu hmono hlim S hS E f) s w hw₀ hw₁ + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelStoppingGreenLimit.lean b/lean/OAI/Probability/SLE/Mass/LevelStoppingGreenLimit.lean index 80b858519d9b489b7bb2ed50fe1c1c1e36a43e59..8f24851b629a72f6fbbd025ba8762413b58a8a44 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelStoppingGreenLimit.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelStoppingGreenLimit.lean @@ -1,131 +1,131 @@ -import OAI.Probability.SLE.Mass.LevelStoppingSpatialKernel -import OAI.Probability.SLE.Mass.LevelGreenLimit - -namespace OAI - -/-! Increasing finite radius levels recover the Green kernel killed at an -arbitrary stopping time. Its expectation is bounded by the initial Green -function, also when the stopping time may be infinite. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -def stoppingGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : ℝ := - if S ω < ∞ then greenDensity κ (fun s => Real.sqrt κ * B s ω) - (γ ω) (G ω) (S ω).toNNReal z else 0 - -lemma levelStoppingCurrentGreen_mono - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (S : Ω → ℝ≥0∞) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v) : - levelStoppingCurrentGreen κ B γ G u S z ω ≤ levelStoppingCurrentGreen κ B γ G v S z ω := by - have hτ := radiusLevelTime_mono (hc.const_mul _) hw z hu huv - unfold levelStoppingCurrentGreen - split_ifs with h₁ h₂ - · exact le_rfl - · exact False.elim (h₂ (h₁.trans_le hτ)) - · exact greenDensity_nonneg hw κ _ z - · exact le_rfl - -lemma levelStoppingCurrentGreen_eventually_eq - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hc : Continuous (fun s => B s ω)) - (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (S : Ω → ℝ≥0∞) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) - (hlim : Tendsto u atTop atTop) : - ∀ᶠ n in atTop, levelStoppingCurrentGreen κ B γ G (u n) S z ω = stoppingGreen κ B γ G S z ω := by - by_cases hfin : S ω < ∞ - · have hcoe : ((S ω).toNNReal : ℝ≥0∞) = S ω := ENNReal.coe_toNNReal hfin.ne - filter_upwards [levelCurrentGreen_eventually_eq ω hc hw z (S ω).toNNReal u hu hlim] with n hn - simpa only [levelCurrentGreen,levelStoppingCurrentGreen,stoppingGreen,hcoe,ite_eq_left hfin] using hn - · have htop : S ω = ∞ := top_le_iff.mp (le_of_not_gt hfin) - filter_upwards [] with n - simp [levelStoppingCurrentGreen,stoppingGreen,htop] - -theorem levelStoppingCurrentGreen_mean_le - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) - (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) : - (∫⁻ ω, ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) ∂P) ≤ - ENNReal.ofReal (initialGreen κ z) := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - have he := levelStoppingRemainderGreen_setLIntegral hB hBm hc hB0 hκ hκ8 - γ G F hw z hz u hu S hS univ MeasurableSet.univ - simp only [Measure.restrict_univ] at he - rw [← he] - calc - _ ≤ ∫⁻ ω, ENNReal.ofReal (stoppedGreen κ (fun s => Real.sqrt κ * B s ω) - (γ ω) (G ω) z u) ∂P := by - apply lintegral_mono - intro ω - unfold levelStoppingRemainderGreen - dsimp only - split_ifs - · exact le_rfl - · simp only [ENNReal.ofReal_zero] - exact bot_le - _ = _ := by - obtain ⟨hi,he⟩ := finite_stopped_green_regular hB hBm hc hB0 κ hκ hκ8 γ G F - (ae_of_all _ hw) z hz u hu - rw [← ofReal_integral_eq_lintegral_ofReal hi - (ae_of_all _ (fun ω => stoppedGreen_nonneg (hw ω) κ z u)),he] - -theorem stoppingGreen_mean_le - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) : - (∫⁻ ω, ENNReal.ofReal (stoppingGreen κ B γ G S z ω) ∂P) ≤ - ENNReal.ofReal (initialGreen κ z) := by - have hm (n : ℕ) : Measurable (fun ω => ENNReal.ofReal - (levelStoppingCurrentGreen κ B γ G n S z ω)) := by - have hj := (levelStoppingCurrentGreen_joint_measurable hBm hc κ n (Nat.cast_nonneg n) - γ G F hw S hS.measurable').ennreal_ofReal - have hh := hj.comp (measurable_id.prodMk (measurable_const (a := z))) - simpa only [Function.comp_def,id_eq] using hh - have hmono : ∀ᵐ ω ∂P, Monotone (fun n : ℕ => ENNReal.ofReal - (levelStoppingCurrentGreen κ B γ G n S z ω)) := by - filter_upwards [] with ω - intro n k hnk - exact ENNReal.ofReal_le_ofReal (levelStoppingCurrentGreen_mono ω (hc ω) (hw ω) - z S (Nat.cast_nonneg n) (Nat.cast_le.mpr hnk)) - have hlim : ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => ENNReal.ofReal - (levelStoppingCurrentGreen κ B γ G n S z ω)) atTop - (𝓝 (ENNReal.ofReal (stoppingGreen κ B γ G S z ω))) := by - filter_upwards [] with ω - apply tendsto_const_nhds.congr' - exact (levelStoppingCurrentGreen_eventually_eq ω (hc ω) (hw ω) z S - (fun n : ℕ => (n:ℝ)) (fun n => Nat.cast_nonneg n) tendsto_natCast_atTop_atTop).mono - (fun _ he => congrArg ENNReal.ofReal he.symm) - exact le_of_tendsto (lintegral_tendsto_of_tendsto_of_monotone - (fun n => (hm n).aemeasurable) hmono hlim) - (Eventually.of_forall (fun n => levelStoppingCurrentGreen_mean_le hB hBm hc hB0 hκ hκ8 - γ G F hw z hz n (Nat.cast_nonneg n) S hS)) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelStoppingSpatialKernel +import OAI.Probability.SLE.Mass.LevelGreenLimit + +namespace OAI + +/-! Increasing finite radius levels recover the Green kernel killed at an +arbitrary stopping time. Its expectation is bounded by the initial Green +function, also when the stopping time may be infinite. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +def stoppingGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : ℝ := + if S ω < ∞ then greenDensity κ (fun s => Real.sqrt κ * B s ω) + (γ ω) (G ω) (S ω).toNNReal z else 0 + +lemma levelStoppingCurrentGreen_mono + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (S : Ω → ℝ≥0∞) {u v : ℝ} (hu : 0 ≤ u) (huv : u ≤ v) : + levelStoppingCurrentGreen κ B γ G u S z ω ≤ levelStoppingCurrentGreen κ B γ G v S z ω := by + have hτ := radiusLevelTime_mono (hc.const_mul _) hw z hu huv + unfold levelStoppingCurrentGreen + split_ifs with h₁ h₂ + · exact le_rfl + · exact False.elim (h₂ (h₁.trans_le hτ)) + · exact greenDensity_nonneg hw κ _ z + · exact le_rfl + +lemma levelStoppingCurrentGreen_eventually_eq + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hc : Continuous (fun s => B s ω)) + (hw : IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (S : Ω → ℝ≥0∞) (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) + (hlim : Tendsto u atTop atTop) : + ∀ᶠ n in atTop, levelStoppingCurrentGreen κ B γ G (u n) S z ω = stoppingGreen κ B γ G S z ω := by + by_cases hfin : S ω < ∞ + · have hcoe : ((S ω).toNNReal : ℝ≥0∞) = S ω := ENNReal.coe_toNNReal hfin.ne + filter_upwards [levelCurrentGreen_eventually_eq ω hc hw z (S ω).toNNReal u hu hlim] with n hn + simpa only [levelCurrentGreen,levelStoppingCurrentGreen,stoppingGreen,hcoe,ite_eq_left hfin] using hn + · have htop : S ω = ∞ := top_le_iff.mp (le_of_not_gt hfin) + filter_upwards [] with n + simp [levelStoppingCurrentGreen,stoppingGreen,htop] + +theorem levelStoppingCurrentGreen_mean_le + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) + (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) : + (∫⁻ ω, ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) ∂P) ≤ + ENNReal.ofReal (initialGreen κ z) := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + have he := levelStoppingRemainderGreen_setLIntegral hB hBm hc hB0 hκ hκ8 + γ G F hw z hz u hu S hS univ MeasurableSet.univ + simp only [Measure.restrict_univ] at he + rw [← he] + calc + _ ≤ ∫⁻ ω, ENNReal.ofReal (stoppedGreen κ (fun s => Real.sqrt κ * B s ω) + (γ ω) (G ω) z u) ∂P := by + apply lintegral_mono + intro ω + unfold levelStoppingRemainderGreen + dsimp only + split_ifs + · exact le_rfl + · simp only [ENNReal.ofReal_zero] + exact bot_le + _ = _ := by + obtain ⟨hi,he⟩ := finite_stopped_green_regular hB hBm hc hB0 κ hκ hκ8 γ G F + (ae_of_all _ hw) z hz u hu + rw [← ofReal_integral_eq_lintegral_ofReal hi + (ae_of_all _ (fun ω => stoppedGreen_nonneg (hw ω) κ z u)),he] + +theorem stoppingGreen_mean_le + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) : + (∫⁻ ω, ENNReal.ofReal (stoppingGreen κ B γ G S z ω) ∂P) ≤ + ENNReal.ofReal (initialGreen κ z) := by + have hm (n : ℕ) : Measurable (fun ω => ENNReal.ofReal + (levelStoppingCurrentGreen κ B γ G n S z ω)) := by + have hj := (levelStoppingCurrentGreen_joint_measurable hBm hc κ n (Nat.cast_nonneg n) + γ G F hw S hS.measurable').ennreal_ofReal + have hh := hj.comp (measurable_id.prodMk (measurable_const (a := z))) + simpa only [Function.comp_def,id_eq] using hh + have hmono : ∀ᵐ ω ∂P, Monotone (fun n : ℕ => ENNReal.ofReal + (levelStoppingCurrentGreen κ B γ G n S z ω)) := by + filter_upwards [] with ω + intro n k hnk + exact ENNReal.ofReal_le_ofReal (levelStoppingCurrentGreen_mono ω (hc ω) (hw ω) + z S (Nat.cast_nonneg n) (Nat.cast_le.mpr hnk)) + have hlim : ∀ᵐ ω ∂P, Tendsto (fun n : ℕ => ENNReal.ofReal + (levelStoppingCurrentGreen κ B γ G n S z ω)) atTop + (𝓝 (ENNReal.ofReal (stoppingGreen κ B γ G S z ω))) := by + filter_upwards [] with ω + apply tendsto_const_nhds.congr' + exact (levelStoppingCurrentGreen_eventually_eq ω (hc ω) (hw ω) z S + (fun n : ℕ => (n:ℝ)) (fun n => Nat.cast_nonneg n) tendsto_natCast_atTop_atTop).mono + (fun _ he => congrArg ENNReal.ofReal he.symm) + exact le_of_tendsto (lintegral_tendsto_of_tendsto_of_monotone + (fun n => (hm n).aemeasurable) hmono hlim) + (Eventually.of_forall (fun n => levelStoppingCurrentGreen_mean_le hB hBm hc hB0 hκ hκ8 + γ G F hw z hz n (Nat.cast_nonneg n) S hS)) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelStoppingKernel.lean b/lean/OAI/Probability/SLE/Mass/LevelStoppingKernel.lean index a534ea323f712b90499add3a80abbea82b91d88b..742181ee213a975f3af2fe68331da0763a9acb18 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelStoppingKernel.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelStoppingKernel.lean @@ -1,230 +1,230 @@ -import OAI.Probability.SLE.Mass.LevelApproximationKernel -import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular -import OAI.Probability.SLE.Gauge.ExtendedExperimentStop - -namespace OAI - -/-! The finite-level remaining Green kernel at any Brownian stopping -time. Both sides vanish at an infinite stopping time. Finite cutoffs use -the existing optional identity for the stopped Green martingale. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -def levelStoppingRemainderGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : ℝ := - if S ω < radiusLevelTime (γ ω) (G ω) z u then - stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u else 0 - -def levelStoppingCurrentGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : ℝ := - if S ω < radiusLevelTime (γ ω) (G ω) z u then - greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (S ω).toNNReal z else 0 - -lemma levelStoppingRemainderGreen_nonneg - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : - 0 ≤ levelStoppingRemainderGreen κ B γ G u S z ω := by - unfold levelStoppingRemainderGreen - split_ifs - · exact stoppedGreen_nonneg (hw ω) κ z u - · exact le_rfl - -lemma levelStoppingCurrentGreen_nonneg - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : - 0 ≤ levelStoppingCurrentGreen κ B γ G u S z ω := by - unfold levelStoppingCurrentGreen - split_ifs - · exact greenDensity_nonneg (hw ω) κ _ z - · exact le_rfl - -theorem levelStoppingRemainderGreen_finite_condExp - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) - (σ : Ω → ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (σ ω : WithTop ℝ≥0))) : - Integrable (levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z) P ∧ - Integrable (levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z) P ∧ - P[levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z | hσ.measurableSpace] =ᵐ[P] - levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - obtain ⟨M,Z,_,hZi,_,_,_,hStop,heq,_⟩ := exists_physical_green_stopping_regular - hB hBm hc hB0 κ hκ hκ8 γ G F (ae_of_all _ hw) z hz u hu - have hτ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u hu - let A : Set Ω := {ω | (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u} - have hA : MeasurableSet[hσ.measurableSpace] A := by - have he : A = {ω | radiusLevelTime (γ ω) (G ω) z u ≤ (σ ω : WithTop ℝ≥0)}ᶜ := by - ext ω - change (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ - ¬ radiusLevelTime (γ ω) (G ω) z u ≤ (σ ω : ℝ≥0∞) - exact lt_iff_not_ge - rw [he] - exact (hτ.measurableSet_stopping_time_le hσ).compl - have hAg : MeasurableSet A := hσ.measurableSpace_le _ hA - have hR : levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z =ᵐ[P] A.indicator Z := by - filter_upwards [heq] with ω he - by_cases ha : ω ∈ A - · have ht : (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha - rw [levelStoppingRemainderGreen,ite_eq_left ht,indicator_of_mem ha,he.2] - · have ht : ¬ (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha - rw [levelStoppingRemainderGreen,ite_eq_right ht,indicator_of_notMem ha] - have hC : levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z =ᵐ[P] - A.indicator (fun ω => M (σ ω) ω) := by - filter_upwards [heq] with ω he - by_cases ha : ω ∈ A - · have ht : (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha - rw [levelStoppingCurrentGreen,ite_eq_left ht,indicator_of_mem ha,ENNReal.toNNReal_coe, - he.1 (σ ω),physicalStoppedGreen,ite_eq_right (not_le.mpr ht)] - · have ht : ¬ (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha - rw [levelStoppingCurrentGreen,ite_eq_right ht,indicator_of_notMem ha] - have hMi : Integrable (fun ω => M (σ ω) ω) P := - integrable_condExp.congr (hStop σ hσ).symm - refine ⟨(hZi.indicator hAg).congr hR.symm,(hMi.indicator hAg).congr hC.symm,?_⟩ - apply (condExp_congr_ae hR).trans - apply (condExp_indicator hZi hA).trans - apply EventuallyEq.trans _ hC.symm - filter_upwards [hStop σ hσ] with ω he - by_cases ha : ω ∈ A - · simpa only [indicator_of_mem ha] using he.symm - · simp only [indicator_of_notMem ha] - -theorem levelStoppingRemainderGreen_finite_setLIntegral - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) - (σ : Ω → ℝ≥0) - (hσ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (σ ω : WithTop ℝ≥0))) - (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : - (∫⁻ ω in A, ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z ω) ∂P) = - ∫⁻ ω in A, ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z ω) ∂P := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - obtain ⟨hR,hC,he⟩ := levelStoppingRemainderGreen_finite_condExp hB hBm hc hB0 hκ hκ8 - γ G F hw z hz u hu σ hσ - have hRn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z ω := - ae_of_all _ (levelStoppingRemainderGreen_nonneg κ B γ G F hw u _ z) - have hCn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z ω := - ae_of_all _ (levelStoppingCurrentGreen_nonneg κ B γ G F hw u _ z) - rw [← ofReal_integral_eq_lintegral_ofReal hR.integrableOn hRn, - ← ofReal_integral_eq_lintegral_ofReal hC.integrableOn hCn] - congr 1 - have : IsFiniteMeasure (P.trim hσ.measurableSpace_le) := isFiniteMeasure_trim hσ.measurableSpace_le - exact (setIntegral_condExp hσ.measurableSpace_le hR hA).symm.trans - (integral_congr_ae (ae_restrict_of_ae he)) - -private lemma stopped_setLIntegral_cutoffs - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) (f : Ω → ℝ≥0∞) - (hz : ∀ ω, S ω = ∞ → f ω = 0) : - (∫⁻ ω in E, f ω ∂P) = - ⨆ n : ℕ, ∫⁻ ω in E ∩ {ω | S ω ≤ (n:ℝ≥0)}, f ω ∂P := by - let A (n : ℕ) := E ∩ {ω | S ω ≤ (n:ℝ≥0)} - have hmono : Monotone A := by - intro n k hnk ω hω - refine ⟨hω.1,?_⟩ - change S ω ≤ ((k:ℝ≥0):ℝ≥0∞) - exact hω.2.trans (ENNReal.coe_le_coe.mpr (by exact_mod_cast hnk)) - have hcover : (⋃ n, A n) = E ∩ {ω | S ω < ∞} := by - ext ω - simp only [mem_iUnion,A,mem_inter_iff,mem_ofPred_eq] - constructor - · rintro ⟨n,hnE,hn⟩ - exact ⟨hnE,hn.trans_lt ENNReal.coe_lt_top⟩ - · rintro ⟨hE,hfin⟩ - obtain ⟨n,hn⟩ := exists_nat_ge ((S ω).toReal) - refine ⟨n,hE,?_⟩ - exact (ENNReal.toReal_le_toReal hfin.ne ENNReal.coe_ne_top).mp (by simpa using hn) - have hsupp : f.support ⊆ {ω | S ω < ∞} := by - intro ω hω - by_contra hn - exact hω (hz ω (top_le_iff.mp (le_of_not_gt hn))) - calc - (∫⁻ ω in E, f ω ∂P) = ∫⁻ ω in {ω | S ω < ∞}, f ω ∂P.restrict E := - (setLIntegral_eq_of_support_subset hsupp).symm - _ = ∫⁻ ω in E ∩ {ω | S ω < ∞}, f ω ∂P := by - rw [Measure.restrict_restrict (measurableSet_lt hS measurable_const),inter_comm] - _ = ∫⁻ ω in ⋃ n, A n, f ω ∂P := by rw [hcover] - _ = _ := setLIntegral_iUnion_of_directed f hmono.directed_le - -theorem levelStoppingRemainderGreen_setLIntegral - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) - (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) - (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : - (∫⁻ ω in E, ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω) ∂P) = - ∫⁻ ω in E, ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) ∂P := by - classical - let ℱ := Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) - have hRzero (ω : Ω) (hω : S ω = ∞) : - ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω) = 0 := by - simp [levelStoppingRemainderGreen,hω] - have hCzero (ω : Ω) (hω : S ω = ∞) : - ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) = 0 := by - simp [levelStoppingCurrentGreen,hω] - rw [stopped_setLIntegral_cutoffs P S hS.measurable' E _ hRzero, - stopped_setLIntegral_cutoffs P S hS.measurable' E _ hCzero] - apply iSup_congr - intro n - let σ := cutoffStop S (n:ℝ≥0) - have hσ := cutoffStop_stopping ℱ S hS (n:ℝ≥0) - have hlocal := levelStoppingRemainderGreen_finite_setLIntegral hB hBm hc hB0 hκ hκ8 - γ G F hw z hz u hu σ hσ (E ∩ {ω | S ω ≤ (n:ℝ≥0)}) - (cutoffStop_success_measurable ℱ S hS (n:ℝ≥0) E hE) - have he (ω : Ω) (hω : ω ∈ E ∩ {ω | S ω ≤ (n:ℝ≥0)}) : (σ ω : ℝ≥0∞) = S ω := by - change (cutoffStop S (n:ℝ≥0) ω : WithTop ℝ≥0) = S ω - rw [cutoffStop_coe,min_eq_left hω.2] - calc - _ = ∫⁻ ω in E ∩ {ω | S ω ≤ (n:ℝ≥0)}, - ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z ω) ∂P := by - apply setLIntegral_congr_fun (hS.measurableSpace_le _ - (hE.inter (hS.measurableSet_le' (n:ℝ≥0)))) - intro ω hω - simp only [levelStoppingRemainderGreen,he ω hω] - split_ifs <;> rfl - _ = _ := hlocal - _ = _ := by - apply setLIntegral_congr_fun (hS.measurableSpace_le _ - (hE.inter (hS.measurableSet_le' (n:ℝ≥0)))) - intro ω hω - simp only [levelStoppingCurrentGreen,he ω hω] - split_ifs <;> rfl - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelApproximationKernel +import OAI.Probability.SLE.Loewner.ActualRadiusStopsRegular +import OAI.Probability.SLE.Gauge.ExtendedExperimentStop + +namespace OAI + +/-! The finite-level remaining Green kernel at any Brownian stopping +time. Both sides vanish at an infinite stopping time. Finite cutoffs use +the existing optional identity for the stopped Green martingale. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +def levelStoppingRemainderGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : ℝ := + if S ω < radiusLevelTime (γ ω) (G ω) z u then + stoppedGreen κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) z u else 0 + +def levelStoppingCurrentGreen {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : ℝ := + if S ω < radiusLevelTime (γ ω) (G ω) z u then + greenDensity κ (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (S ω).toNNReal z else 0 + +lemma levelStoppingRemainderGreen_nonneg + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : + 0 ≤ levelStoppingRemainderGreen κ B γ G u S z ω := by + unfold levelStoppingRemainderGreen + split_ifs + · exact stoppedGreen_nonneg (hw ω) κ z u + · exact le_rfl + +lemma levelStoppingCurrentGreen_nonneg + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) (ω : Ω) : + 0 ≤ levelStoppingCurrentGreen κ B γ G u S z ω := by + unfold levelStoppingCurrentGreen + split_ifs + · exact greenDensity_nonneg (hw ω) κ _ z + · exact le_rfl + +theorem levelStoppingRemainderGreen_finite_condExp + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) + (σ : Ω → ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (σ ω : WithTop ℝ≥0))) : + Integrable (levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z) P ∧ + Integrable (levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z) P ∧ + P[levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z | hσ.measurableSpace] =ᵐ[P] + levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + obtain ⟨M,Z,_,hZi,_,_,_,hStop,heq,_⟩ := exists_physical_green_stopping_regular + hB hBm hc hB0 κ hκ hκ8 γ G F (ae_of_all _ hw) z hz u hu + have hτ := actual_radiusLevelTime_stopping hB hm hBm hc hB0 κ hκ hκ8 γ G F hw z hz u hu + let A : Set Ω := {ω | (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u} + have hA : MeasurableSet[hσ.measurableSpace] A := by + have he : A = {ω | radiusLevelTime (γ ω) (G ω) z u ≤ (σ ω : WithTop ℝ≥0)}ᶜ := by + ext ω + change (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u ↔ + ¬ radiusLevelTime (γ ω) (G ω) z u ≤ (σ ω : ℝ≥0∞) + exact lt_iff_not_ge + rw [he] + exact (hτ.measurableSet_stopping_time_le hσ).compl + have hAg : MeasurableSet A := hσ.measurableSpace_le _ hA + have hR : levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z =ᵐ[P] A.indicator Z := by + filter_upwards [heq] with ω he + by_cases ha : ω ∈ A + · have ht : (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha + rw [levelStoppingRemainderGreen,ite_eq_left ht,indicator_of_mem ha,he.2] + · have ht : ¬ (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha + rw [levelStoppingRemainderGreen,ite_eq_right ht,indicator_of_notMem ha] + have hC : levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z =ᵐ[P] + A.indicator (fun ω => M (σ ω) ω) := by + filter_upwards [heq] with ω he + by_cases ha : ω ∈ A + · have ht : (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha + rw [levelStoppingCurrentGreen,ite_eq_left ht,indicator_of_mem ha,ENNReal.toNNReal_coe, + he.1 (σ ω),physicalStoppedGreen,ite_eq_right (not_le.mpr ht)] + · have ht : ¬ (σ ω : ℝ≥0∞) < radiusLevelTime (γ ω) (G ω) z u := ha + rw [levelStoppingCurrentGreen,ite_eq_right ht,indicator_of_notMem ha] + have hMi : Integrable (fun ω => M (σ ω) ω) P := + integrable_condExp.congr (hStop σ hσ).symm + refine ⟨(hZi.indicator hAg).congr hR.symm,(hMi.indicator hAg).congr hC.symm,?_⟩ + apply (condExp_congr_ae hR).trans + apply (condExp_indicator hZi hA).trans + apply EventuallyEq.trans _ hC.symm + filter_upwards [hStop σ hσ] with ω he + by_cases ha : ω ∈ A + · simpa only [indicator_of_mem ha] using he.symm + · simp only [indicator_of_notMem ha] + +theorem levelStoppingRemainderGreen_finite_setLIntegral + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) + (σ : Ω → ℝ≥0) + (hσ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (σ ω : WithTop ℝ≥0))) + (A : Set Ω) (hA : MeasurableSet[hσ.measurableSpace] A) : + (∫⁻ ω in A, ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z ω) ∂P) = + ∫⁻ ω in A, ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z ω) ∂P := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + obtain ⟨hR,hC,he⟩ := levelStoppingRemainderGreen_finite_condExp hB hBm hc hB0 hκ hκ8 + γ G F hw z hz u hu σ hσ + have hRn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z ω := + ae_of_all _ (levelStoppingRemainderGreen_nonneg κ B γ G F hw u _ z) + have hCn : ∀ᵐ ω ∂P.restrict A, 0 ≤ levelStoppingCurrentGreen κ B γ G u (fun ω => σ ω) z ω := + ae_of_all _ (levelStoppingCurrentGreen_nonneg κ B γ G F hw u _ z) + rw [← ofReal_integral_eq_lintegral_ofReal hR.integrableOn hRn, + ← ofReal_integral_eq_lintegral_ofReal hC.integrableOn hCn] + congr 1 + have : IsFiniteMeasure (P.trim hσ.measurableSpace_le) := isFiniteMeasure_trim hσ.measurableSpace_le + exact (setIntegral_condExp hσ.measurableSpace_le hR hA).symm.trans + (integral_congr_ae (ae_restrict_of_ae he)) + +private lemma stopped_setLIntegral_cutoffs + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) (f : Ω → ℝ≥0∞) + (hz : ∀ ω, S ω = ∞ → f ω = 0) : + (∫⁻ ω in E, f ω ∂P) = + ⨆ n : ℕ, ∫⁻ ω in E ∩ {ω | S ω ≤ (n:ℝ≥0)}, f ω ∂P := by + let A (n : ℕ) := E ∩ {ω | S ω ≤ (n:ℝ≥0)} + have hmono : Monotone A := by + intro n k hnk ω hω + refine ⟨hω.1,?_⟩ + change S ω ≤ ((k:ℝ≥0):ℝ≥0∞) + exact hω.2.trans (ENNReal.coe_le_coe.mpr (by exact_mod_cast hnk)) + have hcover : (⋃ n, A n) = E ∩ {ω | S ω < ∞} := by + ext ω + simp only [mem_iUnion,A,mem_inter_iff,mem_ofPred_eq] + constructor + · rintro ⟨n,hnE,hn⟩ + exact ⟨hnE,hn.trans_lt ENNReal.coe_lt_top⟩ + · rintro ⟨hE,hfin⟩ + obtain ⟨n,hn⟩ := exists_nat_ge ((S ω).toReal) + refine ⟨n,hE,?_⟩ + exact (ENNReal.toReal_le_toReal hfin.ne ENNReal.coe_ne_top).mp (by simpa using hn) + have hsupp : f.support ⊆ {ω | S ω < ∞} := by + intro ω hω + by_contra hn + exact hω (hz ω (top_le_iff.mp (le_of_not_gt hn))) + calc + (∫⁻ ω in E, f ω ∂P) = ∫⁻ ω in {ω | S ω < ∞}, f ω ∂P.restrict E := + (setLIntegral_eq_of_support_subset hsupp).symm + _ = ∫⁻ ω in E ∩ {ω | S ω < ∞}, f ω ∂P := by + rw [Measure.restrict_restrict (measurableSet_lt hS measurable_const),inter_comm] + _ = ∫⁻ ω in ⋃ n, A n, f ω ∂P := by rw [hcover] + _ = _ := setLIntegral_iUnion_of_directed f hmono.directed_le + +theorem levelStoppingRemainderGreen_setLIntegral + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (z : ℂ) (hz : z ∈ upperHalfPlane) (u : ℝ) (hu : 0 ≤ u) + (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) + (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : + (∫⁻ ω in E, ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω) ∂P) = + ∫⁻ ω in E, ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) ∂P := by + classical + let ℱ := Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable) + have hRzero (ω : Ω) (hω : S ω = ∞) : + ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω) = 0 := by + simp [levelStoppingRemainderGreen,hω] + have hCzero (ω : Ω) (hω : S ω = ∞) : + ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) = 0 := by + simp [levelStoppingCurrentGreen,hω] + rw [stopped_setLIntegral_cutoffs P S hS.measurable' E _ hRzero, + stopped_setLIntegral_cutoffs P S hS.measurable' E _ hCzero] + apply iSup_congr + intro n + let σ := cutoffStop S (n:ℝ≥0) + have hσ := cutoffStop_stopping ℱ S hS (n:ℝ≥0) + have hlocal := levelStoppingRemainderGreen_finite_setLIntegral hB hBm hc hB0 hκ hκ8 + γ G F hw z hz u hu σ hσ (E ∩ {ω | S ω ≤ (n:ℝ≥0)}) + (cutoffStop_success_measurable ℱ S hS (n:ℝ≥0) E hE) + have he (ω : Ω) (hω : ω ∈ E ∩ {ω | S ω ≤ (n:ℝ≥0)}) : (σ ω : ℝ≥0∞) = S ω := by + change (cutoffStop S (n:ℝ≥0) ω : WithTop ℝ≥0) = S ω + rw [cutoffStop_coe,min_eq_left hω.2] + calc + _ = ∫⁻ ω in E ∩ {ω | S ω ≤ (n:ℝ≥0)}, + ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u (fun ω => σ ω) z ω) ∂P := by + apply setLIntegral_congr_fun (hS.measurableSpace_le _ + (hE.inter (hS.measurableSet_le' (n:ℝ≥0)))) + intro ω hω + simp only [levelStoppingRemainderGreen,he ω hω] + split_ifs <;> rfl + _ = _ := hlocal + _ = _ := by + apply setLIntegral_congr_fun (hS.measurableSpace_le _ + (hE.inter (hS.measurableSet_le' (n:ℝ≥0)))) + intro ω hω + simp only [levelStoppingCurrentGreen,he ω hω] + split_ifs <;> rfl + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelStoppingMeasures.lean b/lean/OAI/Probability/SLE/Mass/LevelStoppingMeasures.lean index ff84368f19a408293d727d3abe3a94e8699dd7d4..e484b3352cfc17b1514e5297bc9981a17ad44859 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelStoppingMeasures.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelStoppingMeasures.lean @@ -1,280 +1,280 @@ -import OAI.Probability.SLE.Mass.LevelStoppingGreenLimit -import OAI.Probability.SLE.Mass.LevelMeanMeasure - -namespace OAI - -/-! The unreached part of a finite-level approximation at an arbitrary -stopping time, as a positive measure. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -lemma levelApproximationDensity_random_time_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun p : Ω × ℂ => levelApproximationDensity κ B γ G u (S p.1) p.1 p.2) := - Measurable.ite (measurableSet_le - (actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw) (hS.comp measurable_fst)) - (actual_stoppedGreen_joint_measurable hm hc κ u hu γ G F hw).ennreal_ofReal measurable_const - -lemma halfPlaneLevelMeasure_random_time_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun ω => halfPlaneLevelMeasure κ B γ G u (S ω) ω) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneLevelMeasure,levelApproximation, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - exact (levelApproximationDensity_random_time_joint_measurable hm hc κ u hu γ G F hw S hS).lintegral_prod_right' - -def halfPlaneStoppingRemainderMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (S : Ω → ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω))).comap Subtype.val - -def halfPlaneStoppingCurrentMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (S : Ω → ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω))).comap Subtype.val - -def halfPlaneStoppingGreenMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (S : Ω → ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := - (volume.withDensity (fun z => ENNReal.ofReal (stoppingGreen κ B γ G S z ω))).comap Subtype.val - -lemma stoppingGreen_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun p : Ω × ℂ => stoppingGreen κ B γ G S p.2 p.1) := - Measurable.ite (measurableSet_lt (hS.comp measurable_fst) measurable_const) - (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw - (fun ω => (S ω).toNNReal) (ENNReal.measurable_toNNReal.comp hS)) measurable_const - -lemma halfPlaneStoppingGreenMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (halfPlaneStoppingGreenMeasure κ B γ G S) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneStoppingGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (stoppingGreen_joint_measurable hBm hc κ γ G F hw S hS).ennreal_ofReal.lintegral_prod_right' - -lemma halfPlaneStoppingRemainderMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : Measurable (halfPlaneStoppingRemainderMeasure κ B γ G u S) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneStoppingRemainderMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (levelStoppingRemainderGreen_joint_measurable hm hBc κ u hu γ G F hw S hS).ennreal_ofReal.lintegral_prod_right' - -lemma halfPlaneStoppingCurrentMeasure_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : Measurable (halfPlaneStoppingCurrentMeasure κ B γ G u S) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneStoppingCurrentMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, - withDensity_apply _ hAI] - exact (levelStoppingCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw S hS).ennreal_ofReal.lintegral_prod_right' - -lemma halfPlaneLevelMeasure_stopping_split - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) (ω : Ω) : - halfPlaneLevelMeasure κ B γ G u ∞ ω = - halfPlaneLevelMeasure κ B γ G u (S ω) ω + halfPlaneStoppingRemainderMeasure κ B γ G u S ω := by - apply Measure.ext - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [Measure.add_apply,halfPlaneLevelMeasure,halfPlaneStoppingRemainderMeasure,levelApproximation, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - simp_rw [stopping_level_density_split κ B γ G u S ω] - have hd : Measurable (fun z => levelApproximationDensity κ B γ G u (S ω) ω z) := by - have hp : Measurable (fun z : ℂ => (ω,z)) := measurable_const.prodMk measurable_id - have hh := (levelApproximationDensity_random_time_joint_measurable hm hBc κ u hu - γ G F hw S hS).comp hp - simpa only [Function.comp_def] using hh - exact lintegral_add_left hd _ - -theorem halfPlaneStoppingRemainderMeasure_conditional_mean - {Ω : Type*} [MeasurableSpace Ω] - {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) - (E : Set Ω) - (hE : MeasurableSet[hS.measurableSpace] E) : - (P.restrict E).bind (halfPlaneStoppingRemainderMeasure κ B γ G u S) = - (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G u S) := by - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - apply Measure.ext - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by - rintro _ ⟨x,_,rfl⟩; exact x.property - rw [Measure.bind_apply hA (halfPlaneStoppingRemainderMeasure_measurable hm hBc κ u hu γ G F hw S hS.measurable').aemeasurable, - Measure.bind_apply hA (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ u hu γ G F hw S hS.measurable').aemeasurable] - simp only [halfPlaneStoppingRemainderMeasure,halfPlaneStoppingCurrentMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - simpa only [one_mul] using levelStoppingRemainderGreen_spatial_kernel hB hBm hBc hB0 hκ hκ8 γ G F hw - u hu S hS _ hAI hAH (fun _ => 1) measurable_const E hE - -theorem halfPlaneLevelMeasure_stopping_conditional_split - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) - (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : - (P.restrict E).bind (halfPlaneLevelMeasure κ B γ G u ∞) = - (P.restrict E).bind (fun ω => halfPlaneLevelMeasure κ B γ G u (S ω) ω) + - (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G u S) := by - let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) - rw [← halfPlaneStoppingRemainderMeasure_conditional_mean hB hBm hc hB0 hκ hκ8 - γ G F hw u hu S hS E hE] - apply Measure.ext - intro A hA - rw [Measure.add_apply, - Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hc κ u hu γ G F hw ∞).aemeasurable, - Measure.bind_apply hA (halfPlaneLevelMeasure_random_time_measurable hm hc κ u hu - γ G F hw S hS.measurable').aemeasurable, - Measure.bind_apply hA (halfPlaneStoppingRemainderMeasure_measurable hm hc κ u hu - γ G F hw S hS.measurable').aemeasurable] - simp_rw [halfPlaneLevelMeasure_stopping_split hm hc κ u hu γ G F hw S hS.measurable',Measure.add_apply] - exact lintegral_add_left (((Measure.measurable_coe hA).comp - (halfPlaneLevelMeasure_random_time_measurable hm hc κ u hu γ G F hw S hS.measurable'))) _ - -theorem halfPlaneStoppingGreenMeasure_mean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) - (E : Set Ω) : - (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) ≤ initialGreenMeasure κ := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - apply Measure.le_iff.mpr - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by - rintro _ ⟨x,_,rfl⟩ - exact x.property - rw [Measure.bind_apply hA - (halfPlaneStoppingGreenMeasure_measurable hBm hc κ γ G F hw S hS.measurable').aemeasurable] - simp only [halfPlaneStoppingGreenMeasure,initialGreenMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - apply (lintegral_mono' Measure.restrict_le_self (le_refl _)).trans - rw [lintegral_lintegral_swap - (stoppingGreen_joint_measurable hBm hc κ γ G F hw S hS.measurable').ennreal_ofReal.aemeasurable] - exact setLIntegral_mono' hAI (fun z hz => - stoppingGreen_mean_le hB hBm hc hB0 hκ hκ8 γ G F hw z (hAH hz) S hS) - -theorem stoppingCurrentGreen_spatial_mean_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (S : Ω → ℝ≥0∞) (hS : Measurable S) (K : Set ℂ) - (f : ℂ → ℝ≥0∞) (hf : Measurable f) (E : Set Ω) : - Tendsto (fun n => ∫⁻ ω in E, (∫⁻ z in K, - f z * ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G (u n) S z ω) ∂volume) ∂P) - atTop (𝓝 (∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal - (stoppingGreen κ B γ G S z ω) ∂volume) ∂P)) := by - let a (n : ℕ) (q : Ω × ℂ) := - f q.2 * ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G (u n) S q.2 q.1) - have ha (n : ℕ) : Measurable (a n) := - (hf.comp measurable_snd).mul - (levelStoppingCurrentGreen_joint_measurable hBm hBc κ (u n) (hu n) γ G F hw S hS).ennreal_ofReal - have ham (ω : Ω) : Monotone (fun n => fun z => a n (ω,z)) := by - intro n m hnm z - exact mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal - (levelStoppingCurrentGreen_mono ω (hBc ω) (hw ω) z S (hu n) (hmono hnm))) - have hal (ω : Ω) : Tendsto (fun n => ∫⁻ z in K, a n (ω,z)) atTop - (𝓝 (∫⁻ z in K, f z * ENNReal.ofReal - (stoppingGreen κ B γ G S z ω))) := by - apply lintegral_tendsto_of_tendsto_of_monotone (μ := volume.restrict K) - (f := fun n z => a n (ω,z)) - (fun n => ((ha n).comp (measurable_const.prodMk measurable_id)).aemeasurable) - (Eventually.of_forall (fun z n m hnm => ham ω hnm z)) - filter_upwards [] with z - apply tendsto_const_nhds.congr' - exact (levelStoppingCurrentGreen_eventually_eq ω (hBc ω) (hw ω) z S u hu hlim).mono - (fun n hn => congrArg (fun x : ℝ => f z * ENNReal.ofReal x) hn.symm) - exact lintegral_tendsto_of_tendsto_of_monotone - (fun n => (ha n).lintegral_prod_right'.aemeasurable) - (Eventually.of_forall (fun ω n m hnm => lintegral_mono (ham ω hnm))) - (Eventually.of_forall hal) - -theorem halfPlaneStoppingCurrentMeasure_mean_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) - (A : Set upperHalfPlane) (hA : MeasurableSet A) : - Tendsto (fun n => (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u n) S) A) - atTop (𝓝 ((P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) A)) := by - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [Measure.bind_apply hA - (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ (u _) (hu _) γ G F hw S hS).aemeasurable, - Measure.bind_apply hA (halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS).aemeasurable, - halfPlaneStoppingCurrentMeasure,halfPlaneStoppingGreenMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - simpa only [one_mul] using stoppingCurrentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim S hS _ (fun _ => 1) measurable_const E - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelStoppingGreenLimit +import OAI.Probability.SLE.Mass.LevelMeanMeasure + +namespace OAI + +/-! The unreached part of a finite-level approximation at an arbitrary +stopping time, as a positive measure. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +lemma levelApproximationDensity_random_time_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun p : Ω × ℂ => levelApproximationDensity κ B γ G u (S p.1) p.1 p.2) := + Measurable.ite (measurableSet_le + (actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw) (hS.comp measurable_fst)) + (actual_stoppedGreen_joint_measurable hm hc κ u hu γ G F hw).ennreal_ofReal measurable_const + +lemma halfPlaneLevelMeasure_random_time_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun ω => halfPlaneLevelMeasure κ B γ G u (S ω) ω) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneLevelMeasure,levelApproximation, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + exact (levelApproximationDensity_random_time_joint_measurable hm hc κ u hu γ G F hw S hS).lintegral_prod_right' + +def halfPlaneStoppingRemainderMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (S : Ω → ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω))).comap Subtype.val + +def halfPlaneStoppingCurrentMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (S : Ω → ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω))).comap Subtype.val + +def halfPlaneStoppingGreenMeasure {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (S : Ω → ℝ≥0∞) (ω : Ω) : Measure upperHalfPlane := + (volume.withDensity (fun z => ENNReal.ofReal (stoppingGreen κ B γ G S z ω))).comap Subtype.val + +lemma stoppingGreen_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun p : Ω × ℂ => stoppingGreen κ B γ G S p.2 p.1) := + Measurable.ite (measurableSet_lt (hS.comp measurable_fst) measurable_const) + (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw + (fun ω => (S ω).toNNReal) (ENNReal.measurable_toNNReal.comp hS)) measurable_const + +lemma halfPlaneStoppingGreenMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (halfPlaneStoppingGreenMeasure κ B γ G S) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneStoppingGreenMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (stoppingGreen_joint_measurable hBm hc κ γ G F hw S hS).ennreal_ofReal.lintegral_prod_right' + +lemma halfPlaneStoppingRemainderMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : Measurable (halfPlaneStoppingRemainderMeasure κ B γ G u S) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneStoppingRemainderMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (levelStoppingRemainderGreen_joint_measurable hm hBc κ u hu γ G F hw S hS).ennreal_ofReal.lintegral_prod_right' + +lemma halfPlaneStoppingCurrentMeasure_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : Measurable (halfPlaneStoppingCurrentMeasure κ B γ G u S) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneStoppingCurrentMeasure,comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet, + withDensity_apply _ hAI] + exact (levelStoppingCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw S hS).ennreal_ofReal.lintegral_prod_right' + +lemma halfPlaneLevelMeasure_stopping_split + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) (ω : Ω) : + halfPlaneLevelMeasure κ B γ G u ∞ ω = + halfPlaneLevelMeasure κ B γ G u (S ω) ω + halfPlaneStoppingRemainderMeasure κ B γ G u S ω := by + apply Measure.ext + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [Measure.add_apply,halfPlaneLevelMeasure,halfPlaneStoppingRemainderMeasure,levelApproximation, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + simp_rw [stopping_level_density_split κ B γ G u S ω] + have hd : Measurable (fun z => levelApproximationDensity κ B γ G u (S ω) ω z) := by + have hp : Measurable (fun z : ℂ => (ω,z)) := measurable_const.prodMk measurable_id + have hh := (levelApproximationDensity_random_time_joint_measurable hm hBc κ u hu + γ G F hw S hS).comp hp + simpa only [Function.comp_def] using hh + exact lintegral_add_left hd _ + +theorem halfPlaneStoppingRemainderMeasure_conditional_mean + {Ω : Type*} [MeasurableSpace Ω] + {P : Measure Ω} {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) + (E : Set Ω) + (hE : MeasurableSet[hS.measurableSpace] E) : + (P.restrict E).bind (halfPlaneStoppingRemainderMeasure κ B γ G u S) = + (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G u S) := by + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + apply Measure.ext + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by + rintro _ ⟨x,_,rfl⟩; exact x.property + rw [Measure.bind_apply hA (halfPlaneStoppingRemainderMeasure_measurable hm hBc κ u hu γ G F hw S hS.measurable').aemeasurable, + Measure.bind_apply hA (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ u hu γ G F hw S hS.measurable').aemeasurable] + simp only [halfPlaneStoppingRemainderMeasure,halfPlaneStoppingCurrentMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + simpa only [one_mul] using levelStoppingRemainderGreen_spatial_kernel hB hBm hBc hB0 hκ hκ8 γ G F hw + u hu S hS _ hAI hAH (fun _ => 1) measurable_const E hE + +theorem halfPlaneLevelMeasure_stopping_conditional_split + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) + (E : Set Ω) (hE : MeasurableSet[hS.measurableSpace] E) : + (P.restrict E).bind (halfPlaneLevelMeasure κ B γ G u ∞) = + (P.restrict E).bind (fun ω => halfPlaneLevelMeasure κ B γ G u (S ω) ω) + + (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G u S) := by + let hm (s : ℝ≥0) : Measurable (B s) := hBm.comp (measurable_const.prodMk measurable_id) + rw [← halfPlaneStoppingRemainderMeasure_conditional_mean hB hBm hc hB0 hκ hκ8 + γ G F hw u hu S hS E hE] + apply Measure.ext + intro A hA + rw [Measure.add_apply, + Measure.bind_apply hA (halfPlaneLevelMeasure_measurable hm hc κ u hu γ G F hw ∞).aemeasurable, + Measure.bind_apply hA (halfPlaneLevelMeasure_random_time_measurable hm hc κ u hu + γ G F hw S hS.measurable').aemeasurable, + Measure.bind_apply hA (halfPlaneStoppingRemainderMeasure_measurable hm hc κ u hu + γ G F hw S hS.measurable').aemeasurable] + simp_rw [halfPlaneLevelMeasure_stopping_split hm hc κ u hu γ G F hw S hS.measurable',Measure.add_apply] + exact lintegral_add_left (((Measure.measurable_coe hA).comp + (halfPlaneLevelMeasure_random_time_measurable hm hc κ u hu γ G F hw S hS.measurable'))) _ + +theorem halfPlaneStoppingGreenMeasure_mean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) + (E : Set Ω) : + (P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) ≤ initialGreenMeasure κ := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + apply Measure.le_iff.mpr + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + have hAH : (Subtype.val : upperHalfPlane → ℂ) '' A ⊆ upperHalfPlane := by + rintro _ ⟨x,_,rfl⟩ + exact x.property + rw [Measure.bind_apply hA + (halfPlaneStoppingGreenMeasure_measurable hBm hc κ γ G F hw S hS.measurable').aemeasurable] + simp only [halfPlaneStoppingGreenMeasure,initialGreenMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + apply (lintegral_mono' Measure.restrict_le_self (le_refl _)).trans + rw [lintegral_lintegral_swap + (stoppingGreen_joint_measurable hBm hc κ γ G F hw S hS.measurable').ennreal_ofReal.aemeasurable] + exact setLIntegral_mono' hAI (fun z hz => + stoppingGreen_mean_le hB hBm hc hB0 hκ hκ8 γ G F hw z (hAH hz) S hS) + +theorem stoppingCurrentGreen_spatial_mean_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (S : Ω → ℝ≥0∞) (hS : Measurable S) (K : Set ℂ) + (f : ℂ → ℝ≥0∞) (hf : Measurable f) (E : Set Ω) : + Tendsto (fun n => ∫⁻ ω in E, (∫⁻ z in K, + f z * ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G (u n) S z ω) ∂volume) ∂P) + atTop (𝓝 (∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal + (stoppingGreen κ B γ G S z ω) ∂volume) ∂P)) := by + let a (n : ℕ) (q : Ω × ℂ) := + f q.2 * ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G (u n) S q.2 q.1) + have ha (n : ℕ) : Measurable (a n) := + (hf.comp measurable_snd).mul + (levelStoppingCurrentGreen_joint_measurable hBm hBc κ (u n) (hu n) γ G F hw S hS).ennreal_ofReal + have ham (ω : Ω) : Monotone (fun n => fun z => a n (ω,z)) := by + intro n m hnm z + exact mul_le_mul' le_rfl (ENNReal.ofReal_le_ofReal + (levelStoppingCurrentGreen_mono ω (hBc ω) (hw ω) z S (hu n) (hmono hnm))) + have hal (ω : Ω) : Tendsto (fun n => ∫⁻ z in K, a n (ω,z)) atTop + (𝓝 (∫⁻ z in K, f z * ENNReal.ofReal + (stoppingGreen κ B γ G S z ω))) := by + apply lintegral_tendsto_of_tendsto_of_monotone (μ := volume.restrict K) + (f := fun n z => a n (ω,z)) + (fun n => ((ha n).comp (measurable_const.prodMk measurable_id)).aemeasurable) + (Eventually.of_forall (fun z n m hnm => ham ω hnm z)) + filter_upwards [] with z + apply tendsto_const_nhds.congr' + exact (levelStoppingCurrentGreen_eventually_eq ω (hBc ω) (hw ω) z S u hu hlim).mono + (fun n hn => congrArg (fun x : ℝ => f z * ENNReal.ofReal x) hn.symm) + exact lintegral_tendsto_of_tendsto_of_monotone + (fun n => (ha n).lintegral_prod_right'.aemeasurable) + (Eventually.of_forall (fun ω n m hnm => lintegral_mono (ham ω hnm))) + (Eventually.of_forall hal) + +theorem halfPlaneStoppingCurrentMeasure_mean_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) + (A : Set upperHalfPlane) (hA : MeasurableSet A) : + Tendsto (fun n => (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u n) S) A) + atTop (𝓝 ((P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S) A)) := by + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [Measure.bind_apply hA + (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ (u _) (hu _) γ G F hw S hS).aemeasurable, + Measure.bind_apply hA (halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS).aemeasurable, + halfPlaneStoppingCurrentMeasure,halfPlaneStoppingGreenMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + simpa only [one_mul] using stoppingCurrentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim S hS _ (fun _ => 1) measurable_const E + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelStoppingSpatialKernel.lean b/lean/OAI/Probability/SLE/Mass/LevelStoppingSpatialKernel.lean index 9f9f1abfb8d8d28790d962d483748a1038fd7e56..80d255b97c7fc2774b94b105a4841b9098588539 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelStoppingSpatialKernel.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelStoppingSpatialKernel.lean @@ -1,114 +1,114 @@ -import OAI.Probability.SLE.Mass.LevelStoppingKernel - -namespace OAI - -/-! Joint spatial measurability and the spatial form of the finite-level -conditional kernel at an arbitrary, possibly infinite, stopping time. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -lemma actual_greenDensity_random_time_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (t : Ω → ℝ≥0) (ht : Measurable t) : - Measurable (fun p : Ω × ℂ => greenDensity κ - (fun s => Real.sqrt κ * B s p.1) (γ p.1) (G p.1) (t p.1) p.2) := by - let U (s : ℝ≥0) (ω : Ω) := Real.sqrt κ * B s ω - let c := spatialClippedFamily U (hBm.const_mul _) (fun ω => (hc ω).const_mul _) - have hmap : Measurable (fun p : Ω × ℂ => (t p.1,p.2,p.1)) := - (ht.comp measurable_fst).prodMk (measurable_snd.prodMk measurable_fst) - have hj := (c.jointGreen_measurable κ B hBm).comp hmap - have he (p : Ω × ℂ) : c.jointGreen κ B (t p.1,p.2,p.1) = - greenDensity κ (fun s => Real.sqrt κ * B s p.1) (γ p.1) (G p.1) (t p.1) p.2 := - c.jointGreen_eq κ B p.1 ((hc p.1).const_mul _) (fun _ => rfl) (hw p.1) _ _ - simpa only [Function.comp_def,he] using hj - -lemma levelStoppingRemainderGreen_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun p : Ω × ℂ => levelStoppingRemainderGreen κ B γ G u S p.2 p.1) := - Measurable.ite - (measurableSet_lt (hS.comp measurable_fst) - (actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw)) - (actual_stoppedGreen_joint_measurable hm hc κ u hu γ G F hw) measurable_const - -lemma levelStoppingCurrentGreen_joint_measurable - {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} - (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) - (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) : - Measurable (fun p : Ω × ℂ => levelStoppingCurrentGreen κ B γ G u S p.2 p.1) := by - let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - exact Measurable.ite - (measurableSet_lt (hS.comp measurable_fst) - (actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw)) - (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw - (fun ω => (S ω).toNNReal) (ENNReal.measurable_toNNReal.comp hS)) measurable_const - -theorem levelStoppingRemainderGreen_spatial_kernel - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) - {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) - (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) - (f : ℂ → ℝ≥0∞) (hf : Measurable f) (E : Set Ω) - (hE : MeasurableSet[hS.measurableSpace] E) : - (∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal - (levelStoppingRemainderGreen κ B γ G u S z ω) ∂volume) ∂P) = - ∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal - (levelStoppingCurrentGreen κ B γ G u S z ω) ∂volume) ∂P := by - let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure - let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) - have hmR := levelStoppingRemainderGreen_joint_measurable hm hc κ u hu γ G F hw S hS.measurable' - have hmC := levelStoppingCurrentGreen_joint_measurable hBm hc κ u hu γ G F hw S hS.measurable' - have hR := (hf.comp measurable_snd).mul hmR.ennreal_ofReal - have hC := (hf.comp measurable_snd).mul hmC.ennreal_ofReal - conv_lhs => rw [lintegral_lintegral_swap hR.aemeasurable] - conv_rhs => rw [lintegral_lintegral_swap hC.aemeasurable] - apply setLIntegral_congr_fun hK - intro z hz - dsimp only - have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const - have hRz := hmR.ennreal_ofReal.comp hp - have hCz := hmC.ennreal_ofReal.comp hp - simp only [Function.comp_def] at hRz hCz - rw [lintegral_const_mul _ hRz,lintegral_const_mul _ hCz] - rw [levelStoppingRemainderGreen_setLIntegral hB hBm hc hB0 hκ hκ8 - γ G F hw z (hKH hz) u hu S hS E hE] - -lemma stopping_level_density_split - {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) - (u : ℝ) (S : Ω → ℝ≥0∞) (ω : Ω) (z : ℂ) : - levelApproximationDensity κ B γ G u ∞ ω z = - levelApproximationDensity κ B γ G u (S ω) ω z + - ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω) := by - by_cases h : radiusLevelTime (γ ω) (G ω) z u ≤ S ω - · simp only [levelApproximationDensity,le_top,ite_true,ite_eq_left h, - levelStoppingRemainderGreen,ite_eq_right (not_lt.mpr h),ENNReal.ofReal_zero,add_zero] - · simp only [levelApproximationDensity,le_top,ite_true,ite_eq_right h, - levelStoppingRemainderGreen,ite_eq_left (lt_of_not_ge h),zero_add] - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelStoppingKernel + +namespace OAI + +/-! Joint spatial measurability and the spatial form of the finite-level +conditional kernel at an arbitrary, possibly infinite, stopping time. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +lemma actual_greenDensity_random_time_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ : ℝ) (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (t : Ω → ℝ≥0) (ht : Measurable t) : + Measurable (fun p : Ω × ℂ => greenDensity κ + (fun s => Real.sqrt κ * B s p.1) (γ p.1) (G p.1) (t p.1) p.2) := by + let U (s : ℝ≥0) (ω : Ω) := Real.sqrt κ * B s ω + let c := spatialClippedFamily U (hBm.const_mul _) (fun ω => (hc ω).const_mul _) + have hmap : Measurable (fun p : Ω × ℂ => (t p.1,p.2,p.1)) := + (ht.comp measurable_fst).prodMk (measurable_snd.prodMk measurable_fst) + have hj := (c.jointGreen_measurable κ B hBm).comp hmap + have he (p : Ω × ℂ) : c.jointGreen κ B (t p.1,p.2,p.1) = + greenDensity κ (fun s => Real.sqrt κ * B s p.1) (γ p.1) (G p.1) (t p.1) p.2 := + c.jointGreen_eq κ B p.1 ((hc p.1).const_mul _) (fun _ => rfl) (hw p.1) _ _ + simpa only [Function.comp_def,he] using hj + +lemma levelStoppingRemainderGreen_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hm : ∀ t, Measurable (B t)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun p : Ω × ℂ => levelStoppingRemainderGreen κ B γ G u S p.2 p.1) := + Measurable.ite + (measurableSet_lt (hS.comp measurable_fst) + (actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw)) + (actual_stoppedGreen_joint_measurable hm hc κ u hu γ G F hw) measurable_const + +lemma levelStoppingCurrentGreen_joint_measurable + {Ω : Type*} [MeasurableSpace Ω] {B : ℝ≥0 → Ω → ℝ} + (hBm : Measurable (Function.uncurry B)) (hc : ∀ ω, Continuous (fun t => B t ω)) + (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) : + Measurable (fun p : Ω × ℂ => levelStoppingCurrentGreen κ B γ G u S p.2 p.1) := by + let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + exact Measurable.ite + (measurableSet_lt (hS.comp measurable_fst) + (actual_radiusLevel_joint_measurable hm hc κ u hu γ G F hw)) + (actual_greenDensity_random_time_joint_measurable hBm hc κ γ G F hw + (fun ω => (S ω).toNNReal) (ENNReal.measurable_toNNReal.comp hS)) measurable_const + +theorem levelStoppingRemainderGreen_spatial_kernel + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (hB0 : ∀ ω, B 0 ω = 0) + {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) + (K : Set ℂ) (hK : MeasurableSet K) (hKH : K ⊆ upperHalfPlane) + (f : ℂ → ℝ≥0∞) (hf : Measurable f) (E : Set Ω) + (hE : MeasurableSet[hS.measurableSpace] E) : + (∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal + (levelStoppingRemainderGreen κ B γ G u S z ω) ∂volume) ∂P) = + ∫⁻ ω in E, (∫⁻ z in K, f z * ENNReal.ofReal + (levelStoppingCurrentGreen κ B γ G u S z ω) ∂volume) ∂P := by + let : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure + let hm (t : ℝ≥0) : Measurable (B t) := hBm.comp (measurable_const.prodMk measurable_id) + have hmR := levelStoppingRemainderGreen_joint_measurable hm hc κ u hu γ G F hw S hS.measurable' + have hmC := levelStoppingCurrentGreen_joint_measurable hBm hc κ u hu γ G F hw S hS.measurable' + have hR := (hf.comp measurable_snd).mul hmR.ennreal_ofReal + have hC := (hf.comp measurable_snd).mul hmC.ennreal_ofReal + conv_lhs => rw [lintegral_lintegral_swap hR.aemeasurable] + conv_rhs => rw [lintegral_lintegral_swap hC.aemeasurable] + apply setLIntegral_congr_fun hK + intro z hz + dsimp only + have hp : Measurable (fun ω : Ω => (ω,z)) := measurable_id.prodMk measurable_const + have hRz := hmR.ennreal_ofReal.comp hp + have hCz := hmC.ennreal_ofReal.comp hp + simp only [Function.comp_def] at hRz hCz + rw [lintegral_const_mul _ hRz,lintegral_const_mul _ hCz] + rw [levelStoppingRemainderGreen_setLIntegral hB hBm hc hB0 hκ hκ8 + γ G F hw z (hKH hz) u hu S hS E hE] + +lemma stopping_level_density_split + {Ω : Type*} (κ : ℝ) (B : ℝ≥0 → Ω → ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G : Ω → ℝ≥0 → ℂ → ℂ) + (u : ℝ) (S : Ω → ℝ≥0∞) (ω : Ω) (z : ℂ) : + levelApproximationDensity κ B γ G u ∞ ω z = + levelApproximationDensity κ B γ G u (S ω) ω z + + ENNReal.ofReal (levelStoppingRemainderGreen κ B γ G u S z ω) := by + by_cases h : radiusLevelTime (γ ω) (G ω) z u ≤ S ω + · simp only [levelApproximationDensity,le_top,ite_true,ite_eq_left h, + levelStoppingRemainderGreen,ite_eq_right (not_lt.mpr h),ENNReal.ofReal_zero,add_zero] + · simp only [levelApproximationDensity,le_top,ite_true,ite_eq_right h, + levelStoppingRemainderGreen,ite_eq_left (lt_of_not_ge h),zero_add] + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LevelStoppingTests.lean b/lean/OAI/Probability/SLE/Mass/LevelStoppingTests.lean index 40bdda8e95e6eb80f660a317e7b981998b364716..8224e062c4f0691d330ff8d66febe07926012088 100644 --- a/lean/OAI/Probability/SLE/Mass/LevelStoppingTests.lean +++ b/lean/OAI/Probability/SLE/Mass/LevelStoppingTests.lean @@ -1,146 +1,146 @@ -import OAI.Probability.SLE.Mass.LevelStoppingMeasures -import OAI.Probability.SLE.Mass.LevelCurrentTests -import OAI.Probability.SLE.Mass.LevelConditionalMean - -namespace OAI - -/-! Signed compact-test convergence for the Green remainder at arbitrary -stopping times, and its preservation by the chosen convex tails. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped NNReal ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma levelStoppingCurrentGreen_le_green - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hw : IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) : - levelStoppingCurrentGreen κ B γ G u S z ω ≤ stoppingGreen κ B γ G S z ω := by - by_cases hfin : S ω < ∞ - · unfold levelStoppingCurrentGreen stoppingGreen - rw [ite_eq_left hfin] - split_ifs - · exact le_rfl - · exact greenDensity_nonneg hw κ _ z - · have hnot : ¬ S ω < radiusLevelTime (γ ω) (G ω) z u := - fun h => hfin (h.trans_le le_top) - simp only [levelStoppingCurrentGreen,stoppingGreen,ite_eq_right hfin,ite_eq_right hnot] - exact le_rfl - -lemma halfPlaneStoppingCurrentMeasure_le_green - {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} - {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} - (ω : Ω) (hw : IsCapacityTwoWitness - (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (S : Ω → ℝ≥0∞) : - halfPlaneStoppingCurrentMeasure κ B γ G u S ω ≤ halfPlaneStoppingGreenMeasure κ B γ G S ω := by - apply Measure.le_iff.mpr - intro A hA - have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA - simp only [halfPlaneStoppingCurrentMeasure,halfPlaneStoppingGreenMeasure, - comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] - exact lintegral_mono (fun z => ENNReal.ofReal_le_ofReal (levelStoppingCurrentGreen_le_green ω hw u S z)) - -lemma halfPlaneStoppingCurrentMeasure_mean_lintegral_test - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) (f : upperHalfPlane → ℝ) (hf : Measurable f) : - (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneStoppingCurrentMeasure κ B γ G u S)) = - ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * - ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) ∂volume) ∂P := - lintegral_density_kernel_test P _ - (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ u hu γ G F hw S hS) _ - (levelStoppingCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw S hS).ennreal_ofReal - (fun _ => rfl) f hf - -lemma halfPlaneStoppingGreenMeasure_mean_lintegral_test - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (S : Ω → ℝ≥0∞) (hS : Measurable S) (f : upperHalfPlane → ℝ) (hf : Measurable f) : - (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneStoppingGreenMeasure κ B γ G S)) = - ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ENNReal.ofReal - (stoppingGreen κ B γ G S z ω) ∂volume) ∂P := - lintegral_density_kernel_test P _ (halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS) _ - (stoppingGreen_joint_measurable hBm hBc κ γ G F hw S hS).ennreal_ofReal - (fun _ => rfl) f hf - -lemma halfPlaneStoppingCurrentMeasure_mean_lintegral_test_tendsto - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) - (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) - (f : upperHalfPlane → ℝ) (hf : Measurable f) : - Tendsto (fun n => ∫⁻ z, ENNReal.ofReal (f z) - ∂(P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u n) S)) atTop - (𝓝 (∫⁻ z, ENNReal.ofReal (f z) ∂(P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S))) := by - simp_rw [halfPlaneStoppingCurrentMeasure_mean_lintegral_test _ hBm hBc κ _ (hu _) γ G F hw S hS f hf, - halfPlaneStoppingGreenMeasure_mean_lintegral_test _ hBm hBc κ γ G F hw S hS f hf] - simpa only [Measure.restrict_univ] using stoppingCurrentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim S hS univ (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) - (extendHalfPlaneTest_measurable f hf).ennreal_ofReal E - -lemma halfPlaneStoppingCurrentMeasure_mean_le - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) - (E : Set Ω) : - (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G u S) ≤ initialGreenMeasure κ := by - apply le_trans ?_ (halfPlaneStoppingGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw S hS E) - apply Measure.le_iff.mpr - intro A hA - rw [Measure.bind_apply hA (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ u hu γ G F hw S hS.measurable').aemeasurable, - Measure.bind_apply hA (halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS.measurable').aemeasurable] - exact lintegral_mono (fun ω => halfPlaneStoppingCurrentMeasure_le_green ω (hw ω) u S A) - -theorem halfPlaneStoppingCurrentMeasure_mean_test_tendsto - {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) - (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) - (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) - (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) - (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) - (hlim : Tendsto u atTop atTop) (S : Ω → WithTop ℝ≥0) - (hS : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) - (E : Set Ω) - (f : C_c(upperHalfPlane,ℝ)) : - Tendsto (fun n => ∫ z, f z ∂(P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u n) S)) - atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S))) := by - let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ - have hfi : Integrable f (initialGreenMeasure κ) := f.integrable - apply integral_tendsto_of_lintegral_pos_neg - · exact fun n => hfi.mono_measure (halfPlaneStoppingCurrentMeasure_mean_le hB hBm hBc hB0 hκ hκ8 - γ G F hw (u n) (hu n) S hS E) - · exact hfi.mono_measure (halfPlaneStoppingGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw S hS E) - · exact halfPlaneStoppingCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim S hS.measurable' E f f.continuous.measurable - · exact halfPlaneStoppingCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw - u hu hmono hlim S hS.measurable' E (fun z => -f z) f.continuous.measurable.neg - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.LevelStoppingMeasures +import OAI.Probability.SLE.Mass.LevelCurrentTests +import OAI.Probability.SLE.Mass.LevelConditionalMean + +namespace OAI + +/-! Signed compact-test convergence for the Green remainder at arbitrary +stopping times, and its preservation by the chosen convex tails. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped NNReal ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma levelStoppingCurrentGreen_le_green + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hw : IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (S : Ω → ℝ≥0∞) (z : ℂ) : + levelStoppingCurrentGreen κ B γ G u S z ω ≤ stoppingGreen κ B γ G S z ω := by + by_cases hfin : S ω < ∞ + · unfold levelStoppingCurrentGreen stoppingGreen + rw [ite_eq_left hfin] + split_ifs + · exact le_rfl + · exact greenDensity_nonneg hw κ _ z + · have hnot : ¬ S ω < radiusLevelTime (γ ω) (G ω) z u := + fun h => hfin (h.trans_le le_top) + simp only [levelStoppingCurrentGreen,stoppingGreen,ite_eq_right hfin,ite_eq_right hnot] + exact le_rfl + +lemma halfPlaneStoppingCurrentMeasure_le_green + {Ω : Type*} {B : ℝ≥0 → Ω → ℝ} {κ : ℝ} + {γ : Ω → ℝ≥0 → ℂ} {G F : Ω → ℝ≥0 → ℂ → ℂ} + (ω : Ω) (hw : IsCapacityTwoWitness + (fun s => Real.sqrt κ * B s ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (S : Ω → ℝ≥0∞) : + halfPlaneStoppingCurrentMeasure κ B γ G u S ω ≤ halfPlaneStoppingGreenMeasure κ B γ G S ω := by + apply Measure.le_iff.mpr + intro A hA + have hAI := (MeasurableEmbedding.subtype_coe isOpen_upperHalfPlane.measurableSet).measurableSet_image.mpr hA + simp only [halfPlaneStoppingCurrentMeasure,halfPlaneStoppingGreenMeasure, + comap_subtype_coe_apply isOpen_upperHalfPlane.measurableSet,withDensity_apply _ hAI] + exact lintegral_mono (fun z => ENNReal.ofReal_le_ofReal (levelStoppingCurrentGreen_le_green ω hw u S z)) + +lemma halfPlaneStoppingCurrentMeasure_mean_lintegral_test + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ u : ℝ) (hu : 0 ≤ u) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) (f : upperHalfPlane → ℝ) (hf : Measurable f) : + (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneStoppingCurrentMeasure κ B γ G u S)) = + ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * + ENNReal.ofReal (levelStoppingCurrentGreen κ B γ G u S z ω) ∂volume) ∂P := + lintegral_density_kernel_test P _ + (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ u hu γ G F hw S hS) _ + (levelStoppingCurrentGreen_joint_measurable hBm hBc κ u hu γ G F hw S hS).ennreal_ofReal + (fun _ => rfl) f hf + +lemma halfPlaneStoppingGreenMeasure_mean_lintegral_test + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (S : Ω → ℝ≥0∞) (hS : Measurable S) (f : upperHalfPlane → ℝ) (hf : Measurable f) : + (∫⁻ z, ENNReal.ofReal (f z) ∂P.bind (halfPlaneStoppingGreenMeasure κ B γ G S)) = + ∫⁻ ω, (∫⁻ z, ENNReal.ofReal (extendHalfPlaneTest f z) * ENNReal.ofReal + (stoppingGreen κ B γ G S z ω) ∂volume) ∂P := + lintegral_density_kernel_test P _ (halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS) _ + (stoppingGreen_joint_measurable hBm hBc κ γ G F hw S hS).ennreal_ofReal + (fun _ => rfl) f hf + +lemma halfPlaneStoppingCurrentMeasure_mean_lintegral_test_tendsto + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + {B : ℝ≥0 → Ω → ℝ} (hBm : Measurable (Function.uncurry B)) + (hBc : ∀ ω, Continuous (fun t => B t ω)) (κ : ℝ) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (S : Ω → ℝ≥0∞) (hS : Measurable S) (E : Set Ω) + (f : upperHalfPlane → ℝ) (hf : Measurable f) : + Tendsto (fun n => ∫⁻ z, ENNReal.ofReal (f z) + ∂(P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u n) S)) atTop + (𝓝 (∫⁻ z, ENNReal.ofReal (f z) ∂(P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S))) := by + simp_rw [halfPlaneStoppingCurrentMeasure_mean_lintegral_test _ hBm hBc κ _ (hu _) γ G F hw S hS f hf, + halfPlaneStoppingGreenMeasure_mean_lintegral_test _ hBm hBc κ γ G F hw S hS f hf] + simpa only [Measure.restrict_univ] using stoppingCurrentGreen_spatial_mean_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim S hS univ (fun z => ENNReal.ofReal (extendHalfPlaneTest f z)) + (extendHalfPlaneTest_measurable f hf).ennreal_ofReal E + +lemma halfPlaneStoppingCurrentMeasure_mean_le + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℝ) (hu : 0 ≤ u) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) + (E : Set Ω) : + (P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G u S) ≤ initialGreenMeasure κ := by + apply le_trans ?_ (halfPlaneStoppingGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw S hS E) + apply Measure.le_iff.mpr + intro A hA + rw [Measure.bind_apply hA (halfPlaneStoppingCurrentMeasure_measurable hBm hBc κ u hu γ G F hw S hS.measurable').aemeasurable, + Measure.bind_apply hA (halfPlaneStoppingGreenMeasure_measurable hBm hBc κ γ G F hw S hS.measurable').aemeasurable] + exact lintegral_mono (fun ω => halfPlaneStoppingCurrentMeasure_le_green ω (hw ω) u S A) + +theorem halfPlaneStoppingCurrentMeasure_mean_test_tendsto + {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) + (hBm : Measurable (Function.uncurry B)) (hBc : ∀ ω, Continuous (fun t => B t ω)) + (hB0 : ∀ ω, B 0 ω = 0) {κ : ℝ} (hκ : 0 < κ) (hκ8 : κ < 8) + (γ : Ω → ℝ≥0 → ℂ) (G F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ * B t ω) (γ ω) (G ω) (F ω)) + (u : ℕ → ℝ) (hu : ∀ n, 0 ≤ u n) (hmono : Monotone u) + (hlim : Tendsto u atTop atTop) (S : Ω → WithTop ℝ≥0) + (hS : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) S) + (E : Set Ω) + (f : C_c(upperHalfPlane,ℝ)) : + Tendsto (fun n => ∫ z, f z ∂(P.restrict E).bind (halfPlaneStoppingCurrentMeasure κ B γ G (u n) S)) + atTop (𝓝 (∫ z, f z ∂(P.restrict E).bind (halfPlaneStoppingGreenMeasure κ B γ G S))) := by + let : IsFiniteMeasureOnCompacts (initialGreenMeasure κ) := initialGreenMeasure_finite_on_compacts κ + have hfi : Integrable f (initialGreenMeasure κ) := f.integrable + apply integral_tendsto_of_lintegral_pos_neg + · exact fun n => hfi.mono_measure (halfPlaneStoppingCurrentMeasure_mean_le hB hBm hBc hB0 hκ hκ8 + γ G F hw (u n) (hu n) S hS E) + · exact hfi.mono_measure (halfPlaneStoppingGreenMeasure_mean_le hB hBm hBc hB0 hκ hκ8 γ G F hw S hS E) + · exact halfPlaneStoppingCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim S hS.measurable' E f f.continuous.measurable + · exact halfPlaneStoppingCurrentMeasure_mean_lintegral_test_tendsto P hBm hBc κ γ G F hw + u hu hmono hlim S hS.measurable' E (fun z => -f z) f.continuous.measurable.neg + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LocalGridMass.lean b/lean/OAI/Probability/SLE/Mass/LocalGridMass.lean index 637ed44dbc02cf61aaf1d059617665ebc506d79c..f618aa1d8bf92cc25359893204002ff5ed0677d6 100644 --- a/lean/OAI/Probability/SLE/Mass/LocalGridMass.lean +++ b/lean/OAI/Probability/SLE/Mass/LocalGridMass.lean @@ -1,118 +1,118 @@ -import OAI.Probability.SLE.Gauge.CoverGrid - -namespace OAI - -/-! A translated finite grid count for arbitrary sets of bounded diameter. -No regularity of the set being covered is required. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma localGrid_coordinate {r s x y : ℝ} (hr : 0 < r) (_hs : 0 ≤ s) - (hxy : |y-x| ≤ s) : - ⌊x/r⌋-(⌈s/r⌉+2) ≤ ⌊y/r⌋ ∧ ⌊y/r⌋ ≤ ⌊x/r⌋+(⌈s/r⌉+2) := by - have hx0 := Int.floor_le (x/r) - have hx1 := Int.lt_floor_add_one (x/r) - have hy0 := Int.floor_le (y/r) - have hy1 := Int.lt_floor_add_one (y/r) - have hc := Int.le_ceil (s/r) - have hlo : -s/r ≤ (y-x)/r := div_le_div_of_nonneg_right (abs_le.mp hxy).1 hr.le - have hhi : (y-x)/r ≤ s/r := div_le_div_of_nonneg_right (abs_le.mp hxy).2 hr.le - rw [sub_div] at hlo hhi - rw [neg_div] at hlo - constructor - · have hh : ((⌊x/r⌋-(⌈s/r⌉+2):ℤ):ℝ) ≤ (⌊y/r⌋:ℝ) := by - push_cast - linarith - exact_mod_cast hh - · have hh : (⌊y/r⌋:ℝ) ≤ ((⌊x/r⌋+(⌈s/r⌉+2):ℤ):ℝ) := by - push_cast - linarith - exact_mod_cast hh - -def localGridIndex (r s : ℝ) (z : ℂ) : Finset (ℤ×ℤ) := - let N : ℤ := ⌈s/r⌉+2 - (Finset.Icc (⌊z.re/r⌋-N) (⌊z.re/r⌋+N)) ×ˢ - (Finset.Icc (⌊z.im/r⌋-N) (⌊z.im/r⌋+N)) - -lemma localGrid_covers {r s : ℝ} (hr : 0 < r) (hs : 0 ≤ s) (z : ℂ) : - closedBall z s ⊆ ⋃ i∈localGridIndex r s z, closedBall (coverGridCenter r i) (2*r) := by - intro w hw - have hn : ‖w-z‖ ≤ s := by simpa only [mem_closedBall,dist_eq_norm] using hw - have hx := localGrid_coordinate hr hs ((Complex.abs_re_le_norm (w-z)).trans hn) - have hy := localGrid_coordinate hr hs ((Complex.abs_im_le_norm (w-z)).trans hn) - refine mem_iUnion₂.mpr ⟨(⌊w.re/r⌋,⌊w.im/r⌋),?_,coverGrid_floor_distance hr w⟩ - simpa only [localGridIndex,Finset.mem_product,Finset.mem_Icc,Complex.sub_re,Complex.sub_im] - using And.intro hx hy - -lemma localGrid_card_bound {r s : ℝ} (hr : 0 < r) (hs : 0 ≤ s) (z : ℂ) : - ((localGridIndex r s z).card:ℝ) ≤ 81*max 1 ((s/r)^2) := by - let N : ℤ := ⌈s/r⌉+2 - have hN : 0 ≤ N := by - have h := Int.ceil_nonneg (div_nonneg hs hr.le) - dsimp [N] - omega - have hside : 2*(N:ℝ)+1 ≤ 2*(s/r)+7 := by - have hc := Int.ceil_lt_add_one (s/r) - dsimp [N] - push_cast - linarith - have hside0 : 0 ≤ 2*(N:ℝ)+1 := by exact_mod_cast (show (0:ℤ) ≤ 2*N+1 by omega) - have hcard : ((localGridIndex r s z).card:ℝ)=(2*(N:ℝ)+1)^2 := by - have hx := Int.card_Icc_of_le (a:=⌊z.re/r⌋-N) (b:=⌊z.re/r⌋+N) (by omega) - have hy := Int.card_Icc_of_le (a:=⌊z.im/r⌋-N) (b:=⌊z.im/r⌋+N) (by omega) - have hx' : ((Finset.Icc (⌊z.re/r⌋-N) (⌊z.re/r⌋+N)).card:ℝ)=2*(N:ℝ)+1 := by - have h : ((Finset.Icc (⌊z.re/r⌋-N) (⌊z.re/r⌋+N)).card:ℝ)= - ((⌊z.re/r⌋:ℝ)+(N:ℝ))+1-((⌊z.re/r⌋:ℝ)-(N:ℝ)) := by exact_mod_cast hx - linarith - have hy' : ((Finset.Icc (⌊z.im/r⌋-N) (⌊z.im/r⌋+N)).card:ℝ)=2*(N:ℝ)+1 := by - have h : ((Finset.Icc (⌊z.im/r⌋-N) (⌊z.im/r⌋+N)).card:ℝ)= - ((⌊z.im/r⌋:ℝ)+(N:ℝ))+1-((⌊z.im/r⌋:ℝ)-(N:ℝ)) := by exact_mod_cast hy - linarith - change (((Finset.Icc _ _) ×ˢ (Finset.Icc _ _)).card:ℝ)=_ - rw [Finset.card_product,Nat.cast_mul,hx',hy',pow_two] - rw [hcard] - have hratio : 0 ≤ s/r := div_nonneg hs hr.le - by_cases hsmall : s/r ≤ 1 - · have h : 2*(N:ℝ)+1 ≤ 9 := by linarith - have hp := pow_le_pow_left₀ hside0 h 2 - norm_num at hp - exact hp.trans (by nlinarith [le_max_left (1:ℝ) ((s/r)^2)]) - · have hlarge : 1 ≤ s/r := (lt_of_not_ge hsmall).le - have h : 2*(N:ℝ)+1 ≤ 9*(s/r) := by linarith - have hp := pow_le_pow_left₀ hside0 h 2 - calc - _ ≤ (9*(s/r))^2 := hp - _ = 81*(s/r)^2 := by ring - _ ≤ _ := mul_le_mul_of_nonneg_left (le_max_right _ _) (by norm_num) - -lemma localGrid_mass_bound - (μ : Measure ℂ) {r s H : ℝ} (hr : 0 < r) (hs : 0 ≤ s) (hH : 0 ≤ H) - (hcell : ∀ i : ℤ×ℤ, μ (closedBall (coverGridCenter r i) (2*r)) ≤ ENNReal.ofReal H) - (A : Set ℂ) (hA : ∀ x∈A, ∀ y∈A, dist x y ≤ s) : - μ A ≤ ENNReal.ofReal (81*H*max 1 ((s/r)^2)) := by - classical - rcases A.eq_empty_or_nonempty with rfl | ⟨z,hz⟩ - · simp - have hcover : A ⊆ closedBall z s := by - intro w hw - exact hA w hw z hz - calc - μ A ≤ μ (⋃ i∈localGridIndex r s z, closedBall (coverGridCenter r i) (2*r)) := - measure_mono (hcover.trans (localGrid_covers hr hs z)) - _ ≤ ∑ i∈localGridIndex r s z, μ (closedBall (coverGridCenter r i) (2*r)) := measure_biUnion_finset_le _ _ - _ ≤ ∑ _i∈localGridIndex r s z, ENNReal.ofReal H := Finset.sum_le_sum (fun i _ => hcell i) - _ = ENNReal.ofReal (((localGridIndex r s z).card:ℝ)*H) := by - simp only [Finset.sum_const,nsmul_eq_mul,ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast] - _ ≤ _ := ENNReal.ofReal_le_ofReal (by - have h := mul_le_mul_of_nonneg_right (localGrid_card_bound hr hs z) hH - nlinarith) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.CoverGrid + +namespace OAI + +/-! A translated finite grid count for arbitrary sets of bounded diameter. +No regularity of the set being covered is required. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma localGrid_coordinate {r s x y : ℝ} (hr : 0 < r) (_hs : 0 ≤ s) + (hxy : |y-x| ≤ s) : + ⌊x/r⌋-(⌈s/r⌉+2) ≤ ⌊y/r⌋ ∧ ⌊y/r⌋ ≤ ⌊x/r⌋+(⌈s/r⌉+2) := by + have hx0 := Int.floor_le (x/r) + have hx1 := Int.lt_floor_add_one (x/r) + have hy0 := Int.floor_le (y/r) + have hy1 := Int.lt_floor_add_one (y/r) + have hc := Int.le_ceil (s/r) + have hlo : -s/r ≤ (y-x)/r := div_le_div_of_nonneg_right (abs_le.mp hxy).1 hr.le + have hhi : (y-x)/r ≤ s/r := div_le_div_of_nonneg_right (abs_le.mp hxy).2 hr.le + rw [sub_div] at hlo hhi + rw [neg_div] at hlo + constructor + · have hh : ((⌊x/r⌋-(⌈s/r⌉+2):ℤ):ℝ) ≤ (⌊y/r⌋:ℝ) := by + push_cast + linarith + exact_mod_cast hh + · have hh : (⌊y/r⌋:ℝ) ≤ ((⌊x/r⌋+(⌈s/r⌉+2):ℤ):ℝ) := by + push_cast + linarith + exact_mod_cast hh + +def localGridIndex (r s : ℝ) (z : ℂ) : Finset (ℤ×ℤ) := + let N : ℤ := ⌈s/r⌉+2 + (Finset.Icc (⌊z.re/r⌋-N) (⌊z.re/r⌋+N)) ×ˢ + (Finset.Icc (⌊z.im/r⌋-N) (⌊z.im/r⌋+N)) + +lemma localGrid_covers {r s : ℝ} (hr : 0 < r) (hs : 0 ≤ s) (z : ℂ) : + closedBall z s ⊆ ⋃ i∈localGridIndex r s z, closedBall (coverGridCenter r i) (2*r) := by + intro w hw + have hn : ‖w-z‖ ≤ s := by simpa only [mem_closedBall,dist_eq_norm] using hw + have hx := localGrid_coordinate hr hs ((Complex.abs_re_le_norm (w-z)).trans hn) + have hy := localGrid_coordinate hr hs ((Complex.abs_im_le_norm (w-z)).trans hn) + refine mem_iUnion₂.mpr ⟨(⌊w.re/r⌋,⌊w.im/r⌋),?_,coverGrid_floor_distance hr w⟩ + simpa only [localGridIndex,Finset.mem_product,Finset.mem_Icc,Complex.sub_re,Complex.sub_im] + using And.intro hx hy + +lemma localGrid_card_bound {r s : ℝ} (hr : 0 < r) (hs : 0 ≤ s) (z : ℂ) : + ((localGridIndex r s z).card:ℝ) ≤ 81*max 1 ((s/r)^2) := by + let N : ℤ := ⌈s/r⌉+2 + have hN : 0 ≤ N := by + have h := Int.ceil_nonneg (div_nonneg hs hr.le) + dsimp [N] + omega + have hside : 2*(N:ℝ)+1 ≤ 2*(s/r)+7 := by + have hc := Int.ceil_lt_add_one (s/r) + dsimp [N] + push_cast + linarith + have hside0 : 0 ≤ 2*(N:ℝ)+1 := by exact_mod_cast (show (0:ℤ) ≤ 2*N+1 by omega) + have hcard : ((localGridIndex r s z).card:ℝ)=(2*(N:ℝ)+1)^2 := by + have hx := Int.card_Icc_of_le (a:=⌊z.re/r⌋-N) (b:=⌊z.re/r⌋+N) (by omega) + have hy := Int.card_Icc_of_le (a:=⌊z.im/r⌋-N) (b:=⌊z.im/r⌋+N) (by omega) + have hx' : ((Finset.Icc (⌊z.re/r⌋-N) (⌊z.re/r⌋+N)).card:ℝ)=2*(N:ℝ)+1 := by + have h : ((Finset.Icc (⌊z.re/r⌋-N) (⌊z.re/r⌋+N)).card:ℝ)= + ((⌊z.re/r⌋:ℝ)+(N:ℝ))+1-((⌊z.re/r⌋:ℝ)-(N:ℝ)) := by exact_mod_cast hx + linarith + have hy' : ((Finset.Icc (⌊z.im/r⌋-N) (⌊z.im/r⌋+N)).card:ℝ)=2*(N:ℝ)+1 := by + have h : ((Finset.Icc (⌊z.im/r⌋-N) (⌊z.im/r⌋+N)).card:ℝ)= + ((⌊z.im/r⌋:ℝ)+(N:ℝ))+1-((⌊z.im/r⌋:ℝ)-(N:ℝ)) := by exact_mod_cast hy + linarith + change (((Finset.Icc _ _) ×ˢ (Finset.Icc _ _)).card:ℝ)=_ + rw [Finset.card_product,Nat.cast_mul,hx',hy',pow_two] + rw [hcard] + have hratio : 0 ≤ s/r := div_nonneg hs hr.le + by_cases hsmall : s/r ≤ 1 + · have h : 2*(N:ℝ)+1 ≤ 9 := by linarith + have hp := pow_le_pow_left₀ hside0 h 2 + norm_num at hp + exact hp.trans (by nlinarith [le_max_left (1:ℝ) ((s/r)^2)]) + · have hlarge : 1 ≤ s/r := (lt_of_not_ge hsmall).le + have h : 2*(N:ℝ)+1 ≤ 9*(s/r) := by linarith + have hp := pow_le_pow_left₀ hside0 h 2 + calc + _ ≤ (9*(s/r))^2 := hp + _ = 81*(s/r)^2 := by ring + _ ≤ _ := mul_le_mul_of_nonneg_left (le_max_right _ _) (by norm_num) + +lemma localGrid_mass_bound + (μ : Measure ℂ) {r s H : ℝ} (hr : 0 < r) (hs : 0 ≤ s) (hH : 0 ≤ H) + (hcell : ∀ i : ℤ×ℤ, μ (closedBall (coverGridCenter r i) (2*r)) ≤ ENNReal.ofReal H) + (A : Set ℂ) (hA : ∀ x∈A, ∀ y∈A, dist x y ≤ s) : + μ A ≤ ENNReal.ofReal (81*H*max 1 ((s/r)^2)) := by + classical + rcases A.eq_empty_or_nonempty with rfl | ⟨z,hz⟩ + · simp + have hcover : A ⊆ closedBall z s := by + intro w hw + exact hA w hw z hz + calc + μ A ≤ μ (⋃ i∈localGridIndex r s z, closedBall (coverGridCenter r i) (2*r)) := + measure_mono (hcover.trans (localGrid_covers hr hs z)) + _ ≤ ∑ i∈localGridIndex r s z, μ (closedBall (coverGridCenter r i) (2*r)) := measure_biUnion_finset_le _ _ + _ ≤ ∑ _i∈localGridIndex r s z, ENNReal.ofReal H := Finset.sum_le_sum (fun i _ => hcell i) + _ = ENNReal.ofReal (((localGridIndex r s z).card:ℝ)*H) := by + simp only [Finset.sum_const,nsmul_eq_mul,ENNReal.ofReal_mul (Nat.cast_nonneg _),ENNReal.ofReal_natCast] + _ ≤ _ := ENNReal.ofReal_le_ofReal (by + have h := mul_le_mul_of_nonneg_right (localGrid_card_bound hr hs z) hH + nlinarith) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/LocallyFiniteMeasureSubtract.lean b/lean/OAI/Probability/SLE/Mass/LocallyFiniteMeasureSubtract.lean index 2c5692ff4ff424e247fb97e6572a84729be06e12..3c1bf8be615c713d565306696b4a053766716932 100644 --- a/lean/OAI/Probability/SLE/Mass/LocallyFiniteMeasureSubtract.lean +++ b/lean/OAI/Probability/SLE/Mass/LocallyFiniteMeasureSubtract.lean @@ -1,49 +1,49 @@ -import Mathlib.MeasureTheory.Measure.Sub -import Mathlib.MeasureTheory.Measure.Typeclasses.SFinite - -namespace OAI - -/-! Subtraction for locally finite canonical mass. Cancellation is checked -on finite restrictions, so no finite total-mass assumption is needed. -/ - -noncomputable section -open Set MeasureTheory -open scoped ENNReal - -namespace SLEExactGauge - -theorem sigmaFinite_sub_add_cancel_of_le - {X : Type*} [MeasurableSpace X] {μ ν : Measure X} [SigmaFinite ν] - (hle : ν ≤ μ) : μ - ν + ν = μ := by - apply Measure.ext_of_iUnion_eq_univ (iUnion_spanningSets ν) - intro n - rw [Measure.restrict_add,Measure.restrict_sub_eq_restrict_sub_restrict - (measurableSet_spanningSets ν n)] - have : Fact (ν (spanningSets ν n) < ⊤) := ⟨measure_spanningSets_lt_top ν n⟩ - exact Measure.sub_add_cancel_of_le (Measure.restrict_mono le_rfl hle) - -theorem sub_apply_of_finite_restriction - {X : Type*} [MeasurableSpace X] {μ ν : Measure X} (hle : ν ≤ μ) - {A : Set X} (hA : MeasurableSet A) (hνA : ν A ≠ ⊤) : - (μ - ν) A = μ A - ν A := by - have : Fact (ν A < ⊤) := ⟨hνA.lt_top⟩ - calc - (μ - ν) A = ((μ - ν).restrict A) univ := by simp - _ = (μ.restrict A - ν.restrict A) univ := by - rw [Measure.restrict_sub_eq_restrict_sub_restrict hA] - _ = μ.restrict A univ - ν.restrict A univ := - Measure.sub_apply MeasurableSet.univ (Measure.restrict_mono le_rfl hle) - _ = μ A - ν A := by simp - -theorem sub_le_sub_right_of_le - {X : Type*} [MeasurableSpace X] {μ ν ξ : Measure X} [SigmaFinite ξ] - (hξ : ξ ≤ ν) (hμν : μ ≤ ν) : μ - ξ ≤ ν - ξ := by - apply Measure.sub_le_of_le_add - rw [sigmaFinite_sub_add_cancel_of_le hξ] - exact hμν - -end SLEExactGauge - -end - -end OAI +import Mathlib.MeasureTheory.Measure.Sub +import Mathlib.MeasureTheory.Measure.Typeclasses.SFinite + +namespace OAI + +/-! Subtraction for locally finite canonical mass. Cancellation is checked +on finite restrictions, so no finite total-mass assumption is needed. -/ + +noncomputable section +open Set MeasureTheory +open scoped ENNReal + +namespace SLEExactGauge + +theorem sigmaFinite_sub_add_cancel_of_le + {X : Type*} [MeasurableSpace X] {μ ν : Measure X} [SigmaFinite ν] + (hle : ν ≤ μ) : μ - ν + ν = μ := by + apply Measure.ext_of_iUnion_eq_univ (iUnion_spanningSets ν) + intro n + rw [Measure.restrict_add,Measure.restrict_sub_eq_restrict_sub_restrict + (measurableSet_spanningSets ν n)] + have : Fact (ν (spanningSets ν n) < ⊤) := ⟨measure_spanningSets_lt_top ν n⟩ + exact Measure.sub_add_cancel_of_le (Measure.restrict_mono le_rfl hle) + +theorem sub_apply_of_finite_restriction + {X : Type*} [MeasurableSpace X] {μ ν : Measure X} (hle : ν ≤ μ) + {A : Set X} (hA : MeasurableSet A) (hνA : ν A ≠ ⊤) : + (μ - ν) A = μ A - ν A := by + have : Fact (ν A < ⊤) := ⟨hνA.lt_top⟩ + calc + (μ - ν) A = ((μ - ν).restrict A) univ := by simp + _ = (μ.restrict A - ν.restrict A) univ := by + rw [Measure.restrict_sub_eq_restrict_sub_restrict hA] + _ = μ.restrict A univ - ν.restrict A univ := + Measure.sub_apply MeasurableSet.univ (Measure.restrict_mono le_rfl hle) + _ = μ A - ν A := by simp + +theorem sub_le_sub_right_of_le + {X : Type*} [MeasurableSpace X] {μ ν ξ : Measure X} [SigmaFinite ξ] + (hξ : ξ ≤ ν) (hμν : μ ≤ ν) : μ - ξ ≤ ν - ξ := by + apply Measure.sub_le_of_le_add + rw [sigmaFinite_sub_add_cancel_of_le hξ] + exact hμν + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MarkedTubeHarmonicMeasure.lean b/lean/OAI/Probability/SLE/Mass/MarkedTubeHarmonicMeasure.lean index a1e51fa893c3d03f63f90e6c105057c4fe5a2777..d997a27f67de0896507df3332373b81493a41dc8 100644 --- a/lean/OAI/Probability/SLE/Mass/MarkedTubeHarmonicMeasure.lean +++ b/lean/OAI/Probability/SLE/Mass/MarkedTubeHarmonicMeasure.lean @@ -1,103 +1,103 @@ -import OAI.Probability.SLE.Geometry.MarkedWindowRoutes -import OAI.Probability.SLE.Geometry.FiniteExitArcAt - -namespace OAI - -/-! Convert the two explicit marked-point Brownian tubes into lower -bounds for both conformal exit arcs. Every geometric condition concerns -actual paths in the surviving domain; the only probabilistic input here -is the generic published conformal Brownian exit law. -/ - -noncomputable section -open Set Filter MeasureTheory Metric -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Serpentine -open RibbonCoordinates CrosscutSides Target - -lemma marked_window_tubes_opposite_sides - (hopen : PublishedPlanarDiskOpenness) (w : ℂ) {a : ℝ} (ha : 0 < a) - (f : ℂ → ℂ) (j : ℂ → ℝ) - (hf : ContinuousOn f (ball (0 : ℂ) (a / 12))) - (hfi : InjOn f (ball (0 : ℂ) (a / 12))) - (hzero : ∀ z ∈ ball (0 : ℂ) (a / 12), (f z).re = 0 ↔ z.im = 0) - (hrel : ∀ z ∈ ball (0 : ℂ) (a / 12), j (markedCornerChart w z) = (f z).re) : - ∃ right : Bool, - (∀ y : ℂ, dist y (markedWindowRoute right w a (1 / 64)) < a / 10000 → 0 < j y) ∧ - (∀ y : ℂ, dist y (markedWindowRoute (!right) w a (1 / 64)) < a / 10000 → j y < 0) := by - rcases marked_physical_tubes_opposite_sides hopen w ha f j hf hfi hzero hrel with h | h - · refine ⟨false, ?_⟩ - simpa only [markedWindowRoute_small, Bool.false_eq_true, ↓reduceIte, Bool.not_false] using h - · refine ⟨true, ?_⟩ - simpa only [markedWindowRoute_small, ↓reduceIte, Bool.not_true, Bool.false_eq_true] using And.symm h - -theorem marked_tubes_harmonic_bounds - (hopen : PublishedPlanarDiskOpenness) (hexitLaw : PublishedHalfPlaneConformalExitLaw) - (W : WienerLaw) (hW : IsWiener W) - (D : Set ℂ) (hD : IsOpen D) (hDH : D ⊆ upperHalfPlane) - (g F : ℂ → ℂ) (hg : DifferentiableOn ℂ g D) - (hF : DifferentiableOn ℂ F upperHalfPlane) - (hgH : MapsTo g D upperHalfPlane) (hFD : MapsTo F upperHalfPlane D) - (hFg : ∀ z ∈ D, F (g z) = z) (hgF : ∀ z ∈ upperHalfPlane, g (F z) = z) - (w : ℂ) (hwD : w ∈ D) {a : ℝ} (ha : 0 < a) - (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) - (hloc : ContinuousOn (fun z => h (g (markedCornerChart w z))) (ball (0 : ℂ) (a / 12))) - (hloci : InjOn (fun z => h (g (markedCornerChart w z))) (ball (0 : ℂ) (a / 12))) - (hlocz : ∀ z ∈ ball (0 : ℂ) (a / 12), - (h (g (markedCornerChart w z))).re = 0 ↔ z.im = 0) - (q : ℝ≥0∞) - (hprob : ∀ right : Bool, q ≤ (W.prod W) {b | ∀ t ∈ Icc (0 : ℝ) 3, - dist (physicalUnitBrownianPath w b t) (markedWindowRoute right w a t) < a / 10000}) - (hgeom : ∀ (right : Bool) (b : Driver × Driver), - (∀ t ∈ Icc (0 : ℝ) 3, - dist (physicalUnitBrownianPath w b t) (markedWindowRoute right w a t) < a / 10000) → - ∃ τ : ℝ, UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ (1 / 64 : ℝ) < τ ∧ - ∀ t ∈ Ico (1 / 64 : ℝ) τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0) : - q ≤ ENNReal.ofReal ((g w).arg / Real.pi) ∧ - q ≤ ENNReal.ofReal (1 - (g w).arg / Real.pi) := by - let E := fun right : Bool => {b | ∀ t ∈ Icc (0 : ℝ) 3, - dist (physicalUnitBrownianPath w b t) (markedWindowRoute right w a t) < a / 10000} - obtain ⟨hae, hnegBound, hposBound⟩ := hexitLaw W hW D g F hD hDH - hg hF hgH hFD hFg hgF w hwD - obtain ⟨right, hpos, hneg⟩ := marked_window_tubes_opposite_sides hopen w ha - (fun z => h (g (markedCornerChart w z))) (fun z => (h (g z)).re) - hloc hloci hlocz (fun _ _ => rfl) - have hposProb : q ≤ (W.prod W) (unitWindowExitArc D g w 1) := by - refine (hprob right).trans (finite_exit_event_measure_le_arc_at (W.prod W) D g w h hreal - hg.continuousOn (E right) (Or.inr rfl) (by norm_num : (0 : ℝ) ≤ 1 / 64) hae - (hgeom right) ?_) - intro b hb - simpa only [one_mul] using hpos _ (hb _ (by norm_num)) - have hnegProb : q ≤ (W.prod W) (unitWindowExitArc D g w (-1)) := by - refine (hprob (!right)).trans (finite_exit_event_measure_le_arc_at (W.prod W) D g w h hreal - hg.continuousOn (E (!right)) (Or.inl rfl) (by norm_num : (0 : ℝ) ≤ 1 / 64) hae - (hgeom (!right)) ?_) - intro b hb - have hh := hneg _ (hb _ (by norm_num)) - nlinarith - exact ⟨hnegProb.trans hnegBound, hposProb.trans hposBound⟩ - -/-- Early survival, a final exit and avoidance of the actual complete -guide imply the geometric hypotheses in the harmonic-measure theorem. -/ -theorem marked_tube_first_exit_geometry - (D : Set ℂ) (hD : IsOpen D) (g : ℂ → ℂ) (h : ℂ ≃ₜ ℂ) - (α : ℝ≥0 → ℂ) - (hzero : ∀ z ∈ D, (h (g z)).re = 0 → ∃ u : ℝ≥0, 0 < u ∧ α u = z) - (ζ : ℝ → ℂ) (hc : Continuous ζ) - (hearly : MapsTo ζ (Icc (0 : ℝ) (1 / 64)) D) - (hlast : ζ 3 ∉ D) - (havoid : ∀ t ∈ Icc (1 / 64 : ℝ) 3, ∀ u : ℝ≥0, 0 < u → ζ t ≠ α u) : - ∃ τ : ℝ, UnitWindowFirstExit D ζ τ ∧ (1 / 64 : ℝ) < τ ∧ - ∀ t ∈ Ico (1 / 64 : ℝ) τ, (h (g (ζ t))).re ≠ 0 := by - obtain ⟨τ, hτ, hout, hin⟩ := exists_first_domain_exit (L := 3) D hD ζ hc.continuousOn hearly - ⟨3, by norm_num, hlast⟩ - refine ⟨τ, ⟨⟨by linarith [hτ.1], hτ.2⟩, hout, hin⟩, hτ.1, ?_⟩ - intro t ht he - obtain ⟨u, hu, hue⟩ := hzero (ζ t) (hin ⟨by linarith [ht.1], ht.2⟩) he - exact havoid t ⟨ht.1, ht.2.le.trans hτ.2⟩ u hu hue.symm - -end SLEExactGauge.Serpentine - -end - -end OAI +import OAI.Probability.SLE.Geometry.MarkedWindowRoutes +import OAI.Probability.SLE.Geometry.FiniteExitArcAt + +namespace OAI + +/-! Convert the two explicit marked-point Brownian tubes into lower +bounds for both conformal exit arcs. Every geometric condition concerns +actual paths in the surviving domain; the only probabilistic input here +is the generic published conformal Brownian exit law. -/ + +noncomputable section +open Set Filter MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Serpentine +open RibbonCoordinates CrosscutSides Target + +lemma marked_window_tubes_opposite_sides + (hopen : PublishedPlanarDiskOpenness) (w : ℂ) {a : ℝ} (ha : 0 < a) + (f : ℂ → ℂ) (j : ℂ → ℝ) + (hf : ContinuousOn f (ball (0 : ℂ) (a / 12))) + (hfi : InjOn f (ball (0 : ℂ) (a / 12))) + (hzero : ∀ z ∈ ball (0 : ℂ) (a / 12), (f z).re = 0 ↔ z.im = 0) + (hrel : ∀ z ∈ ball (0 : ℂ) (a / 12), j (markedCornerChart w z) = (f z).re) : + ∃ right : Bool, + (∀ y : ℂ, dist y (markedWindowRoute right w a (1 / 64)) < a / 10000 → 0 < j y) ∧ + (∀ y : ℂ, dist y (markedWindowRoute (!right) w a (1 / 64)) < a / 10000 → j y < 0) := by + rcases marked_physical_tubes_opposite_sides hopen w ha f j hf hfi hzero hrel with h | h + · refine ⟨false, ?_⟩ + simpa only [markedWindowRoute_small, Bool.false_eq_true, ↓reduceIte, Bool.not_false] using h + · refine ⟨true, ?_⟩ + simpa only [markedWindowRoute_small, ↓reduceIte, Bool.not_true, Bool.false_eq_true] using And.symm h + +theorem marked_tubes_harmonic_bounds + (hopen : PublishedPlanarDiskOpenness) (hexitLaw : PublishedHalfPlaneConformalExitLaw) + (W : WienerLaw) (hW : IsWiener W) + (D : Set ℂ) (hD : IsOpen D) (hDH : D ⊆ upperHalfPlane) + (g F : ℂ → ℂ) (hg : DifferentiableOn ℂ g D) + (hF : DifferentiableOn ℂ F upperHalfPlane) + (hgH : MapsTo g D upperHalfPlane) (hFD : MapsTo F upperHalfPlane D) + (hFg : ∀ z ∈ D, F (g z) = z) (hgF : ∀ z ∈ upperHalfPlane, g (F z) = z) + (w : ℂ) (hwD : w ∈ D) {a : ℝ} (ha : 0 < a) + (h : ℂ ≃ₜ ℂ) (hreal : ∀ x : ℝ, h (x : ℂ) = (x : ℂ)) + (hloc : ContinuousOn (fun z => h (g (markedCornerChart w z))) (ball (0 : ℂ) (a / 12))) + (hloci : InjOn (fun z => h (g (markedCornerChart w z))) (ball (0 : ℂ) (a / 12))) + (hlocz : ∀ z ∈ ball (0 : ℂ) (a / 12), + (h (g (markedCornerChart w z))).re = 0 ↔ z.im = 0) + (q : ℝ≥0∞) + (hprob : ∀ right : Bool, q ≤ (W.prod W) {b | ∀ t ∈ Icc (0 : ℝ) 3, + dist (physicalUnitBrownianPath w b t) (markedWindowRoute right w a t) < a / 10000}) + (hgeom : ∀ (right : Bool) (b : Driver × Driver), + (∀ t ∈ Icc (0 : ℝ) 3, + dist (physicalUnitBrownianPath w b t) (markedWindowRoute right w a t) < a / 10000) → + ∃ τ : ℝ, UnitWindowFirstExit D (physicalUnitBrownianPath w b) τ ∧ (1 / 64 : ℝ) < τ ∧ + ∀ t ∈ Ico (1 / 64 : ℝ) τ, (h (g (physicalUnitBrownianPath w b t))).re ≠ 0) : + q ≤ ENNReal.ofReal ((g w).arg / Real.pi) ∧ + q ≤ ENNReal.ofReal (1 - (g w).arg / Real.pi) := by + let E := fun right : Bool => {b | ∀ t ∈ Icc (0 : ℝ) 3, + dist (physicalUnitBrownianPath w b t) (markedWindowRoute right w a t) < a / 10000} + obtain ⟨hae, hnegBound, hposBound⟩ := hexitLaw W hW D g F hD hDH + hg hF hgH hFD hFg hgF w hwD + obtain ⟨right, hpos, hneg⟩ := marked_window_tubes_opposite_sides hopen w ha + (fun z => h (g (markedCornerChart w z))) (fun z => (h (g z)).re) + hloc hloci hlocz (fun _ _ => rfl) + have hposProb : q ≤ (W.prod W) (unitWindowExitArc D g w 1) := by + refine (hprob right).trans (finite_exit_event_measure_le_arc_at (W.prod W) D g w h hreal + hg.continuousOn (E right) (Or.inr rfl) (by norm_num : (0 : ℝ) ≤ 1 / 64) hae + (hgeom right) ?_) + intro b hb + simpa only [one_mul] using hpos _ (hb _ (by norm_num)) + have hnegProb : q ≤ (W.prod W) (unitWindowExitArc D g w (-1)) := by + refine (hprob (!right)).trans (finite_exit_event_measure_le_arc_at (W.prod W) D g w h hreal + hg.continuousOn (E (!right)) (Or.inl rfl) (by norm_num : (0 : ℝ) ≤ 1 / 64) hae + (hgeom (!right)) ?_) + intro b hb + have hh := hneg _ (hb _ (by norm_num)) + nlinarith + exact ⟨hnegProb.trans hnegBound, hposProb.trans hposBound⟩ + +/-- Early survival, a final exit and avoidance of the actual complete +guide imply the geometric hypotheses in the harmonic-measure theorem. -/ +theorem marked_tube_first_exit_geometry + (D : Set ℂ) (hD : IsOpen D) (g : ℂ → ℂ) (h : ℂ ≃ₜ ℂ) + (α : ℝ≥0 → ℂ) + (hzero : ∀ z ∈ D, (h (g z)).re = 0 → ∃ u : ℝ≥0, 0 < u ∧ α u = z) + (ζ : ℝ → ℂ) (hc : Continuous ζ) + (hearly : MapsTo ζ (Icc (0 : ℝ) (1 / 64)) D) + (hlast : ζ 3 ∉ D) + (havoid : ∀ t ∈ Icc (1 / 64 : ℝ) 3, ∀ u : ℝ≥0, 0 < u → ζ t ≠ α u) : + ∃ τ : ℝ, UnitWindowFirstExit D ζ τ ∧ (1 / 64 : ℝ) < τ ∧ + ∀ t ∈ Ico (1 / 64 : ℝ) τ, (h (g (ζ t))).re ≠ 0 := by + obtain ⟨τ, hτ, hout, hin⟩ := exists_first_domain_exit (L := 3) D hD ζ hc.continuousOn hearly + ⟨3, by norm_num, hlast⟩ + refine ⟨τ, ⟨⟨by linarith [hτ.1], hτ.2⟩, hout, hin⟩, hτ.1, ?_⟩ + intro t ht he + obtain ⟨u, hu, hue⟩ := hzero (ζ t) (hin ⟨by linarith [ht.1], ht.2⟩) he + exact havoid t ⟨ht.1, ht.2.le.trans hτ.2⟩ u hu hue.symm + +end SLEExactGauge.Serpentine + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassCompensatorEndpoint.lean b/lean/OAI/Probability/SLE/Mass/MassCompensatorEndpoint.lean index f603c038536009687f862d1432c46824edf0b0ba..cd20cfccf0f7cf38f41bcce252376ace961ca6a0 100644 --- a/lean/OAI/Probability/SLE/Mass/MassCompensatorEndpoint.lean +++ b/lean/OAI/Probability/SLE/Mass/MassCompensatorEndpoint.lean @@ -1,117 +1,117 @@ -import OAI.Probability.SLE.Mass.MassCompensatorMesh - -namespace OAI - -/-! The last full dyadic cell before a finite or infinite time. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -lemma compensatorMeshTime_eq_div (n i : ℕ) : - compensatorMeshTime n i = (i:ℝ≥0)/(2:ℝ≥0)^n := by - simp only [compensatorMeshTime,zpow_neg,zpow_natCast,div_eq_mul_inv] - -@[simp] lemma compensatorMeshTime_end (n : ℕ) : - compensatorMeshTime n (n*2^n) = n := by - rw [compensatorMeshTime_eq_div] - push_cast - exact mul_div_cancel_right₀ _ (by positivity) - -def compensatorFiniteIndex (n : ℕ) (t : ℝ≥0) : ℕ := - min (n*2^n) ⌊t*(2:ℝ≥0)^n⌋₊ - -lemma compensatorFiniteIndex_le (n : ℕ) (t : ℝ≥0) : - compensatorFiniteIndex n t ≤ n*2^n := min_le_left _ _ - -lemma compensator_cell_before_iff (n i : ℕ) (t : ℝ≥0) : - compensatorMeshTime n (i+1) ≤ t ↔ i < ⌊t*(2:ℝ≥0)^n⌋₊ := by - rw [compensatorMeshTime_eq_div,div_le_iff₀ (by positivity), - ← Nat.le_floor_iff (show 0 ≤ t*(2:ℝ≥0)^n from zero_le)] - exact Nat.succ_le_iff - -lemma compensator_before_sum (n : ℕ) (t : ℝ≥0) (q : ℕ → ℝ) : - (∑ i ∈ Finset.range (n*2^n), - if compensatorMeshTime n (i+1) ≤ t then q i else 0) = - ∑ i ∈ Finset.range (compensatorFiniteIndex n t), q i := by - simp only [compensator_cell_before_iff] - rw [← Finset.sum_filter] - congr 1 - ext i - simp only [Finset.mem_filter,Finset.mem_range,compensatorFiniteIndex,lt_min_iff] - -lemma compensatorFiniteIndex_eq_floor {n : ℕ} {t : ℝ≥0} (ht : t ≤ n) : - compensatorFiniteIndex n t = ⌊t*(2:ℝ≥0)^n⌋₊ := by - apply min_eq_right - apply Nat.floor_le_of_le - push_cast - exact mul_le_mul_of_nonneg_right ht (by positivity) - -lemma compensatorEndpoint_le (n : ℕ) (t : ℝ≥0) : - compensatorMeshTime n (compensatorFiniteIndex n t) ≤ t := by - rw [compensatorMeshTime_eq_div,div_le_iff₀ (by positivity)] - exact (Nat.cast_le.mpr (min_le_right _ _)).trans (Nat.floor_le (show 0 ≤ t*(2:ℝ≥0)^n from zero_le)) - -lemma compensatorEndpoint_lower {n : ℕ} {t : ℝ≥0} (ht : t ≤ n) : - t < compensatorMeshTime n (compensatorFiniteIndex n t)+(2:ℝ≥0)^(-(n:ℤ)) := by - rw [compensatorFiniteIndex_eq_floor ht,compensatorMeshTime_eq_div, - zpow_neg,zpow_natCast,← one_div,← add_div,lt_div_iff₀ (by positivity)] - exact Nat.lt_floor_add_one _ - -/-- A finite endpoint is approached from below even though the grid -horizon grows. -/ -theorem compensatorEndpoint_tendsto (t : ℝ≥0) : - Tendsto (fun n => compensatorMeshTime n (compensatorFiniteIndex n t)) atTop (𝓝 t) := by - apply NNReal.tendsto_coe.mp - apply tendsto_of_tendsto_of_tendsto_of_le_of_le' - (show Tendsto (fun n : ℕ => (t:ℝ)-(2:ℝ)^(-(n:ℤ))) atTop (𝓝 (t:ℝ)) from - by simpa only [sub_zero,zpow_neg,zpow_natCast,inv_pow] using - tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_lt_one - (show 0 ≤ (2:ℝ)⁻¹ by positivity) (show (2:ℝ)⁻¹<1 by norm_num))) - tendsto_const_nhds - · have hn : ∀ᶠ n : ℕ in atTop, t ≤ (n:ℝ≥0) := - (tendsto_natCast_atTop_atTop (R := ℝ≥0)).eventually (eventually_ge_atTop t) - filter_upwards [hn] with n hn - have h := compensatorEndpoint_lower hn - have hc : (t:ℝ) < (compensatorMeshTime n (compensatorFiniteIndex n t):ℝ)+ - (2:ℝ)^(-(n:ℤ)) := by exact_mod_cast h - linarith - · exact Eventually.of_forall (fun n => by exact_mod_cast compensatorEndpoint_le n t) - -def compensatorStopIndex (n : ℕ) (T : ℝ≥0∞) : ℕ := - if T=∞ then n*2^n else compensatorFiniteIndex n T.toNNReal - -lemma compensatorStopIndex_measurable {Ω : Type*} [MeasurableSpace Ω] - (T : Ω → ℝ≥0∞) (hT : Measurable T) (n : ℕ) : - Measurable (fun ω => compensatorStopIndex n (T ω)) := by - apply Measurable.ite (measurableSet_eq_fun hT measurable_const) measurable_const - exact measurable_const.min ((hT.ennreal_toNNReal.mul_const ((2:ℝ≥0)^n)).nat_floor) - -lemma compensatorStopIndex_le (n : ℕ) (T : ℝ≥0∞) : - compensatorStopIndex n T ≤ n*2^n := by - unfold compensatorStopIndex - split_ifs - · exact le_rfl - · exact compensatorFiniteIndex_le _ _ - -def massAtExtendedTime (A : ℝ≥0 → ℝ) (L : ℝ) (T : ℝ≥0∞) : ℝ := - if T=∞ then L else A T.toNNReal - -theorem massAt_compensatorEndpoint_tendsto - (A : ℝ≥0 → ℝ) (L : ℝ) (hA : Continuous A) - (hL : Tendsto A atTop (𝓝 L)) (T : ℝ≥0∞) : - Tendsto (fun n => A (compensatorMeshTime n (compensatorStopIndex n T))) - atTop (𝓝 (massAtExtendedTime A L T)) := by - by_cases hT : T=∞ - · simpa only [compensatorStopIndex,massAtExtendedTime,hT,ite_true, - compensatorMeshTime_end,Function.comp_def] using hL.comp (tendsto_natCast_atTop_atTop (R := ℝ≥0)) - · simpa only [compensatorStopIndex,massAtExtendedTime,hT,ite_false,Function.comp_def] using - hA.continuousAt.tendsto.comp (compensatorEndpoint_tendsto T.toNNReal) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassCompensatorMesh + +namespace OAI + +/-! The last full dyadic cell before a finite or infinite time. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +lemma compensatorMeshTime_eq_div (n i : ℕ) : + compensatorMeshTime n i = (i:ℝ≥0)/(2:ℝ≥0)^n := by + simp only [compensatorMeshTime,zpow_neg,zpow_natCast,div_eq_mul_inv] + +@[simp] lemma compensatorMeshTime_end (n : ℕ) : + compensatorMeshTime n (n*2^n) = n := by + rw [compensatorMeshTime_eq_div] + push_cast + exact mul_div_cancel_right₀ _ (by positivity) + +def compensatorFiniteIndex (n : ℕ) (t : ℝ≥0) : ℕ := + min (n*2^n) ⌊t*(2:ℝ≥0)^n⌋₊ + +lemma compensatorFiniteIndex_le (n : ℕ) (t : ℝ≥0) : + compensatorFiniteIndex n t ≤ n*2^n := min_le_left _ _ + +lemma compensator_cell_before_iff (n i : ℕ) (t : ℝ≥0) : + compensatorMeshTime n (i+1) ≤ t ↔ i < ⌊t*(2:ℝ≥0)^n⌋₊ := by + rw [compensatorMeshTime_eq_div,div_le_iff₀ (by positivity), + ← Nat.le_floor_iff (show 0 ≤ t*(2:ℝ≥0)^n from zero_le)] + exact Nat.succ_le_iff + +lemma compensator_before_sum (n : ℕ) (t : ℝ≥0) (q : ℕ → ℝ) : + (∑ i ∈ Finset.range (n*2^n), + if compensatorMeshTime n (i+1) ≤ t then q i else 0) = + ∑ i ∈ Finset.range (compensatorFiniteIndex n t), q i := by + simp only [compensator_cell_before_iff] + rw [← Finset.sum_filter] + congr 1 + ext i + simp only [Finset.mem_filter,Finset.mem_range,compensatorFiniteIndex,lt_min_iff] + +lemma compensatorFiniteIndex_eq_floor {n : ℕ} {t : ℝ≥0} (ht : t ≤ n) : + compensatorFiniteIndex n t = ⌊t*(2:ℝ≥0)^n⌋₊ := by + apply min_eq_right + apply Nat.floor_le_of_le + push_cast + exact mul_le_mul_of_nonneg_right ht (by positivity) + +lemma compensatorEndpoint_le (n : ℕ) (t : ℝ≥0) : + compensatorMeshTime n (compensatorFiniteIndex n t) ≤ t := by + rw [compensatorMeshTime_eq_div,div_le_iff₀ (by positivity)] + exact (Nat.cast_le.mpr (min_le_right _ _)).trans (Nat.floor_le (show 0 ≤ t*(2:ℝ≥0)^n from zero_le)) + +lemma compensatorEndpoint_lower {n : ℕ} {t : ℝ≥0} (ht : t ≤ n) : + t < compensatorMeshTime n (compensatorFiniteIndex n t)+(2:ℝ≥0)^(-(n:ℤ)) := by + rw [compensatorFiniteIndex_eq_floor ht,compensatorMeshTime_eq_div, + zpow_neg,zpow_natCast,← one_div,← add_div,lt_div_iff₀ (by positivity)] + exact Nat.lt_floor_add_one _ + +/-- A finite endpoint is approached from below even though the grid +horizon grows. -/ +theorem compensatorEndpoint_tendsto (t : ℝ≥0) : + Tendsto (fun n => compensatorMeshTime n (compensatorFiniteIndex n t)) atTop (𝓝 t) := by + apply NNReal.tendsto_coe.mp + apply tendsto_of_tendsto_of_tendsto_of_le_of_le' + (show Tendsto (fun n : ℕ => (t:ℝ)-(2:ℝ)^(-(n:ℤ))) atTop (𝓝 (t:ℝ)) from + by simpa only [sub_zero,zpow_neg,zpow_natCast,inv_pow] using + tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_lt_one + (show 0 ≤ (2:ℝ)⁻¹ by positivity) (show (2:ℝ)⁻¹<1 by norm_num))) + tendsto_const_nhds + · have hn : ∀ᶠ n : ℕ in atTop, t ≤ (n:ℝ≥0) := + (tendsto_natCast_atTop_atTop (R := ℝ≥0)).eventually (eventually_ge_atTop t) + filter_upwards [hn] with n hn + have h := compensatorEndpoint_lower hn + have hc : (t:ℝ) < (compensatorMeshTime n (compensatorFiniteIndex n t):ℝ)+ + (2:ℝ)^(-(n:ℤ)) := by exact_mod_cast h + linarith + · exact Eventually.of_forall (fun n => by exact_mod_cast compensatorEndpoint_le n t) + +def compensatorStopIndex (n : ℕ) (T : ℝ≥0∞) : ℕ := + if T=∞ then n*2^n else compensatorFiniteIndex n T.toNNReal + +lemma compensatorStopIndex_measurable {Ω : Type*} [MeasurableSpace Ω] + (T : Ω → ℝ≥0∞) (hT : Measurable T) (n : ℕ) : + Measurable (fun ω => compensatorStopIndex n (T ω)) := by + apply Measurable.ite (measurableSet_eq_fun hT measurable_const) measurable_const + exact measurable_const.min ((hT.ennreal_toNNReal.mul_const ((2:ℝ≥0)^n)).nat_floor) + +lemma compensatorStopIndex_le (n : ℕ) (T : ℝ≥0∞) : + compensatorStopIndex n T ≤ n*2^n := by + unfold compensatorStopIndex + split_ifs + · exact le_rfl + · exact compensatorFiniteIndex_le _ _ + +def massAtExtendedTime (A : ℝ≥0 → ℝ) (L : ℝ) (T : ℝ≥0∞) : ℝ := + if T=∞ then L else A T.toNNReal + +theorem massAt_compensatorEndpoint_tendsto + (A : ℝ≥0 → ℝ) (L : ℝ) (hA : Continuous A) + (hL : Tendsto A atTop (𝓝 L)) (T : ℝ≥0∞) : + Tendsto (fun n => A (compensatorMeshTime n (compensatorStopIndex n T))) + atTop (𝓝 (massAtExtendedTime A L T)) := by + by_cases hT : T=∞ + · simpa only [compensatorStopIndex,massAtExtendedTime,hT,ite_true, + compensatorMeshTime_end,Function.comp_def] using hL.comp (tendsto_natCast_atTop_atTop (R := ℝ≥0)) + · simpa only [compensatorStopIndex,massAtExtendedTime,hT,ite_false,Function.comp_def] using + hA.continuousAt.tendsto.comp (compensatorEndpoint_tendsto T.toNNReal) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassCompensatorMesh.lean b/lean/OAI/Probability/SLE/Mass/MassCompensatorMesh.lean index 705940a2dd567c0463621c56b87b4ae90f174270..a343eee7feb2e9ef497e6c8bd03ed4f31bc8e9be 100644 --- a/lean/OAI/Probability/SLE/Mass/MassCompensatorMesh.lean +++ b/lean/OAI/Probability/SLE/Mass/MassCompensatorMesh.lean @@ -1,146 +1,146 @@ -import OAI.Probability.SLE.Mass.MassUniquenessMesh -import OAI.Probability.SLE.MomentGauge -import Mathlib.Topology.Order.Compact - -namespace OAI - -/-! The deterministic quadratic-error estimate for the prescribed dyadic -potential compensator. A finite terminal limit controls the growing horizon. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -/-- A continuous mass path with a finite terminal limit is uniformly -continuous even though the compensator mesh horizon grows to infinity. -/ -lemma mass_path_uniformContinuous (A : ℝ≥0 → ℝ) (L : ℝ) - (hc : Continuous A) (hl : Tendsto A atTop (𝓝 L)) : UniformContinuous A := by - apply hc.uniformContinuous_of_tendsto_cocompact - simpa only [cocompact_eq_atTop] using hl - -def compensatorMeshTime (n i : ℕ) : ℝ≥0 := - (i : ℝ≥0) * (2 : ℝ≥0)^(-(n:ℤ)) - -@[simp] lemma compensatorMeshTime_zero (n : ℕ) : compensatorMeshTime n 0=0 := by - simp [compensatorMeshTime] - -lemma compensatorMeshTime_mono (n : ℕ) : Monotone (compensatorMeshTime n) := by - intro i j hij - exact mul_le_mul_of_nonneg_right (by exact_mod_cast hij) (by positivity) - -lemma compensatorMeshTime_dist (n i : ℕ) : - dist (compensatorMeshTime n (i+1)) (compensatorMeshTime n i) = (2:ℝ)^(-(n:ℤ)) := by - rw [NNReal.dist_eq] - simp only [compensatorMeshTime,NNReal.coe_mul,NNReal.coe_zpow,NNReal.coe_ofNat, - NNReal.coe_natCast,NNReal.coe_add,NNReal.coe_one,Nat.cast_add,Nat.cast_one] - rw [show ((i:ℝ)+1)*(2:ℝ)^(-(n:ℤ))-(i:ℝ)*(2:ℝ)^(-(n:ℤ)) = - (2:ℝ)^(-(n:ℤ)) by ring, abs_of_pos (by positivity)] - -lemma compensatorMeshTime_small_increments (A : ℝ≥0 → ℝ) (L : ℝ) - (hc : Continuous A) (hl : Tendsto A atTop (𝓝 L)) : - ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, ∀ i : ℕ, - |A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i)| ≤ ε := by - intro ε hε - obtain ⟨δ,hδ,hAδ⟩ := Metric.uniformContinuous_iff.mp - (mass_path_uniformContinuous A L hc hl) ε hε - have hmesh : Tendsto (fun n : ℕ => (2:ℝ)^(-(n:ℤ))) atTop (𝓝 0) := by - simpa only [zpow_neg, zpow_natCast, inv_pow] using - (tendsto_pow_atTop_nhds_zero_of_lt_one (show 0 ≤ (2:ℝ)⁻¹ by positivity) - (show (2:ℝ)⁻¹ < 1 by norm_num)) - filter_upwards [hmesh.eventually (gt_mem_nhds hδ)] with n hn i - have h := hAδ (a := compensatorMeshTime n (i+1)) (b := compensatorMeshTime n i) - (by simpa only [compensatorMeshTime_dist] using hn) - exact (by simpa only [Real.dist_eq] using h.le) - -def compensatorQuadraticError (A : ℝ≥0 → ℝ) (n : ℕ) : ℝ := - ∑ i ∈ Finset.range (n*2^n), - (A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i))^2 - -lemma compensatorQuadraticError_nonneg (A : ℝ≥0 → ℝ) (n : ℕ) : - 0 ≤ compensatorQuadraticError A n := - Finset.sum_nonneg (fun _ _ => sq_nonneg _) - -lemma compensatorQuadraticError_le (A : ℝ≥0 → ℝ) (L : ℝ) - (hm : Monotone A) (h0 : A 0=0) (hb : ∀ t, A t ≤ L) - (n : ℕ) {ε : ℝ} (hε : 0 ≤ ε) - (hsmall : ∀ i : ℕ, A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i) ≤ ε) : - compensatorQuadraticError A n ≤ ε*L := by - have hsum : (∑ i ∈ Finset.range (n*2^n), - (A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i))) ≤ L := by - rw [Finset.sum_range_sub (fun i => A (compensatorMeshTime n i)) (n*2^n), - compensatorMeshTime_zero,h0,sub_zero] - exact hb _ - calc - compensatorQuadraticError A n ≤ ∑ i ∈ Finset.range (n*2^n), - ε*(A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i)) := by - apply Finset.sum_le_sum - intro i _ - have hnonneg := sub_nonneg.mpr (hm (compensatorMeshTime_mono n (Nat.le_succ i))) - nlinarith [hsmall i] - _ = ε*∑ i ∈ Finset.range (n*2^n), - (A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i)) := - (Finset.mul_sum _ _ _).symm - _ ≤ ε*L := mul_le_mul_of_nonneg_left hsum hε - -lemma compensatorQuadraticError_le_terminal_sq (A : ℝ≥0 → ℝ) (L : ℝ) - (hm : Monotone A) (h0 : A 0=0) (hb : ∀ t, A t ≤ L) (n : ℕ) : - compensatorQuadraticError A n ≤ L^2 := by - have hL : 0 ≤ L := by simpa only [h0] using hb 0 - simpa only [pow_two] using compensatorQuadraticError_le A L hm h0 hb n hL (fun i => by - have hAi : 0 ≤ A (compensatorMeshTime n i) := by - rw [← h0] - exact hm (show 0 ≤ _ from zero_le) - linarith [hb (compensatorMeshTime n (i+1))]) - -/-- The accumulated squared mass increments vanish for the exact growing -horizon and dyadic mesh used by the paper's potential formula. -/ -theorem compensatorQuadraticError_tendsto_zero (A : ℝ≥0 → ℝ) (L : ℝ) - (hm : Monotone A) (h0 : A 0=0) (hc : Continuous A) - (hl : Tendsto A atTop (𝓝 L)) (hb : ∀ t, A t ≤ L) : - Tendsto (compensatorQuadraticError A) atTop (𝓝 0) := by - have hL : 0 ≤ L := by simpa only [h0] using hb 0 - apply tendsto_order.mpr - constructor - · intro a ha - exact Eventually.of_forall (fun n => ha.trans_le (compensatorQuadraticError_nonneg A n)) - · intro b hb0 - have hd : 0 < b/(L+1) := div_pos hb0 (by linarith) - filter_upwards [compensatorMeshTime_small_increments A L hc hl _ hd] with n hn - have hbound := compensatorQuadraticError_le A L hm h0 hb n hd.le - (fun i => (le_abs_self _).trans (hn i)) - have he : b/(L+1)*(L+1)=b := div_mul_cancel₀ b (by linarith) - nlinarith - -/-- Dominated convergence for the exact quadratic mesh error; the bound -requires only the second terminal moment, not a uniform time horizon. -/ -theorem compensatorQuadraticError_mean_tendsto_zero - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) - (hA : ∀ t, Measurable (A t)) (hL : Integrable (fun ω => L ω^2) P) - (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ - Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ - ∀ t, A t ω ≤ L ω) : - Tendsto (fun n => ∫ ω, compensatorQuadraticError (fun t => A t ω) n ∂P) - atTop (𝓝 0) := by - have hm (n : ℕ) : Measurable (fun ω => compensatorQuadraticError (fun t => A t ω) n) := by - apply Finset.measurable_sum - intro i _ - exact ((hA _).sub (hA _)).pow_const 2 - have hb (n : ℕ) : ∀ᵐ ω ∂P, ‖compensatorQuadraticError (fun t => A t ω) n‖ ≤ L ω^2 := by - filter_upwards [hp] with ω hω - rw [Real.norm_of_nonneg (compensatorQuadraticError_nonneg _ _)] - exact compensatorQuadraticError_le_terminal_sq _ _ hω.1 hω.2.1 hω.2.2.2.2 n - have hl : ∀ᵐ ω ∂P, Tendsto (fun n => compensatorQuadraticError (fun t => A t ω) n) - atTop (𝓝 0) := by - filter_upwards [hp] with ω hω - exact compensatorQuadraticError_tendsto_zero _ _ hω.1 hω.2.1 hω.2.2.1 hω.2.2.2.1 hω.2.2.2.2 - simpa only [integral_zero] using tendsto_integral_of_dominated_convergence - (fun ω => L ω^2) (fun n => (hm n).aestronglyMeasurable) hL hb hl - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassUniquenessMesh +import OAI.Probability.SLE.MomentGauge +import Mathlib.Topology.Order.Compact + +namespace OAI + +/-! The deterministic quadratic-error estimate for the prescribed dyadic +potential compensator. A finite terminal limit controls the growing horizon. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +/-- A continuous mass path with a finite terminal limit is uniformly +continuous even though the compensator mesh horizon grows to infinity. -/ +lemma mass_path_uniformContinuous (A : ℝ≥0 → ℝ) (L : ℝ) + (hc : Continuous A) (hl : Tendsto A atTop (𝓝 L)) : UniformContinuous A := by + apply hc.uniformContinuous_of_tendsto_cocompact + simpa only [cocompact_eq_atTop] using hl + +def compensatorMeshTime (n i : ℕ) : ℝ≥0 := + (i : ℝ≥0) * (2 : ℝ≥0)^(-(n:ℤ)) + +@[simp] lemma compensatorMeshTime_zero (n : ℕ) : compensatorMeshTime n 0=0 := by + simp [compensatorMeshTime] + +lemma compensatorMeshTime_mono (n : ℕ) : Monotone (compensatorMeshTime n) := by + intro i j hij + exact mul_le_mul_of_nonneg_right (by exact_mod_cast hij) (by positivity) + +lemma compensatorMeshTime_dist (n i : ℕ) : + dist (compensatorMeshTime n (i+1)) (compensatorMeshTime n i) = (2:ℝ)^(-(n:ℤ)) := by + rw [NNReal.dist_eq] + simp only [compensatorMeshTime,NNReal.coe_mul,NNReal.coe_zpow,NNReal.coe_ofNat, + NNReal.coe_natCast,NNReal.coe_add,NNReal.coe_one,Nat.cast_add,Nat.cast_one] + rw [show ((i:ℝ)+1)*(2:ℝ)^(-(n:ℤ))-(i:ℝ)*(2:ℝ)^(-(n:ℤ)) = + (2:ℝ)^(-(n:ℤ)) by ring, abs_of_pos (by positivity)] + +lemma compensatorMeshTime_small_increments (A : ℝ≥0 → ℝ) (L : ℝ) + (hc : Continuous A) (hl : Tendsto A atTop (𝓝 L)) : + ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, ∀ i : ℕ, + |A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i)| ≤ ε := by + intro ε hε + obtain ⟨δ,hδ,hAδ⟩ := Metric.uniformContinuous_iff.mp + (mass_path_uniformContinuous A L hc hl) ε hε + have hmesh : Tendsto (fun n : ℕ => (2:ℝ)^(-(n:ℤ))) atTop (𝓝 0) := by + simpa only [zpow_neg, zpow_natCast, inv_pow] using + (tendsto_pow_atTop_nhds_zero_of_lt_one (show 0 ≤ (2:ℝ)⁻¹ by positivity) + (show (2:ℝ)⁻¹ < 1 by norm_num)) + filter_upwards [hmesh.eventually (gt_mem_nhds hδ)] with n hn i + have h := hAδ (a := compensatorMeshTime n (i+1)) (b := compensatorMeshTime n i) + (by simpa only [compensatorMeshTime_dist] using hn) + exact (by simpa only [Real.dist_eq] using h.le) + +def compensatorQuadraticError (A : ℝ≥0 → ℝ) (n : ℕ) : ℝ := + ∑ i ∈ Finset.range (n*2^n), + (A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i))^2 + +lemma compensatorQuadraticError_nonneg (A : ℝ≥0 → ℝ) (n : ℕ) : + 0 ≤ compensatorQuadraticError A n := + Finset.sum_nonneg (fun _ _ => sq_nonneg _) + +lemma compensatorQuadraticError_le (A : ℝ≥0 → ℝ) (L : ℝ) + (hm : Monotone A) (h0 : A 0=0) (hb : ∀ t, A t ≤ L) + (n : ℕ) {ε : ℝ} (hε : 0 ≤ ε) + (hsmall : ∀ i : ℕ, A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i) ≤ ε) : + compensatorQuadraticError A n ≤ ε*L := by + have hsum : (∑ i ∈ Finset.range (n*2^n), + (A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i))) ≤ L := by + rw [Finset.sum_range_sub (fun i => A (compensatorMeshTime n i)) (n*2^n), + compensatorMeshTime_zero,h0,sub_zero] + exact hb _ + calc + compensatorQuadraticError A n ≤ ∑ i ∈ Finset.range (n*2^n), + ε*(A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i)) := by + apply Finset.sum_le_sum + intro i _ + have hnonneg := sub_nonneg.mpr (hm (compensatorMeshTime_mono n (Nat.le_succ i))) + nlinarith [hsmall i] + _ = ε*∑ i ∈ Finset.range (n*2^n), + (A (compensatorMeshTime n (i+1))-A (compensatorMeshTime n i)) := + (Finset.mul_sum _ _ _).symm + _ ≤ ε*L := mul_le_mul_of_nonneg_left hsum hε + +lemma compensatorQuadraticError_le_terminal_sq (A : ℝ≥0 → ℝ) (L : ℝ) + (hm : Monotone A) (h0 : A 0=0) (hb : ∀ t, A t ≤ L) (n : ℕ) : + compensatorQuadraticError A n ≤ L^2 := by + have hL : 0 ≤ L := by simpa only [h0] using hb 0 + simpa only [pow_two] using compensatorQuadraticError_le A L hm h0 hb n hL (fun i => by + have hAi : 0 ≤ A (compensatorMeshTime n i) := by + rw [← h0] + exact hm (show 0 ≤ _ from zero_le) + linarith [hb (compensatorMeshTime n (i+1))]) + +/-- The accumulated squared mass increments vanish for the exact growing +horizon and dyadic mesh used by the paper's potential formula. -/ +theorem compensatorQuadraticError_tendsto_zero (A : ℝ≥0 → ℝ) (L : ℝ) + (hm : Monotone A) (h0 : A 0=0) (hc : Continuous A) + (hl : Tendsto A atTop (𝓝 L)) (hb : ∀ t, A t ≤ L) : + Tendsto (compensatorQuadraticError A) atTop (𝓝 0) := by + have hL : 0 ≤ L := by simpa only [h0] using hb 0 + apply tendsto_order.mpr + constructor + · intro a ha + exact Eventually.of_forall (fun n => ha.trans_le (compensatorQuadraticError_nonneg A n)) + · intro b hb0 + have hd : 0 < b/(L+1) := div_pos hb0 (by linarith) + filter_upwards [compensatorMeshTime_small_increments A L hc hl _ hd] with n hn + have hbound := compensatorQuadraticError_le A L hm h0 hb n hd.le + (fun i => (le_abs_self _).trans (hn i)) + have he : b/(L+1)*(L+1)=b := div_mul_cancel₀ b (by linarith) + nlinarith + +/-- Dominated convergence for the exact quadratic mesh error; the bound +requires only the second terminal moment, not a uniform time horizon. -/ +theorem compensatorQuadraticError_mean_tendsto_zero + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) + (hA : ∀ t, Measurable (A t)) (hL : Integrable (fun ω => L ω^2) P) + (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ + Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ + ∀ t, A t ω ≤ L ω) : + Tendsto (fun n => ∫ ω, compensatorQuadraticError (fun t => A t ω) n ∂P) + atTop (𝓝 0) := by + have hm (n : ℕ) : Measurable (fun ω => compensatorQuadraticError (fun t => A t ω) n) := by + apply Finset.measurable_sum + intro i _ + exact ((hA _).sub (hA _)).pow_const 2 + have hb (n : ℕ) : ∀ᵐ ω ∂P, ‖compensatorQuadraticError (fun t => A t ω) n‖ ≤ L ω^2 := by + filter_upwards [hp] with ω hω + rw [Real.norm_of_nonneg (compensatorQuadraticError_nonneg _ _)] + exact compensatorQuadraticError_le_terminal_sq _ _ hω.1 hω.2.1 hω.2.2.2.2 n + have hl : ∀ᵐ ω ∂P, Tendsto (fun n => compensatorQuadraticError (fun t => A t ω) n) + atTop (𝓝 0) := by + filter_upwards [hp] with ω hω + exact compensatorQuadraticError_tendsto_zero _ _ hω.1 hω.2.1 hω.2.2.1 hω.2.2.2.1 hω.2.2.2.2 + simpa only [integral_zero] using tendsto_integral_of_dominated_convergence + (fun ω => L ω^2) (fun n => (hm n).aestronglyMeasurable) hL hb hl + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassCompensatorRecovery.lean b/lean/OAI/Probability/SLE/Mass/MassCompensatorRecovery.lean index 88ef900b892d00bade52bf1e4f02254602e50c61..dab8ce7a1b6a41e3631df0a738b570b743cf82e2 100644 --- a/lean/OAI/Probability/SLE/Mass/MassCompensatorRecovery.lean +++ b/lean/OAI/Probability/SLE/Mass/MassCompensatorRecovery.lean @@ -1,144 +1,144 @@ -import OAI.Probability.SLE.Mass.MassCompensatorEndpoint -import OAI.Probability.SLE.Mass.MassCompensatorSelection - -namespace OAI - -/-! Recovery of a continuous finite mass path from its discrete -predictable increments, with the paper's growing dyadic horizon. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - -def meshMassIncrement (A : ℝ≥0 → Ω → ℝ) (n i : ℕ) (ω : Ω) : ℝ := - A (compensatorMeshTime n (i+1)) ω-A (compensatorMeshTime n i) ω - -def meshMassCompensator (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) - (n j : ℕ) (ω : Ω) : ℝ := - ∑ i ∈ Finset.range j, condExp (ℱ n i) P (meshMassIncrement A n i) ω - -omit [IsProbabilityMeasure P] in -lemma meshMassIncrement_memLp - (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) (hL : MemLp L 2 P) - (hA : ∀ t, Measurable (A t)) - (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ ∀ t, A t ω ≤ L ω) - (n i : ℕ) : MemLp (meshMassIncrement A n i) 2 P := by - apply hL.mono' (((hA _).sub (hA _)).aestronglyMeasurable) - filter_upwards [hp] with ω hω - have hn : 0 ≤ meshMassIncrement A n i ω := - sub_nonneg.mpr (hω.1 (compensatorMeshTime_mono n (Nat.le_succ i))) - have hz : 0 ≤ A (compensatorMeshTime n i) ω := by - rw [← hω.2.1] - exact hω.1 (show 0 ≤ _ from zero_le) - simp only [Pi.sub_apply] - change 0 ≤ A (compensatorMeshTime n (i+1)) ω-A (compensatorMeshTime n i) ω at hn - rw [Real.norm_of_nonneg hn] - linarith [hω.2.2 (compensatorMeshTime n (i+1))] - -omit [IsProbabilityMeasure P] in -lemma meshMassCompensator_decomposition - (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (n j : ℕ) (ω : Ω) : - meshMassCompensator P ℱ A n j ω = - A (compensatorMeshTime n j) ω-A 0 ω- - centeredMassPartialSum P (ℱ n) (meshMassIncrement A n) j ω := by - unfold centeredMassPartialSum meshMassCompensator - rw [Finset.sum_sub_distrib] - have he : (∑ i ∈ Finset.range j, meshMassIncrement A n i ω) = - A (compensatorMeshTime n j) ω-A 0 ω := by - simp only [meshMassIncrement] - rw [Finset.sum_range_sub (fun i => A (compensatorMeshTime n i) ω) j,compensatorMeshTime_zero] - rw [he] - ring - -theorem meshMassCentered_selected_tendstoInMeasure - (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) - (hA : ∀ t, Measurable (A t)) - (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i))) - (hL : MemLp L 2 P) - (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ - Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ - ∀ t, A t ω ≤ L ω) - (J : ℕ → Ω → ℕ) (hJ : ∀ n ω, J n ω ≤ n*2^n) : - TendstoInMeasure P - (fun n ω => centeredMassPartialSum P (ℱ n) (meshMassIncrement A n) (J n ω) ω) - atTop 0 := by - apply centeredMassPartialSum_selected_tendstoInMeasure P ℱ (meshMassIncrement A) - (fun n i => meshMassIncrement_memLp P A L hL hA - (hp.mono (fun ω hω => ⟨hω.1,hω.2.1,hω.2.2.2.2⟩)) n i) - (fun n i => (hgrid n (i+1)).sub ((hgrid n i).mono (ℱ n |>.mono (Nat.le_succ i)))) - (fun n => n*2^n) J hJ - exact compensatorQuadraticError_mean_tendsto_zero P A L hA hL.integrable_sq hp - -omit [IsProbabilityMeasure P] in -lemma meshMassCompensator_selected_measurable - (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) - (n : ℕ) (J : Ω → ℕ) (hJ : Measurable J) : - Measurable (fun ω => meshMassCompensator P ℱ A n (J ω) ω) := by - have hjoint : Measurable (fun z : ℕ × Ω => meshMassCompensator P ℱ A n z.1 z.2) := by - apply measurable_from_prod_countable_right - intro j - change Measurable (fun ω => ∑ i ∈ Finset.range j, condExp (ℱ n i) P (meshMassIncrement A n i) ω) - apply Finset.measurable_sum - intro i _ - exact stronglyMeasurable_condExp.measurable.le (ℱ n |>.le i) - exact hjoint.comp (hJ.prodMk measurable_id) - -/-- Any measurable choice of endpoints where the sampled mass has its -pathwise limit recovers that same mass from the predictable increments. -/ -theorem meshMassCompensator_selected_tendstoInMeasure - (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L V : Ω → ℝ) - (hA : ∀ t, Measurable (A t)) - (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i))) - (hL : MemLp L 2 P) - (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ - Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ - ∀ t, A t ω ≤ L ω) - (J : ℕ → Ω → ℕ) (hJm : ∀ n, Measurable (J n)) - (hJ : ∀ n ω, J n ω ≤ n*2^n) - (hJV : ∀ᵐ ω ∂P, Tendsto (fun n => A (compensatorMeshTime n (J n ω)) ω) - atTop (𝓝 (V ω))) : - TendstoInMeasure P (fun n ω => meshMassCompensator P ℱ A n (J n ω) ω) atTop V := by - have hZ := meshMassCentered_selected_tendstoInMeasure P ℱ A L hA hgrid hL hp J hJ - apply (exists_seq_tendstoInMeasure_atTop_iff - (fun n => (meshMassCompensator_selected_measurable P ℱ A n (J n) (hJm n)).aestronglyMeasurable)).mpr - intro ns hns - obtain ⟨ms,hms,hlim⟩ := (hZ.comp hns.tendsto_atTop).exists_seq_tendsto_ae - refine ⟨ms,hms,?_⟩ - filter_upwards [hp,hJV,hlim] with ω hω hV hZω - have ht := (hV.comp (hns.tendsto_atTop.comp hms.tendsto_atTop)).sub hZω - simp only [Pi.zero_apply,sub_zero,Function.comp_def] at ht - convert ht using 1 - funext n - rw [meshMassCompensator_decomposition,hω.2.1,sub_zero] - -/-- The exact last-full-cell convention, including an infinite endpoint, -recovers the stopped mass in probability. -/ -theorem meshMassCompensator_stopped_tendstoInMeasure - (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) - (hA : ∀ t, Measurable (A t)) - (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i))) - (hL : MemLp L 2 P) - (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ - Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ - ∀ t, A t ω ≤ L ω) - (T : Ω → ℝ≥0∞) (hT : Measurable T) : - TendstoInMeasure P - (fun n ω => meshMassCompensator P ℱ A n (compensatorStopIndex n (T ω)) ω) - atTop (fun ω => massAtExtendedTime (fun t => A t ω) (L ω) (T ω)) := by - apply meshMassCompensator_selected_tendstoInMeasure P ℱ A L _ hA hgrid hL hp - (fun n ω => compensatorStopIndex n (T ω)) - (fun n => compensatorStopIndex_measurable T hT n) - (fun n ω => compensatorStopIndex_le n (T ω)) - filter_upwards [hp] with ω hω - exact massAt_compensatorEndpoint_tendsto (fun t => A t ω) (L ω) - hω.2.2.1 hω.2.2.2.1 (T ω) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassCompensatorEndpoint +import OAI.Probability.SLE.Mass.MassCompensatorSelection + +namespace OAI + +/-! Recovery of a continuous finite mass path from its discrete +predictable increments, with the paper's growing dyadic horizon. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + +def meshMassIncrement (A : ℝ≥0 → Ω → ℝ) (n i : ℕ) (ω : Ω) : ℝ := + A (compensatorMeshTime n (i+1)) ω-A (compensatorMeshTime n i) ω + +def meshMassCompensator (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) + (n j : ℕ) (ω : Ω) : ℝ := + ∑ i ∈ Finset.range j, condExp (ℱ n i) P (meshMassIncrement A n i) ω + +omit [IsProbabilityMeasure P] in +lemma meshMassIncrement_memLp + (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) (hL : MemLp L 2 P) + (hA : ∀ t, Measurable (A t)) + (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ ∀ t, A t ω ≤ L ω) + (n i : ℕ) : MemLp (meshMassIncrement A n i) 2 P := by + apply hL.mono' (((hA _).sub (hA _)).aestronglyMeasurable) + filter_upwards [hp] with ω hω + have hn : 0 ≤ meshMassIncrement A n i ω := + sub_nonneg.mpr (hω.1 (compensatorMeshTime_mono n (Nat.le_succ i))) + have hz : 0 ≤ A (compensatorMeshTime n i) ω := by + rw [← hω.2.1] + exact hω.1 (show 0 ≤ _ from zero_le) + simp only [Pi.sub_apply] + change 0 ≤ A (compensatorMeshTime n (i+1)) ω-A (compensatorMeshTime n i) ω at hn + rw [Real.norm_of_nonneg hn] + linarith [hω.2.2 (compensatorMeshTime n (i+1))] + +omit [IsProbabilityMeasure P] in +lemma meshMassCompensator_decomposition + (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (n j : ℕ) (ω : Ω) : + meshMassCompensator P ℱ A n j ω = + A (compensatorMeshTime n j) ω-A 0 ω- + centeredMassPartialSum P (ℱ n) (meshMassIncrement A n) j ω := by + unfold centeredMassPartialSum meshMassCompensator + rw [Finset.sum_sub_distrib] + have he : (∑ i ∈ Finset.range j, meshMassIncrement A n i ω) = + A (compensatorMeshTime n j) ω-A 0 ω := by + simp only [meshMassIncrement] + rw [Finset.sum_range_sub (fun i => A (compensatorMeshTime n i) ω) j,compensatorMeshTime_zero] + rw [he] + ring + +theorem meshMassCentered_selected_tendstoInMeasure + (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) + (hA : ∀ t, Measurable (A t)) + (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i))) + (hL : MemLp L 2 P) + (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ + Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ + ∀ t, A t ω ≤ L ω) + (J : ℕ → Ω → ℕ) (hJ : ∀ n ω, J n ω ≤ n*2^n) : + TendstoInMeasure P + (fun n ω => centeredMassPartialSum P (ℱ n) (meshMassIncrement A n) (J n ω) ω) + atTop 0 := by + apply centeredMassPartialSum_selected_tendstoInMeasure P ℱ (meshMassIncrement A) + (fun n i => meshMassIncrement_memLp P A L hL hA + (hp.mono (fun ω hω => ⟨hω.1,hω.2.1,hω.2.2.2.2⟩)) n i) + (fun n i => (hgrid n (i+1)).sub ((hgrid n i).mono (ℱ n |>.mono (Nat.le_succ i)))) + (fun n => n*2^n) J hJ + exact compensatorQuadraticError_mean_tendsto_zero P A L hA hL.integrable_sq hp + +omit [IsProbabilityMeasure P] in +lemma meshMassCompensator_selected_measurable + (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) + (n : ℕ) (J : Ω → ℕ) (hJ : Measurable J) : + Measurable (fun ω => meshMassCompensator P ℱ A n (J ω) ω) := by + have hjoint : Measurable (fun z : ℕ × Ω => meshMassCompensator P ℱ A n z.1 z.2) := by + apply measurable_from_prod_countable_right + intro j + change Measurable (fun ω => ∑ i ∈ Finset.range j, condExp (ℱ n i) P (meshMassIncrement A n i) ω) + apply Finset.measurable_sum + intro i _ + exact stronglyMeasurable_condExp.measurable.le (ℱ n |>.le i) + exact hjoint.comp (hJ.prodMk measurable_id) + +/-- Any measurable choice of endpoints where the sampled mass has its +pathwise limit recovers that same mass from the predictable increments. -/ +theorem meshMassCompensator_selected_tendstoInMeasure + (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L V : Ω → ℝ) + (hA : ∀ t, Measurable (A t)) + (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i))) + (hL : MemLp L 2 P) + (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ + Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ + ∀ t, A t ω ≤ L ω) + (J : ℕ → Ω → ℕ) (hJm : ∀ n, Measurable (J n)) + (hJ : ∀ n ω, J n ω ≤ n*2^n) + (hJV : ∀ᵐ ω ∂P, Tendsto (fun n => A (compensatorMeshTime n (J n ω)) ω) + atTop (𝓝 (V ω))) : + TendstoInMeasure P (fun n ω => meshMassCompensator P ℱ A n (J n ω) ω) atTop V := by + have hZ := meshMassCentered_selected_tendstoInMeasure P ℱ A L hA hgrid hL hp J hJ + apply (exists_seq_tendstoInMeasure_atTop_iff + (fun n => (meshMassCompensator_selected_measurable P ℱ A n (J n) (hJm n)).aestronglyMeasurable)).mpr + intro ns hns + obtain ⟨ms,hms,hlim⟩ := (hZ.comp hns.tendsto_atTop).exists_seq_tendsto_ae + refine ⟨ms,hms,?_⟩ + filter_upwards [hp,hJV,hlim] with ω hω hV hZω + have ht := (hV.comp (hns.tendsto_atTop.comp hms.tendsto_atTop)).sub hZω + simp only [Pi.zero_apply,sub_zero,Function.comp_def] at ht + convert ht using 1 + funext n + rw [meshMassCompensator_decomposition,hω.2.1,sub_zero] + +/-- The exact last-full-cell convention, including an infinite endpoint, +recovers the stopped mass in probability. -/ +theorem meshMassCompensator_stopped_tendstoInMeasure + (ℱ : ℕ → Filtration ℕ mΩ) (A : ℝ≥0 → Ω → ℝ) (L : Ω → ℝ) + (hA : ∀ t, Measurable (A t)) + (hgrid : ∀ n i, StronglyMeasurable[ℱ n i] (A (compensatorMeshTime n i))) + (hL : MemLp L 2 P) + (hp : ∀ᵐ ω ∂P, Monotone (fun t => A t ω) ∧ A 0 ω=0 ∧ + Continuous (fun t => A t ω) ∧ Tendsto (fun t => A t ω) atTop (𝓝 (L ω)) ∧ + ∀ t, A t ω ≤ L ω) + (T : Ω → ℝ≥0∞) (hT : Measurable T) : + TendstoInMeasure P + (fun n ω => meshMassCompensator P ℱ A n (compensatorStopIndex n (T ω)) ω) + atTop (fun ω => massAtExtendedTime (fun t => A t ω) (L ω) (T ω)) := by + apply meshMassCompensator_selected_tendstoInMeasure P ℱ A L _ hA hgrid hL hp + (fun n ω => compensatorStopIndex n (T ω)) + (fun n => compensatorStopIndex_measurable T hT n) + (fun n ω => compensatorStopIndex_le n (T ω)) + filter_upwards [hp] with ω hω + exact massAt_compensatorEndpoint_tendsto (fun t => A t ω) (L ω) + hω.2.2.1 hω.2.2.2.1 (T ω) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassCompensatorSelection.lean b/lean/OAI/Probability/SLE/Mass/MassCompensatorSelection.lean index 598e99832c292edf2eea2a6e87a32174ca92002b..9a331cbb709fa2b207c3b5e55929db625889c67b 100644 --- a/lean/OAI/Probability/SLE/Mass/MassCompensatorSelection.lean +++ b/lean/OAI/Probability/SLE/Mass/MassCompensatorSelection.lean @@ -1,81 +1,81 @@ -import OAI.Probability.SLE.Mass.MassCompensatorVariance -import Mathlib.MeasureTheory.Function.ConvergenceInMeasure - -namespace OAI - -/-! The compensator error at a selected grid endpoint. The endpoint may -depend on the whole path; the maximal bound does not require it to be a -discrete stopping time. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - -lemma centeredMassPartialSum_selected_bound - (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (hD : ∀ i, MemLp (D i) 2 P) - (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) - (N : ℕ) (J : Ω → ℕ) (hJ : ∀ ω, J ω ≤ N) (ε : ℝ) (hε : 0 < ε) : - ENNReal.ofReal (ε^2) * P {ω | ε ≤ |centeredMassPartialSum P ℱ D (J ω) ω|} ≤ - ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range N, D i ω^2 ∂P) := by - have hs : {ω | ε ≤ |centeredMassPartialSum P ℱ D (J ω) ω|} ⊆ - {ω | ε^2 ≤ (Finset.range (N+1)).sup' Finset.nonempty_range_add_one - (fun j => centeredMassPartialSum P ℱ D j ω^2)} := by - intro ω hω - change ε^2 ≤ _ - apply Finset.le_sup'_of_le (fun j => centeredMassPartialSum P ℱ D j ω^2) - (Finset.mem_range.mpr (Nat.lt_succ_of_le (hJ ω))) - have hh := mul_self_le_mul_self hε.le hω - simpa only [sq_abs,← pow_two] using hh - have hb := centeredMassPartialSum_maximal_error P ℱ D hD hDm - (NNReal.mk (ε^2) (sq_nonneg ε)) N - have hb' : ENNReal.ofReal (ε^2) * P {ω | ε^2 ≤ - (Finset.range (N+1)).sup' Finset.nonempty_range_add_one - (fun j => centeredMassPartialSum P ℱ D j ω^2)} ≤ - ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range N, D i ω^2 ∂P) := by - simpa only [ENNReal.ofReal_eq_coe_nnreal (sq_nonneg ε),NNReal.coe_mk] using hb - exact (mul_le_mul_of_nonneg_left (measure_mono hs) - (show 0 ≤ ENNReal.ofReal (ε^2) from zero_le)).trans hb' - -/-- Vanishing expected quadratic increments imply that the centered -compensator vanishes in probability at every bounded selected index. -/ -theorem centeredMassPartialSum_selected_tendstoInMeasure - (ℱ : ℕ → Filtration ℕ mΩ) (D : ℕ → ℕ → Ω → ℝ) - (hD : ∀ n i, MemLp (D n i) 2 P) - (hDm : ∀ n i, StronglyMeasurable[ℱ n (i+1)] (D n i)) - (N : ℕ → ℕ) (J : ℕ → Ω → ℕ) (hJ : ∀ n ω, J n ω ≤ N n) - (hE : Tendsto (fun n => ∫ ω, ∑ i ∈ Finset.range (N n), D n i ω^2 ∂P) - atTop (𝓝 0)) : - TendstoInMeasure P (fun n ω => centeredMassPartialSum P (ℱ n) (D n) (J n ω) ω) - atTop 0 := by - rw [tendstoInMeasure_iff_norm] - intro ε hε - let c : ℝ≥0∞ := ENNReal.ofReal (ε^2) - have hc0 : c ≠ 0 := (ENNReal.ofReal_pos.mpr (sq_pos_of_pos hε)).ne' - have hcT : c ≠ ∞ := ENNReal.ofReal_ne_top - have hb (n : ℕ) : P {ω | ε ≤ |centeredMassPartialSum P (ℱ n) (D n) (J n ω) ω|} ≤ - c⁻¹*ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range (N n), D n i ω^2 ∂P) := by - calc - _ = c⁻¹*(c*P {ω | ε ≤ |centeredMassPartialSum P (ℱ n) (D n) (J n ω) ω|}) := by - rw [← mul_assoc,ENNReal.inv_mul_cancel hc0 hcT,one_mul] - _ ≤ _ := mul_le_mul_of_nonneg_left - (centeredMassPartialSum_selected_bound P (ℱ n) (D n) (hD n) (hDm n) - (N n) (J n) (hJ n) ε hε) (show 0 ≤ c⁻¹ from zero_le) - have ht : Tendsto (fun n => c⁻¹*ENNReal.ofReal - (∫ ω, ∑ i ∈ Finset.range (N n), D n i ω^2 ∂P)) atTop (𝓝 0) := by - simpa only [ENNReal.ofReal_zero,mul_zero] using - ENNReal.Tendsto.const_mul (ENNReal.tendsto_ofReal hE) - (Or.inr (ENNReal.inv_ne_top.mpr hc0)) - simpa only [Pi.zero_apply,sub_zero,Real.norm_eq_abs] using - tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht - (fun _ => bot_le) hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassCompensatorVariance +import Mathlib.MeasureTheory.Function.ConvergenceInMeasure + +namespace OAI + +/-! The compensator error at a selected grid endpoint. The endpoint may +depend on the whole path; the maximal bound does not require it to be a +discrete stopping time. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + +lemma centeredMassPartialSum_selected_bound + (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (hD : ∀ i, MemLp (D i) 2 P) + (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) + (N : ℕ) (J : Ω → ℕ) (hJ : ∀ ω, J ω ≤ N) (ε : ℝ) (hε : 0 < ε) : + ENNReal.ofReal (ε^2) * P {ω | ε ≤ |centeredMassPartialSum P ℱ D (J ω) ω|} ≤ + ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range N, D i ω^2 ∂P) := by + have hs : {ω | ε ≤ |centeredMassPartialSum P ℱ D (J ω) ω|} ⊆ + {ω | ε^2 ≤ (Finset.range (N+1)).sup' Finset.nonempty_range_add_one + (fun j => centeredMassPartialSum P ℱ D j ω^2)} := by + intro ω hω + change ε^2 ≤ _ + apply Finset.le_sup'_of_le (fun j => centeredMassPartialSum P ℱ D j ω^2) + (Finset.mem_range.mpr (Nat.lt_succ_of_le (hJ ω))) + have hh := mul_self_le_mul_self hε.le hω + simpa only [sq_abs,← pow_two] using hh + have hb := centeredMassPartialSum_maximal_error P ℱ D hD hDm + (NNReal.mk (ε^2) (sq_nonneg ε)) N + have hb' : ENNReal.ofReal (ε^2) * P {ω | ε^2 ≤ + (Finset.range (N+1)).sup' Finset.nonempty_range_add_one + (fun j => centeredMassPartialSum P ℱ D j ω^2)} ≤ + ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range N, D i ω^2 ∂P) := by + simpa only [ENNReal.ofReal_eq_coe_nnreal (sq_nonneg ε),NNReal.coe_mk] using hb + exact (mul_le_mul_of_nonneg_left (measure_mono hs) + (show 0 ≤ ENNReal.ofReal (ε^2) from zero_le)).trans hb' + +/-- Vanishing expected quadratic increments imply that the centered +compensator vanishes in probability at every bounded selected index. -/ +theorem centeredMassPartialSum_selected_tendstoInMeasure + (ℱ : ℕ → Filtration ℕ mΩ) (D : ℕ → ℕ → Ω → ℝ) + (hD : ∀ n i, MemLp (D n i) 2 P) + (hDm : ∀ n i, StronglyMeasurable[ℱ n (i+1)] (D n i)) + (N : ℕ → ℕ) (J : ℕ → Ω → ℕ) (hJ : ∀ n ω, J n ω ≤ N n) + (hE : Tendsto (fun n => ∫ ω, ∑ i ∈ Finset.range (N n), D n i ω^2 ∂P) + atTop (𝓝 0)) : + TendstoInMeasure P (fun n ω => centeredMassPartialSum P (ℱ n) (D n) (J n ω) ω) + atTop 0 := by + rw [tendstoInMeasure_iff_norm] + intro ε hε + let c : ℝ≥0∞ := ENNReal.ofReal (ε^2) + have hc0 : c ≠ 0 := (ENNReal.ofReal_pos.mpr (sq_pos_of_pos hε)).ne' + have hcT : c ≠ ∞ := ENNReal.ofReal_ne_top + have hb (n : ℕ) : P {ω | ε ≤ |centeredMassPartialSum P (ℱ n) (D n) (J n ω) ω|} ≤ + c⁻¹*ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range (N n), D n i ω^2 ∂P) := by + calc + _ = c⁻¹*(c*P {ω | ε ≤ |centeredMassPartialSum P (ℱ n) (D n) (J n ω) ω|}) := by + rw [← mul_assoc,ENNReal.inv_mul_cancel hc0 hcT,one_mul] + _ ≤ _ := mul_le_mul_of_nonneg_left + (centeredMassPartialSum_selected_bound P (ℱ n) (D n) (hD n) (hDm n) + (N n) (J n) (hJ n) ε hε) (show 0 ≤ c⁻¹ from zero_le) + have ht : Tendsto (fun n => c⁻¹*ENNReal.ofReal + (∫ ω, ∑ i ∈ Finset.range (N n), D n i ω^2 ∂P)) atTop (𝓝 0) := by + simpa only [ENNReal.ofReal_zero,mul_zero] using + ENNReal.Tendsto.const_mul (ENNReal.tendsto_ofReal hE) + (Or.inr (ENNReal.inv_ne_top.mpr hc0)) + simpa only [Pi.zero_apply,sub_zero,Real.norm_eq_abs] using + tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds ht + (fun _ => bot_le) hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassCompensatorVariance.lean b/lean/OAI/Probability/SLE/Mass/MassCompensatorVariance.lean index b479aefd0b6a2d3856df50b160a97f2b512fc035..a865a79331c8a5bdef79ececc7bced3ceae75d58 100644 --- a/lean/OAI/Probability/SLE/Mass/MassCompensatorVariance.lean +++ b/lean/OAI/Probability/SLE/Mass/MassCompensatorVariance.lean @@ -1,192 +1,192 @@ -import OAI.Probability.SLE.Mass.MassCompensatorMesh -import Mathlib.Probability.Martingale.Centering -import Mathlib.Probability.Martingale.OptionalStopping -import Mathlib.Probability.CondVar - -namespace OAI - -/-! Orthogonality and variance bounds for the discrete potential -compensator used in the prescribed moment coefficients. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - -section Centered -variable (m : MeasurableSpace Ω) (hm : m ≤ mΩ) - -include hm in -lemma centered_mass_increment_condExp (X : Ω → ℝ) (hX : MemLp X 2 P) : - P[X-P[X|m]|m] =ᵐ[P] 0 := by - have hi := hX.integrable one_le_two - have h := condExp_sub hi (integrable_condExp (μ := P) (m := m) (f := X)) m - rw [condExp_of_stronglyMeasurable hm stronglyMeasurable_condExp integrable_condExp] at h - simpa only [sub_self] using h - -include hm in -lemma integral_mul_centered_mass_increment - (X Y : Ω → ℝ) (hX : MemLp X 2 P) (hY : MemLp Y 2 P) - (hYm : StronglyMeasurable[m] Y) : - (∫ ω, Y ω*(X ω-P[X|m] ω) ∂P)=0 := by - have hZ : MemLp (X-P[X|m]) 2 P := hX.sub (hX.condExp one_le_two) - have hprod : Integrable (Y*(X-P[X|m])) P := memLp_one_iff_integrable.mp (hY.mul hZ) - have he := condExp_mul_of_stronglyMeasurable_left hYm hprod (hZ.integrable one_le_two) - have hz := centered_mass_increment_condExp (mΩ := mΩ) P m hm X hX - have he0 : P[Y*(X-P[X|m])|m] =ᵐ[P] 0 := by - filter_upwards [he,hz] with ω hp hz - simp only [Pi.mul_apply,Pi.zero_apply,hz,mul_zero] at hp ⊢ - exact hp - calc - _ = ∫ ω, P[Y*(X-P[X|m])|m] ω ∂P := (integral_condExp hm).symm - _ = 0 := by simpa only [Pi.zero_apply,integral_zero] using integral_congr_ae he0 - -include hm in -lemma centered_mass_increment_variance_le (X : Ω → ℝ) (hX : MemLp X 2 P) : - (∫ ω, (X ω-P[X|m] ω)^2 ∂P) ≤ ∫ ω, X ω^2 ∂P := by - have hZ : Integrable (fun ω => (X ω-P[X|m] ω)^2) P := - (hX.sub (hX.condExp one_le_two)).integrable_sq - have he := setIntegral_condVar (hm := hm) hZ (MeasurableSet.univ : MeasurableSet[m] univ) - simp only [Measure.restrict_univ] at he - rw [← he] - calc - _ ≤ ∫ ω, P[X^2|m] ω ∂P := - integral_mono_ae integrable_condVar integrable_condExp (condVar_ae_le_condExp_sq hm hX) - _ = _ := integral_condExp hm - -end Centered - -lemma martingale_square_submartingale - {ι : Type*} [Preorder ι] (ℱ : Filtration ι mΩ) - (M : ι → Ω → ℝ) (hM : Martingale M ℱ P) (h2 : ∀ i, MemLp (M i) 2 P) : - Submartingale (fun i ω => M i ω^2) ℱ P := by - refine ⟨fun i => (hM.stronglyMeasurable i).pow 2,?_,fun i => (h2 i).integrable_sq⟩ - intro i j hij - have hv := condVar_ae_eq_condExp_sq_sub_sq_condExp (ℱ.le i) (h2 j) - have hn : 0 ≤ᵐ[P] condVar (ℱ i) (M j) P := - condExp_nonneg (ae_of_all _ (fun ω => sq_nonneg (M j ω-P[M j|ℱ i] ω))) - filter_upwards [hv,hn,hM.condExp_ae_eq hij] with ω hv hn he - simp only [Pi.sub_apply,Pi.pow_apply] at hv - rw [he] at hv - change 0 ≤ condVar (ℱ i) (M j) P ω at hn - change condVar (ℱ i) (M j) P ω = P[fun ω => M j ω^2|ℱ i] ω - M i ω^2 at hv - linarith - -lemma martingale_square_maximal_ineq - (ℱ : Filtration ℕ mΩ) (M : ℕ → Ω → ℝ) - (hM : Martingale M ℱ P) (h2 : ∀ i, MemLp (M i) 2 P) - (ε : ℝ≥0) (N : ℕ) : - (ε:ℝ≥0∞) * P {ω | (ε:ℝ) ≤ (Finset.range (N+1)).sup' - Finset.nonempty_range_add_one (fun j => M j ω^2)} ≤ - ENNReal.ofReal (∫ ω, M N ω^2 ∂P) := by - have hsub := martingale_square_submartingale P ℱ M hM h2 - have h := maximal_ineq hsub (fun i ω => sq_nonneg (M i ω)) (ε := ε) N - refine h.trans (ENNReal.ofReal_le_ofReal ?_) - exact setIntegral_le_integral (h2 N).integrable_sq (ae_of_all _ (fun ω => sq_nonneg (M N ω))) - -noncomputable def centeredMassPartialSum (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (n : ℕ) (ω : Ω) : ℝ := ∑ i ∈ Finset.range n, (D i ω - condExp (ℱ i) P (D i) ω) - -omit [IsProbabilityMeasure P] in -lemma centeredMassPartialSum_memLp (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (hD : ∀ i, MemLp (D i) 2 P) (n : ℕ) : - MemLp (centeredMassPartialSum P ℱ D n) 2 P := - memLp_finsetSum _ (fun i _ => (hD i).sub ((hD i).condExp one_le_two)) - -omit [IsProbabilityMeasure P] in -lemma centeredMassPartialSum_stronglyMeasurable - (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (hD : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (n : ℕ) : - StronglyMeasurable[ℱ n] (centeredMassPartialSum P ℱ D n) := by - apply Finset.stronglyMeasurable_fun_sum - intro i hi - have hle : i+1 ≤ n := Nat.succ_le_of_lt (Finset.mem_range.mp hi) - exact ((hD i).mono (ℱ.mono hle)).sub - (stronglyMeasurable_condExp.mono (ℱ.mono (Nat.le_of_lt (Finset.mem_range.mp hi)))) - -lemma centeredMassPartialSum_variance - (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (hD : ∀ i, MemLp (D i) 2 P) - (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (n : ℕ) : - (∫ ω, centeredMassPartialSum P ℱ D n ω^2 ∂P) = - ∑ i ∈ Finset.range n, ∫ ω, (D i ω-P[D i|ℱ i] ω)^2 ∂P := by - induction n with - | zero => simp [centeredMassPartialSum] - | succ n ih => - have hS := centeredMassPartialSum_memLp P ℱ D hD n - have hX : MemLp (D n-P[D n|ℱ n]) 2 P := (hD n).sub ((hD n).condExp one_le_two) - have hSX : Integrable (fun ω => centeredMassPartialSum P ℱ D n ω* - (D n ω-P[D n|ℱ n] ω)) P := memLp_one_iff_integrable.mp (hS.mul hX) - have hcross := integral_mul_centered_mass_increment P (ℱ n) (ℱ.le n) - (D n) (centeredMassPartialSum P ℱ D n) (hD n) hS - (centeredMassPartialSum_stronglyMeasurable P ℱ D hDm n) - have he : (fun ω => centeredMassPartialSum P ℱ D (n+1) ω^2) = - (fun ω => centeredMassPartialSum P ℱ D n ω^2+ - 2*(centeredMassPartialSum P ℱ D n ω*(D n ω-P[D n|ℱ n] ω))+ - (D n ω-P[D n|ℱ n] ω)^2) := by - funext ω - simp only [centeredMassPartialSum,Finset.sum_range_succ] - ring - have hi₁ := integral_add (hS.integrable_sq.add (hSX.const_mul 2)) hX.integrable_sq - have hi₂ := integral_add hS.integrable_sq (hSX.const_mul 2) - simp only [Pi.add_apply,Pi.sub_apply] at hi₁ hi₂ - rw [he,hi₁,hi₂,integral_const_mul,hcross,mul_zero,add_zero,ih,Finset.sum_range_succ] - -lemma centeredMassPartialSum_variance_le - (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (hD : ∀ i, MemLp (D i) 2 P) - (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (n : ℕ) : - (∫ ω, centeredMassPartialSum P ℱ D n ω^2 ∂P) ≤ - ∫ ω, ∑ i ∈ Finset.range n, D i ω^2 ∂P := by - rw [centeredMassPartialSum_variance P ℱ D hD hDm n, - integral_finsetSum _ (fun i _ => (hD i).integrable_sq)] - exact Finset.sum_le_sum (fun i _ => centered_mass_increment_variance_le P (ℱ i) (ℱ.le i) (D i) (hD i)) - -lemma centeredMassPartialSum_martingale - (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (hD : ∀ i, MemLp (D i) 2 P) - (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) : - Martingale (centeredMassPartialSum P ℱ D) ℱ P := by - let A : ℕ → Ω → ℝ := fun n ω => ∑ i ∈ Finset.range n, D i ω - have hAm : StronglyAdapted ℱ A := by - intro n - apply Finset.stronglyMeasurable_fun_sum - intro i hi - exact (hDm i).mono (ℱ.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hi))) - have hAI (n : ℕ) : Integrable (A n) P := - (memLp_finsetSum _ (fun i _ => hD i)).integrable one_le_two - have hd (i : ℕ) : A (i+1)-A i=D i := by - funext ω - simp only [A,Finset.sum_range_succ,Pi.sub_apply,add_sub_cancel_left] - have he : martingalePart A ℱ P = centeredMassPartialSum P ℱ D := by - rw [martingalePart_eq_sum] - funext n ω - simp only [Pi.add_apply,Finset.sum_apply,Pi.sub_apply] - change A 0 ω+(∑ i ∈ Finset.range n, - ((A (i+1)-A i) ω-condExp (ℱ i) P (A (i+1)-A i) ω)) = _ - simp only [hd] - simp only [A,Finset.range_zero,Finset.sum_empty,zero_add,centeredMassPartialSum] - simpa only [he] using martingale_martingalePart hAm hAI - -/-- A maximal estimate valid at every selected grid index, without -requiring that the selected index itself be a stopping time. -/ -theorem centeredMassPartialSum_maximal_error - (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) - (hD : ∀ i, MemLp (D i) 2 P) - (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (ε : ℝ≥0) (N : ℕ) : - (ε:ℝ≥0∞)*P {ω | (ε:ℝ) ≤ (Finset.range (N+1)).sup' - Finset.nonempty_range_add_one (fun j => centeredMassPartialSum P ℱ D j ω^2)} ≤ - ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range N, D i ω^2 ∂P) := by - exact (martingale_square_maximal_ineq P ℱ (centeredMassPartialSum P ℱ D) - (centeredMassPartialSum_martingale P ℱ D hD hDm) - (centeredMassPartialSum_memLp P ℱ D hD) ε N).trans - (ENNReal.ofReal_le_ofReal (centeredMassPartialSum_variance_le P ℱ D hD hDm N)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassCompensatorMesh +import Mathlib.Probability.Martingale.Centering +import Mathlib.Probability.Martingale.OptionalStopping +import Mathlib.Probability.CondVar + +namespace OAI + +/-! Orthogonality and variance bounds for the discrete potential +compensator used in the prescribed moment coefficients. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + +section Centered +variable (m : MeasurableSpace Ω) (hm : m ≤ mΩ) + +include hm in +lemma centered_mass_increment_condExp (X : Ω → ℝ) (hX : MemLp X 2 P) : + P[X-P[X|m]|m] =ᵐ[P] 0 := by + have hi := hX.integrable one_le_two + have h := condExp_sub hi (integrable_condExp (μ := P) (m := m) (f := X)) m + rw [condExp_of_stronglyMeasurable hm stronglyMeasurable_condExp integrable_condExp] at h + simpa only [sub_self] using h + +include hm in +lemma integral_mul_centered_mass_increment + (X Y : Ω → ℝ) (hX : MemLp X 2 P) (hY : MemLp Y 2 P) + (hYm : StronglyMeasurable[m] Y) : + (∫ ω, Y ω*(X ω-P[X|m] ω) ∂P)=0 := by + have hZ : MemLp (X-P[X|m]) 2 P := hX.sub (hX.condExp one_le_two) + have hprod : Integrable (Y*(X-P[X|m])) P := memLp_one_iff_integrable.mp (hY.mul hZ) + have he := condExp_mul_of_stronglyMeasurable_left hYm hprod (hZ.integrable one_le_two) + have hz := centered_mass_increment_condExp (mΩ := mΩ) P m hm X hX + have he0 : P[Y*(X-P[X|m])|m] =ᵐ[P] 0 := by + filter_upwards [he,hz] with ω hp hz + simp only [Pi.mul_apply,Pi.zero_apply,hz,mul_zero] at hp ⊢ + exact hp + calc + _ = ∫ ω, P[Y*(X-P[X|m])|m] ω ∂P := (integral_condExp hm).symm + _ = 0 := by simpa only [Pi.zero_apply,integral_zero] using integral_congr_ae he0 + +include hm in +lemma centered_mass_increment_variance_le (X : Ω → ℝ) (hX : MemLp X 2 P) : + (∫ ω, (X ω-P[X|m] ω)^2 ∂P) ≤ ∫ ω, X ω^2 ∂P := by + have hZ : Integrable (fun ω => (X ω-P[X|m] ω)^2) P := + (hX.sub (hX.condExp one_le_two)).integrable_sq + have he := setIntegral_condVar (hm := hm) hZ (MeasurableSet.univ : MeasurableSet[m] univ) + simp only [Measure.restrict_univ] at he + rw [← he] + calc + _ ≤ ∫ ω, P[X^2|m] ω ∂P := + integral_mono_ae integrable_condVar integrable_condExp (condVar_ae_le_condExp_sq hm hX) + _ = _ := integral_condExp hm + +end Centered + +lemma martingale_square_submartingale + {ι : Type*} [Preorder ι] (ℱ : Filtration ι mΩ) + (M : ι → Ω → ℝ) (hM : Martingale M ℱ P) (h2 : ∀ i, MemLp (M i) 2 P) : + Submartingale (fun i ω => M i ω^2) ℱ P := by + refine ⟨fun i => (hM.stronglyMeasurable i).pow 2,?_,fun i => (h2 i).integrable_sq⟩ + intro i j hij + have hv := condVar_ae_eq_condExp_sq_sub_sq_condExp (ℱ.le i) (h2 j) + have hn : 0 ≤ᵐ[P] condVar (ℱ i) (M j) P := + condExp_nonneg (ae_of_all _ (fun ω => sq_nonneg (M j ω-P[M j|ℱ i] ω))) + filter_upwards [hv,hn,hM.condExp_ae_eq hij] with ω hv hn he + simp only [Pi.sub_apply,Pi.pow_apply] at hv + rw [he] at hv + change 0 ≤ condVar (ℱ i) (M j) P ω at hn + change condVar (ℱ i) (M j) P ω = P[fun ω => M j ω^2|ℱ i] ω - M i ω^2 at hv + linarith + +lemma martingale_square_maximal_ineq + (ℱ : Filtration ℕ mΩ) (M : ℕ → Ω → ℝ) + (hM : Martingale M ℱ P) (h2 : ∀ i, MemLp (M i) 2 P) + (ε : ℝ≥0) (N : ℕ) : + (ε:ℝ≥0∞) * P {ω | (ε:ℝ) ≤ (Finset.range (N+1)).sup' + Finset.nonempty_range_add_one (fun j => M j ω^2)} ≤ + ENNReal.ofReal (∫ ω, M N ω^2 ∂P) := by + have hsub := martingale_square_submartingale P ℱ M hM h2 + have h := maximal_ineq hsub (fun i ω => sq_nonneg (M i ω)) (ε := ε) N + refine h.trans (ENNReal.ofReal_le_ofReal ?_) + exact setIntegral_le_integral (h2 N).integrable_sq (ae_of_all _ (fun ω => sq_nonneg (M N ω))) + +noncomputable def centeredMassPartialSum (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (n : ℕ) (ω : Ω) : ℝ := ∑ i ∈ Finset.range n, (D i ω - condExp (ℱ i) P (D i) ω) + +omit [IsProbabilityMeasure P] in +lemma centeredMassPartialSum_memLp (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (hD : ∀ i, MemLp (D i) 2 P) (n : ℕ) : + MemLp (centeredMassPartialSum P ℱ D n) 2 P := + memLp_finsetSum _ (fun i _ => (hD i).sub ((hD i).condExp one_le_two)) + +omit [IsProbabilityMeasure P] in +lemma centeredMassPartialSum_stronglyMeasurable + (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (hD : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (n : ℕ) : + StronglyMeasurable[ℱ n] (centeredMassPartialSum P ℱ D n) := by + apply Finset.stronglyMeasurable_fun_sum + intro i hi + have hle : i+1 ≤ n := Nat.succ_le_of_lt (Finset.mem_range.mp hi) + exact ((hD i).mono (ℱ.mono hle)).sub + (stronglyMeasurable_condExp.mono (ℱ.mono (Nat.le_of_lt (Finset.mem_range.mp hi)))) + +lemma centeredMassPartialSum_variance + (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (hD : ∀ i, MemLp (D i) 2 P) + (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (n : ℕ) : + (∫ ω, centeredMassPartialSum P ℱ D n ω^2 ∂P) = + ∑ i ∈ Finset.range n, ∫ ω, (D i ω-P[D i|ℱ i] ω)^2 ∂P := by + induction n with + | zero => simp [centeredMassPartialSum] + | succ n ih => + have hS := centeredMassPartialSum_memLp P ℱ D hD n + have hX : MemLp (D n-P[D n|ℱ n]) 2 P := (hD n).sub ((hD n).condExp one_le_two) + have hSX : Integrable (fun ω => centeredMassPartialSum P ℱ D n ω* + (D n ω-P[D n|ℱ n] ω)) P := memLp_one_iff_integrable.mp (hS.mul hX) + have hcross := integral_mul_centered_mass_increment P (ℱ n) (ℱ.le n) + (D n) (centeredMassPartialSum P ℱ D n) (hD n) hS + (centeredMassPartialSum_stronglyMeasurable P ℱ D hDm n) + have he : (fun ω => centeredMassPartialSum P ℱ D (n+1) ω^2) = + (fun ω => centeredMassPartialSum P ℱ D n ω^2+ + 2*(centeredMassPartialSum P ℱ D n ω*(D n ω-P[D n|ℱ n] ω))+ + (D n ω-P[D n|ℱ n] ω)^2) := by + funext ω + simp only [centeredMassPartialSum,Finset.sum_range_succ] + ring + have hi₁ := integral_add (hS.integrable_sq.add (hSX.const_mul 2)) hX.integrable_sq + have hi₂ := integral_add hS.integrable_sq (hSX.const_mul 2) + simp only [Pi.add_apply,Pi.sub_apply] at hi₁ hi₂ + rw [he,hi₁,hi₂,integral_const_mul,hcross,mul_zero,add_zero,ih,Finset.sum_range_succ] + +lemma centeredMassPartialSum_variance_le + (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (hD : ∀ i, MemLp (D i) 2 P) + (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (n : ℕ) : + (∫ ω, centeredMassPartialSum P ℱ D n ω^2 ∂P) ≤ + ∫ ω, ∑ i ∈ Finset.range n, D i ω^2 ∂P := by + rw [centeredMassPartialSum_variance P ℱ D hD hDm n, + integral_finsetSum _ (fun i _ => (hD i).integrable_sq)] + exact Finset.sum_le_sum (fun i _ => centered_mass_increment_variance_le P (ℱ i) (ℱ.le i) (D i) (hD i)) + +lemma centeredMassPartialSum_martingale + (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (hD : ∀ i, MemLp (D i) 2 P) + (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) : + Martingale (centeredMassPartialSum P ℱ D) ℱ P := by + let A : ℕ → Ω → ℝ := fun n ω => ∑ i ∈ Finset.range n, D i ω + have hAm : StronglyAdapted ℱ A := by + intro n + apply Finset.stronglyMeasurable_fun_sum + intro i hi + exact (hDm i).mono (ℱ.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hi))) + have hAI (n : ℕ) : Integrable (A n) P := + (memLp_finsetSum _ (fun i _ => hD i)).integrable one_le_two + have hd (i : ℕ) : A (i+1)-A i=D i := by + funext ω + simp only [A,Finset.sum_range_succ,Pi.sub_apply,add_sub_cancel_left] + have he : martingalePart A ℱ P = centeredMassPartialSum P ℱ D := by + rw [martingalePart_eq_sum] + funext n ω + simp only [Pi.add_apply,Finset.sum_apply,Pi.sub_apply] + change A 0 ω+(∑ i ∈ Finset.range n, + ((A (i+1)-A i) ω-condExp (ℱ i) P (A (i+1)-A i) ω)) = _ + simp only [hd] + simp only [A,Finset.range_zero,Finset.sum_empty,zero_add,centeredMassPartialSum] + simpa only [he] using martingale_martingalePart hAm hAI + +/-- A maximal estimate valid at every selected grid index, without +requiring that the selected index itself be a stopping time. -/ +theorem centeredMassPartialSum_maximal_error + (ℱ : Filtration ℕ mΩ) (D : ℕ → Ω → ℝ) + (hD : ∀ i, MemLp (D i) 2 P) + (hDm : ∀ i, StronglyMeasurable[ℱ (i+1)] (D i)) (ε : ℝ≥0) (N : ℕ) : + (ε:ℝ≥0∞)*P {ω | (ε:ℝ) ≤ (Finset.range (N+1)).sup' + Finset.nonempty_range_add_one (fun j => centeredMassPartialSum P ℱ D j ω^2)} ≤ + ENNReal.ofReal (∫ ω, ∑ i ∈ Finset.range N, D i ω^2 ∂P) := by + exact (martingale_square_maximal_ineq P ℱ (centeredMassPartialSum P ℱ D) + (centeredMassPartialSum_martingale P ℱ D hD hDm) + (centeredMassPartialSum_memLp P ℱ D hD) ε N).trans + (ENNReal.ofReal_le_ofReal (centeredMassPartialSum_variance_le P ℱ D hD hDm N)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassGridCutoffs.lean b/lean/OAI/Probability/SLE/Mass/MassGridCutoffs.lean index f1dcf0be8ff7ef7d2b380d50d37d856891f6cfca..508a6997d17473494f35de9d12b8d8897ac7c705 100644 --- a/lean/OAI/Probability/SLE/Mass/MassGridCutoffs.lean +++ b/lean/OAI/Probability/SLE/Mass/MassGridCutoffs.lean @@ -1,105 +1,105 @@ -import OAI.Probability.SLE.Mass.MassGridOverlap -import OAI.Probability.SLE.Mass.LevelRadonMoments - -namespace OAI - -/-! Compact spatial test functions adapted to the mass grid. A fixed -compact cutoff keeps every enlarged grid cell inside the open half-plane. -/ - -noncomputable section -open Set Metric Filter MeasureTheory Classical -open scoped ENNReal Topology CompactlySupported - -namespace SLEExactGauge.Multipoint - -lemma exists_unit_compact_test_cutoff (K : Set upperHalfPlane) (hK : IsCompact K) : - ∃ f : C_c(upperHalfPlane,ℝ), (∀ z, 0 ≤ f z ∧ f z ≤ 1) ∧ ∀ z ∈ K, f z = 1 := by - let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - obtain ⟨f,hf,hfc,_,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen - hK isOpen_univ (subset_univ K) - exact ⟨⟨f,hfc⟩,h01,fun z hz => hf hz⟩ - -def massGridBump (r : ℝ) (i : ℤ×ℤ) : C(upperHalfPlane,ℝ) := - ⟨fun z => max 0 (min 1 ((3*r-dist (z:ℂ) (coverGridCenter r i))/r)), by fun_prop⟩ - -def massGridCutoff (χ : C_c(upperHalfPlane,ℝ)) (r : ℝ) (i : ℤ×ℤ) : - C_c(upperHalfPlane,ℝ) := massGridBump r i • χ - -lemma massGridBump_bounds (r : ℝ) (i : ℤ×ℤ) (z : upperHalfPlane) : - 0 ≤ massGridBump r i z ∧ massGridBump r i z ≤ 1 := by - exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩ - -lemma massGridBump_eq_one {r : ℝ} (hr : 0 < r) (i : ℤ×ℤ) (z : upperHalfPlane) - (hz : dist (z:ℂ) (coverGridCenter r i) ≤ 2*r) : massGridBump r i z = 1 := by - have hh : 1 ≤ (3*r-dist (z:ℂ) (coverGridCenter r i))/r := (le_div_iff₀ hr).mpr (by linarith) - simp only [massGridBump,ContinuousMap.coe_mk,min_eq_left hh,max_eq_right (by norm_num : (0:ℝ)≤1)] - -lemma massGridBump_eq_zero {r : ℝ} (hr : 0 < r) (i : ℤ×ℤ) (z : upperHalfPlane) - (hz : 3*r < dist (z:ℂ) (coverGridCenter r i)) : massGridBump r i z = 0 := by - have hh : (3*r-dist (z:ℂ) (coverGridCenter r i))/r ≤ 0 := - div_nonpos_of_nonpos_of_nonneg (by linarith) hr.le - apply max_eq_left - exact (min_le_right _ _).trans hh - -lemma massGridCutoff_bounds (χ : C_c(upperHalfPlane,ℝ)) - (hχ : ∀ z, 0 ≤ χ z ∧ χ z ≤ 1) (r : ℝ) (i : ℤ×ℤ) (z : upperHalfPlane) : - 0 ≤ massGridCutoff χ r i z ∧ massGridCutoff χ r i z ≤ 1 := by - have hb := massGridBump_bounds r i z - change 0 ≤ massGridBump r i z * χ z ∧ massGridBump r i z * χ z ≤ 1 - exact ⟨mul_nonneg hb.1 (hχ z).1,(mul_le_mul_of_nonneg_right hb.2 (hχ z).1).trans (by simpa only [one_mul] using (hχ z).2)⟩ - -lemma massGridCutoff_eq_one (χ : C_c(upperHalfPlane,ℝ)) {r : ℝ} (hr : 0 < r) - (i : ℤ×ℤ) (z : upperHalfPlane) (hχ : χ z = 1) - (hz : dist (z:ℂ) (coverGridCenter r i) ≤ 2*r) : massGridCutoff χ r i z = 1 := by - change massGridBump r i z * χ z = 1 - rw [massGridBump_eq_one hr i z hz,hχ,one_mul] - -lemma massGridCutoff_support (χ : C_c(upperHalfPlane,ℝ)) {r : ℝ} (hr : 0 < r) - (i : ℤ×ℤ) (z : ℂ) (hz : extendHalfPlaneTest (massGridCutoff χ r i) z ≠ 0) : - dist z (coverGridCenter r i) ≤ 3*r := by - by_cases hH : z ∈ upperHalfPlane - · by_contra h - have he := massGridBump_eq_zero hr i ⟨z,hH⟩ (lt_of_not_ge h) - apply hz - rw [show z = (⟨z,hH⟩ : upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] - change massGridBump r i ⟨z,hH⟩ * χ ⟨z,hH⟩ = 0 - rw [he,zero_mul] - · exact False.elim (hz (extendHalfPlaneTest_of_not_mem _ z hH)) - -lemma massGridCutoff_common_support (χ : C_c(upperHalfPlane,ℝ)) (r : ℝ) (i : ℤ×ℤ) - (z : ℂ) (hz : z ∉ (Subtype.val : upperHalfPlane → ℂ) '' tsupport χ) : - extendHalfPlaneTest (massGridCutoff χ r i) z = 0 := by - by_cases hH : z ∈ upperHalfPlane - · let x : upperHalfPlane := ⟨z,hH⟩ - have hχ : χ x = 0 := image_eq_zero_of_notMem_tsupport (fun h => hz ⟨x,h,rfl⟩) - change extendHalfPlaneTest (massGridCutoff χ r i) x = 0 - rw [extendHalfPlaneTest_apply] - change massGridBump r i x * χ x = 0 - rw [hχ,mul_zero] - · exact extendHalfPlaneTest_of_not_mem _ z hH - -lemma massGridCutoff_sum_bound (χ : C_c(upperHalfPlane,ℝ)) - (hχ : ∀ z, 0 ≤ χ z ∧ χ z ≤ 1) {r : ℝ} (hr : 0 < r) (I : Finset (ℤ×ℤ)) - (x y : ℂ) : - (∑ i ∈ I, ENNReal.ofReal (extendHalfPlaneTest (massGridCutoff χ r i) x) * - ENNReal.ofReal (extendHalfPlaneTest (massGridCutoff χ r i) y)) ≤ - if dist x y ≤ 6*r then 81 else 0 := by - refine massGrid_cutoff_product_sum_le hr I - (fun i z => ENNReal.ofReal (extendHalfPlaneTest (massGridCutoff χ r i) z)) ?_ ?_ x y - · intro i z - apply ENNReal.ofReal_le_one.mpr - by_cases hH : z ∈ upperHalfPlane - · rw [show z = (⟨z,hH⟩ : upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] - exact (massGridCutoff_bounds χ hχ r i ⟨z,hH⟩).2 - · rw [extendHalfPlaneTest_of_not_mem _ z hH] - norm_num - · intro i z hz - apply massGridCutoff_support χ hr i z - intro he - exact hz (by rw [he,ENNReal.ofReal_zero]) - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.MassGridOverlap +import OAI.Probability.SLE.Mass.LevelRadonMoments + +namespace OAI + +/-! Compact spatial test functions adapted to the mass grid. A fixed +compact cutoff keeps every enlarged grid cell inside the open half-plane. -/ + +noncomputable section +open Set Metric Filter MeasureTheory Classical +open scoped ENNReal Topology CompactlySupported + +namespace SLEExactGauge.Multipoint + +lemma exists_unit_compact_test_cutoff (K : Set upperHalfPlane) (hK : IsCompact K) : + ∃ f : C_c(upperHalfPlane,ℝ), (∀ z, 0 ≤ f z ∧ f z ≤ 1) ∧ ∀ z ∈ K, f z = 1 := by + let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + obtain ⟨f,hf,hfc,_,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen + hK isOpen_univ (subset_univ K) + exact ⟨⟨f,hfc⟩,h01,fun z hz => hf hz⟩ + +def massGridBump (r : ℝ) (i : ℤ×ℤ) : C(upperHalfPlane,ℝ) := + ⟨fun z => max 0 (min 1 ((3*r-dist (z:ℂ) (coverGridCenter r i))/r)), by fun_prop⟩ + +def massGridCutoff (χ : C_c(upperHalfPlane,ℝ)) (r : ℝ) (i : ℤ×ℤ) : + C_c(upperHalfPlane,ℝ) := massGridBump r i • χ + +lemma massGridBump_bounds (r : ℝ) (i : ℤ×ℤ) (z : upperHalfPlane) : + 0 ≤ massGridBump r i z ∧ massGridBump r i z ≤ 1 := by + exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩ + +lemma massGridBump_eq_one {r : ℝ} (hr : 0 < r) (i : ℤ×ℤ) (z : upperHalfPlane) + (hz : dist (z:ℂ) (coverGridCenter r i) ≤ 2*r) : massGridBump r i z = 1 := by + have hh : 1 ≤ (3*r-dist (z:ℂ) (coverGridCenter r i))/r := (le_div_iff₀ hr).mpr (by linarith) + simp only [massGridBump,ContinuousMap.coe_mk,min_eq_left hh,max_eq_right (by norm_num : (0:ℝ)≤1)] + +lemma massGridBump_eq_zero {r : ℝ} (hr : 0 < r) (i : ℤ×ℤ) (z : upperHalfPlane) + (hz : 3*r < dist (z:ℂ) (coverGridCenter r i)) : massGridBump r i z = 0 := by + have hh : (3*r-dist (z:ℂ) (coverGridCenter r i))/r ≤ 0 := + div_nonpos_of_nonpos_of_nonneg (by linarith) hr.le + apply max_eq_left + exact (min_le_right _ _).trans hh + +lemma massGridCutoff_bounds (χ : C_c(upperHalfPlane,ℝ)) + (hχ : ∀ z, 0 ≤ χ z ∧ χ z ≤ 1) (r : ℝ) (i : ℤ×ℤ) (z : upperHalfPlane) : + 0 ≤ massGridCutoff χ r i z ∧ massGridCutoff χ r i z ≤ 1 := by + have hb := massGridBump_bounds r i z + change 0 ≤ massGridBump r i z * χ z ∧ massGridBump r i z * χ z ≤ 1 + exact ⟨mul_nonneg hb.1 (hχ z).1,(mul_le_mul_of_nonneg_right hb.2 (hχ z).1).trans (by simpa only [one_mul] using (hχ z).2)⟩ + +lemma massGridCutoff_eq_one (χ : C_c(upperHalfPlane,ℝ)) {r : ℝ} (hr : 0 < r) + (i : ℤ×ℤ) (z : upperHalfPlane) (hχ : χ z = 1) + (hz : dist (z:ℂ) (coverGridCenter r i) ≤ 2*r) : massGridCutoff χ r i z = 1 := by + change massGridBump r i z * χ z = 1 + rw [massGridBump_eq_one hr i z hz,hχ,one_mul] + +lemma massGridCutoff_support (χ : C_c(upperHalfPlane,ℝ)) {r : ℝ} (hr : 0 < r) + (i : ℤ×ℤ) (z : ℂ) (hz : extendHalfPlaneTest (massGridCutoff χ r i) z ≠ 0) : + dist z (coverGridCenter r i) ≤ 3*r := by + by_cases hH : z ∈ upperHalfPlane + · by_contra h + have he := massGridBump_eq_zero hr i ⟨z,hH⟩ (lt_of_not_ge h) + apply hz + rw [show z = (⟨z,hH⟩ : upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] + change massGridBump r i ⟨z,hH⟩ * χ ⟨z,hH⟩ = 0 + rw [he,zero_mul] + · exact False.elim (hz (extendHalfPlaneTest_of_not_mem _ z hH)) + +lemma massGridCutoff_common_support (χ : C_c(upperHalfPlane,ℝ)) (r : ℝ) (i : ℤ×ℤ) + (z : ℂ) (hz : z ∉ (Subtype.val : upperHalfPlane → ℂ) '' tsupport χ) : + extendHalfPlaneTest (massGridCutoff χ r i) z = 0 := by + by_cases hH : z ∈ upperHalfPlane + · let x : upperHalfPlane := ⟨z,hH⟩ + have hχ : χ x = 0 := image_eq_zero_of_notMem_tsupport (fun h => hz ⟨x,h,rfl⟩) + change extendHalfPlaneTest (massGridCutoff χ r i) x = 0 + rw [extendHalfPlaneTest_apply] + change massGridBump r i x * χ x = 0 + rw [hχ,mul_zero] + · exact extendHalfPlaneTest_of_not_mem _ z hH + +lemma massGridCutoff_sum_bound (χ : C_c(upperHalfPlane,ℝ)) + (hχ : ∀ z, 0 ≤ χ z ∧ χ z ≤ 1) {r : ℝ} (hr : 0 < r) (I : Finset (ℤ×ℤ)) + (x y : ℂ) : + (∑ i ∈ I, ENNReal.ofReal (extendHalfPlaneTest (massGridCutoff χ r i) x) * + ENNReal.ofReal (extendHalfPlaneTest (massGridCutoff χ r i) y)) ≤ + if dist x y ≤ 6*r then 81 else 0 := by + refine massGrid_cutoff_product_sum_le hr I + (fun i z => ENNReal.ofReal (extendHalfPlaneTest (massGridCutoff χ r i) z)) ?_ ?_ x y + · intro i z + apply ENNReal.ofReal_le_one.mpr + by_cases hH : z ∈ upperHalfPlane + · rw [show z = (⟨z,hH⟩ : upperHalfPlane).val from rfl,extendHalfPlaneTest_apply] + exact (massGridCutoff_bounds χ hχ r i ⟨z,hH⟩).2 + · rw [extendHalfPlaneTest_of_not_mem _ z hH] + norm_num + · intro i z hz + apply massGridCutoff_support χ hr i z + intro he + exact hz (by rw [he,ENNReal.ofReal_zero]) + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassGridOverlap.lean b/lean/OAI/Probability/SLE/Mass/MassGridOverlap.lean index 532a8125098f8de274361345035f2d2b2a9bf623..8a448d4a4b0fe66110b8376d1443586f887e9505 100644 --- a/lean/OAI/Probability/SLE/Mass/MassGridOverlap.lean +++ b/lean/OAI/Probability/SLE/Mass/MassGridOverlap.lean @@ -1,91 +1,91 @@ -import OAI.Probability.SLE.Gauge.CoverGrid - -namespace OAI - -/-! A fixed overlap bound for planar mass cutoffs on the physical grid. -/ - -noncomputable section -open Set Metric Filter MeasureTheory Classical -open scoped ENNReal - -namespace SLEExactGauge - -lemma massGrid_coordinate_bounds {r x : ℝ} (hr : 0 < r) (i : ℤ) - (hi : |x-r*(i:ℝ)| ≤ 3*r) : - ⌊x/r⌋-4 ≤ i ∧ i ≤ ⌊x/r⌋+4 := by - have hf : r*(⌊x/r⌋:ℝ) ≤ x := by - have hh := mul_le_mul_of_nonneg_left (Int.floor_le (x/r)) hr.le - rwa [mul_div_cancel₀ _ hr.ne'] at hh - have hg : x < r*((⌊x/r⌋:ℝ)+1) := by - have hh := mul_lt_mul_of_pos_left (Int.lt_floor_add_one (x/r)) hr - rwa [mul_div_cancel₀ _ hr.ne'] at hh - have hlo : ((⌊x/r⌋-4:ℤ):ℝ) ≤ (i:ℝ) := by - push_cast - nlinarith [abs_le.mp hi] - have hhi : (i:ℝ) ≤ ((⌊x/r⌋+4:ℤ):ℝ) := by - push_cast - nlinarith [abs_le.mp hi] - exact ⟨by exact_mod_cast hlo,by exact_mod_cast hhi⟩ - -lemma massGrid_nearby_card_le {r : ℝ} (hr : 0 < r) (I : Finset (ℤ×ℤ)) (z : ℂ) : - (I.filter (fun ij => dist z (coverGridCenter r ij) ≤ 3*r)).card ≤ 81 := by - let J := (Finset.Icc (⌊z.re/r⌋-4) (⌊z.re/r⌋+4)) ×ˢ - (Finset.Icc (⌊z.im/r⌋-4) (⌊z.im/r⌋+4)) - have hsub : I.filter (fun ij => dist z (coverGridCenter r ij) ≤ 3*r) ⊆ J := by - intro ij hij - have hd := (Finset.mem_filter.mp hij).2 - have hre : |z.re-r*(ij.1:ℝ)| ≤ 3*r := by - have h := (Complex.abs_re_le_norm (z-coverGridCenter r ij)).trans - (by simpa only [dist_eq_norm] using hd) - simpa [coverGridCenter] using h - have him : |z.im-r*(ij.2:ℝ)| ≤ 3*r := by - have h := (Complex.abs_im_le_norm (z-coverGridCenter r ij)).trans - (by simpa only [dist_eq_norm] using hd) - simpa [coverGridCenter] using h - simpa only [J,Finset.mem_product,Finset.mem_Icc] using - And.intro (massGrid_coordinate_bounds hr ij.1 hre) (massGrid_coordinate_bounds hr ij.2 him) - have hcard : J.card = 81 := by - simp only [J,Finset.card_product,Int.card_Icc] - rw [show ⌊z.re/r⌋+4+1-(⌊z.re/r⌋-4) = (9:ℤ) by omega, - show ⌊z.im/r⌋+4+1-(⌊z.im/r⌋-4) = (9:ℤ) by omega] - rfl - exact (Finset.card_le_card hsub).trans_eq hcard - -lemma massGrid_cutoff_product_sum_le {r : ℝ} (hr : 0 < r) (I : Finset (ℤ×ℤ)) - (f : (ℤ×ℤ) → ℂ → ℝ≥0∞) (hf : ∀ i z, f i z ≤ 1) - (hs : ∀ i z, f i z ≠ 0 → dist z (coverGridCenter r i) ≤ 3*r) - (x y : ℂ) : - (∑ i ∈ I, f i x * f i y) ≤ if dist x y ≤ 6*r then 81 else 0 := by - by_cases hxy : dist x y ≤ 6*r - · rw [ite_eq_left hxy] - let J := I.filter (fun i => dist x (coverGridCenter r i) ≤ 3*r) - have he : (∑ i ∈ I, f i x * f i y) = ∑ i ∈ J, f i x * f i y := by - symm - apply Finset.sum_subset (Finset.filter_subset _ _) - intro i hi hij - have hx : f i x = 0 := by - by_contra h - exact hij (Finset.mem_filter.mpr ⟨hi,hs i x h⟩) - simp only [hx,zero_mul] - rw [he] - calc - _ ≤ ∑ _i ∈ J, (1 : ℝ≥0∞) := - Finset.sum_le_sum (fun i _ => (mul_le_mul' (hf i x) (hf i y)).trans_eq (one_mul 1)) - _ = J.card := by simp - _ ≤ 81 := by exact_mod_cast massGrid_nearby_card_le hr I x - · rw [ite_eq_right hxy] - apply le_of_eq - apply Finset.sum_eq_zero - intro i hi - by_cases hx : f i x = 0 - · simp only [hx,zero_mul] - by_cases hy : f i y = 0 - · simp only [hy,mul_zero] - exact False.elim (hxy ((dist_triangle_right x y (coverGridCenter r i)).trans - (by linarith [hs i x hx,hs i y hy]))) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Gauge.CoverGrid + +namespace OAI + +/-! A fixed overlap bound for planar mass cutoffs on the physical grid. -/ + +noncomputable section +open Set Metric Filter MeasureTheory Classical +open scoped ENNReal + +namespace SLEExactGauge + +lemma massGrid_coordinate_bounds {r x : ℝ} (hr : 0 < r) (i : ℤ) + (hi : |x-r*(i:ℝ)| ≤ 3*r) : + ⌊x/r⌋-4 ≤ i ∧ i ≤ ⌊x/r⌋+4 := by + have hf : r*(⌊x/r⌋:ℝ) ≤ x := by + have hh := mul_le_mul_of_nonneg_left (Int.floor_le (x/r)) hr.le + rwa [mul_div_cancel₀ _ hr.ne'] at hh + have hg : x < r*((⌊x/r⌋:ℝ)+1) := by + have hh := mul_lt_mul_of_pos_left (Int.lt_floor_add_one (x/r)) hr + rwa [mul_div_cancel₀ _ hr.ne'] at hh + have hlo : ((⌊x/r⌋-4:ℤ):ℝ) ≤ (i:ℝ) := by + push_cast + nlinarith [abs_le.mp hi] + have hhi : (i:ℝ) ≤ ((⌊x/r⌋+4:ℤ):ℝ) := by + push_cast + nlinarith [abs_le.mp hi] + exact ⟨by exact_mod_cast hlo,by exact_mod_cast hhi⟩ + +lemma massGrid_nearby_card_le {r : ℝ} (hr : 0 < r) (I : Finset (ℤ×ℤ)) (z : ℂ) : + (I.filter (fun ij => dist z (coverGridCenter r ij) ≤ 3*r)).card ≤ 81 := by + let J := (Finset.Icc (⌊z.re/r⌋-4) (⌊z.re/r⌋+4)) ×ˢ + (Finset.Icc (⌊z.im/r⌋-4) (⌊z.im/r⌋+4)) + have hsub : I.filter (fun ij => dist z (coverGridCenter r ij) ≤ 3*r) ⊆ J := by + intro ij hij + have hd := (Finset.mem_filter.mp hij).2 + have hre : |z.re-r*(ij.1:ℝ)| ≤ 3*r := by + have h := (Complex.abs_re_le_norm (z-coverGridCenter r ij)).trans + (by simpa only [dist_eq_norm] using hd) + simpa [coverGridCenter] using h + have him : |z.im-r*(ij.2:ℝ)| ≤ 3*r := by + have h := (Complex.abs_im_le_norm (z-coverGridCenter r ij)).trans + (by simpa only [dist_eq_norm] using hd) + simpa [coverGridCenter] using h + simpa only [J,Finset.mem_product,Finset.mem_Icc] using + And.intro (massGrid_coordinate_bounds hr ij.1 hre) (massGrid_coordinate_bounds hr ij.2 him) + have hcard : J.card = 81 := by + simp only [J,Finset.card_product,Int.card_Icc] + rw [show ⌊z.re/r⌋+4+1-(⌊z.re/r⌋-4) = (9:ℤ) by omega, + show ⌊z.im/r⌋+4+1-(⌊z.im/r⌋-4) = (9:ℤ) by omega] + rfl + exact (Finset.card_le_card hsub).trans_eq hcard + +lemma massGrid_cutoff_product_sum_le {r : ℝ} (hr : 0 < r) (I : Finset (ℤ×ℤ)) + (f : (ℤ×ℤ) → ℂ → ℝ≥0∞) (hf : ∀ i z, f i z ≤ 1) + (hs : ∀ i z, f i z ≠ 0 → dist z (coverGridCenter r i) ≤ 3*r) + (x y : ℂ) : + (∑ i ∈ I, f i x * f i y) ≤ if dist x y ≤ 6*r then 81 else 0 := by + by_cases hxy : dist x y ≤ 6*r + · rw [ite_eq_left hxy] + let J := I.filter (fun i => dist x (coverGridCenter r i) ≤ 3*r) + have he : (∑ i ∈ I, f i x * f i y) = ∑ i ∈ J, f i x * f i y := by + symm + apply Finset.sum_subset (Finset.filter_subset _ _) + intro i hi hij + have hx : f i x = 0 := by + by_contra h + exact hij (Finset.mem_filter.mpr ⟨hi,hs i x h⟩) + simp only [hx,zero_mul] + rw [he] + calc + _ ≤ ∑ _i ∈ J, (1 : ℝ≥0∞) := + Finset.sum_le_sum (fun i _ => (mul_le_mul' (hf i x) (hf i y)).trans_eq (one_mul 1)) + _ = J.card := by simp + _ ≤ 81 := by exact_mod_cast massGrid_nearby_card_le hr I x + · rw [ite_eq_right hxy] + apply le_of_eq + apply Finset.sum_eq_zero + intro i hi + by_cases hx : f i x = 0 + · simp only [hx,zero_mul] + by_cases hy : f i y = 0 + · simp only [hy,mul_zero] + exact False.elim (hxy ((dist_triangle_right x y (coverGridCenter r i)).trans + (by linarith [hs i x hx,hs i y hy]))) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassKernelUniqueness.lean b/lean/OAI/Probability/SLE/Mass/MassKernelUniqueness.lean index 6a545c4ab0ed8b47b53c39397b7889a34151c383..b74faaf193ae6d32cdd59b75d76faac2ea1f1f37 100644 --- a/lean/OAI/Probability/SLE/Mass/MassKernelUniqueness.lean +++ b/lean/OAI/Probability/SLE/Mass/MassKernelUniqueness.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.Mass.MassMartingaleUniqueness - -namespace OAI - -/-! Equal terminal remainder kernels yield a martingale difference; continuity -and increasing integrable masses then imply uniqueness. -/ - -noncomputable section -open MeasureTheory Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] - {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} - -lemma mass_difference_martingale_of_terminal_mean - (A B : ℝ≥0 → Ω → ℝ) (Z : Ω → ℝ) (hZ : Integrable Z P) - (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B) - (hAI : ∀ t, Integrable (A t) P) (hBI : ∀ t, Integrable (B t) P) - (hmean : ∀ t E, MeasurableSet[ℱ t] E → - (∫ ω in E, A t ω-B t ω ∂P) = ∫ ω in E, Z ω ∂P) : - Martingale (fun t ω => A t ω-B t ω) ℱ P := by - have he (t : ℝ≥0) : (fun ω => A t ω-B t ω) =ᵐ[P] P[Z | ℱ t] := by - apply ae_eq_condExp_of_forall_setIntegral_eq (ℱ.le t) hZ - (fun E _ _ => ((hAI t).sub (hBI t)).integrableOn) - (fun E hE _ => hmean t E hE) - exact ((hAm t).sub (hBm t)).aestronglyMeasurable - exact (martingale_condExp Z ℱ P).congr (hAm.sub hBm) - (fun t => (he t).symm) - -theorem increasing_mass_unique_of_terminal_mean - (A B : ℝ≥0 → Ω → ℝ) (Z : Ω → ℝ) (hZ : Integrable Z P) - (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B) - (hAI : ∀ t, Integrable (A t) P) (hBI : ∀ t, Integrable (B t) P) - (hmean : ∀ t E, MeasurableSet[ℱ t] E → - (∫ ω in E, A t ω-B t ω ∂P) = ∫ ω in E, Z ω ∂P) - (hA : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) - (hB : ∀ᵐ ω ∂P, Monotone (fun t => B t ω)) - (hA0 : A 0 =ᵐ[P] 0) (hB0 : B 0 =ᵐ[P] 0) - (hcont : ∀ᵐ ω ∂P, Continuous (fun t => A t ω-B t ω)) : - ∀ᵐ ω ∂P, ∀ t, A t ω = B t ω := - increasing_mass_martingale_indistinguishable A B - (mass_difference_martingale_of_terminal_mean A B Z hZ hAm hBm hAI hBI hmean) - hA hB hA0 hB0 hAI hBI hcont - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassMartingaleUniqueness + +namespace OAI + +/-! Equal terminal remainder kernels yield a martingale difference; continuity +and increasing integrable masses then imply uniqueness. -/ + +noncomputable section +open MeasureTheory Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] + {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} + +lemma mass_difference_martingale_of_terminal_mean + (A B : ℝ≥0 → Ω → ℝ) (Z : Ω → ℝ) (hZ : Integrable Z P) + (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B) + (hAI : ∀ t, Integrable (A t) P) (hBI : ∀ t, Integrable (B t) P) + (hmean : ∀ t E, MeasurableSet[ℱ t] E → + (∫ ω in E, A t ω-B t ω ∂P) = ∫ ω in E, Z ω ∂P) : + Martingale (fun t ω => A t ω-B t ω) ℱ P := by + have he (t : ℝ≥0) : (fun ω => A t ω-B t ω) =ᵐ[P] P[Z | ℱ t] := by + apply ae_eq_condExp_of_forall_setIntegral_eq (ℱ.le t) hZ + (fun E _ _ => ((hAI t).sub (hBI t)).integrableOn) + (fun E hE _ => hmean t E hE) + exact ((hAm t).sub (hBm t)).aestronglyMeasurable + exact (martingale_condExp Z ℱ P).congr (hAm.sub hBm) + (fun t => (he t).symm) + +theorem increasing_mass_unique_of_terminal_mean + (A B : ℝ≥0 → Ω → ℝ) (Z : Ω → ℝ) (hZ : Integrable Z P) + (hAm : StronglyAdapted ℱ A) (hBm : StronglyAdapted ℱ B) + (hAI : ∀ t, Integrable (A t) P) (hBI : ∀ t, Integrable (B t) P) + (hmean : ∀ t E, MeasurableSet[ℱ t] E → + (∫ ω in E, A t ω-B t ω ∂P) = ∫ ω in E, Z ω ∂P) + (hA : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) + (hB : ∀ᵐ ω ∂P, Monotone (fun t => B t ω)) + (hA0 : A 0 =ᵐ[P] 0) (hB0 : B 0 =ᵐ[P] 0) + (hcont : ∀ᵐ ω ∂P, Continuous (fun t => A t ω-B t ω)) : + ∀ᵐ ω ∂P, ∀ t, A t ω = B t ω := + increasing_mass_martingale_indistinguishable A B + (mass_difference_martingale_of_terminal_mean A B Z hZ hAm hBm hAI hBI hmean) + hA hB hA0 hB0 hAI hBI hcont + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassMartingaleTesting.lean b/lean/OAI/Probability/SLE/Mass/MassMartingaleTesting.lean index e83d61b25c4a08d35f37a0475ead53b4f27f7240..7c052504820c3bab290ecfc23767306ad9f8d24e 100644 --- a/lean/OAI/Probability/SLE/Mass/MassMartingaleTesting.lean +++ b/lean/OAI/Probability/SLE/Mass/MassMartingaleTesting.lean @@ -1,75 +1,75 @@ -import OAI.Probability.SLE.Mass.MassUniquenessSeparation -import Mathlib.Probability.Martingale.Basic -import Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut - -namespace OAI - -/-! Bounded predictable testing of the increments in the mass uniqueness proof. -/ - -noncomputable section -open MeasureTheory Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -lemma massSeparationTest_continuous : Continuous massSeparationTest := - continuous_iff_continuousAt.mpr (fun x => (massSeparationTest_hasDerivAt x).continuousAt) - -lemma massSeparationSlope_continuous : Continuous massSeparationSlope := - continuous_iff_continuousAt.mpr (fun x => (massSeparationSlope_hasDerivAt x).continuousAt) - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] - {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} {M : ℝ≥0 → Ω → ℝ} - -omit [IsFiniteMeasure P] in -lemma martingale_mass_test_integrable (hM : Martingale M ℱ P) (t : ℝ≥0) : - Integrable (fun ω => massSeparationTest (M t ω)) P := by - apply ((hM.integrable t).norm.const_mul 2).mono' - (massSeparationTest_continuous.comp_stronglyMeasurable - ((hM.stronglyMeasurable t).mono (ℱ.le t))).aestronglyMeasurable - filter_upwards [] with ω - simpa only [Real.norm_eq_abs,massSeparationTest_zero,sub_zero] using - massSeparationTest_lipschitz 0 (M t ω) - -omit [IsFiniteMeasure P] in -lemma martingale_mass_weighted_increment_integrable (hM : Martingale M ℱ P) - (s t : ℝ≥0) : - Integrable (fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) P := by - exact ((hM.integrable t).sub (hM.integrable s)).bdd_mul - (massSeparationSlope_continuous.comp_stronglyMeasurable - ((hM.stronglyMeasurable s).mono (ℱ.le s))).aestronglyMeasurable - (ae_of_all _ (fun ω => by simpa only [Real.norm_eq_abs] using - massSeparationSlope_abs_le (M s ω))) - -lemma martingale_mass_weighted_increment_mean (hM : Martingale M ℱ P) - {s t : ℝ≥0} (hst : s ≤ t) : - (∫ ω, massSeparationSlope (M s ω)*(M t ω-M s ω) ∂P) = 0 := by - have hg := (hM.integrable t).sub (hM.integrable s) - have hfg := martingale_mass_weighted_increment_integrable hM s t - have hCE : P[(fun ω => M t ω-M s ω) | ℱ s] =ᵐ[P] 0 := by - have hsub := condExp_sub (hM.integrable t) (hM.integrable s) (ℱ s) - have hs : P[M s | ℱ s] = M s := - condExp_of_stronglyMeasurable (ℱ.le s) (hM.stronglyMeasurable s) (hM.integrable s) - filter_upwards [hsub,hM.condExp_ae_eq hst] with ω hω htω - simp only [Pi.sub_apply,hs,htω,sub_self,Pi.zero_apply] at hω ⊢ - exact hω - have hpull := condExp_mul_of_stronglyMeasurable_left - (massSeparationSlope_continuous.comp_stronglyMeasurable (hM.stronglyMeasurable s)) hfg hg - calc - _ = ∫ ω, P[(fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) | ℱ s] ω ∂P := - (integral_condExp (ℱ.le s)).symm - _ = 0 := by - have he : P[(fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) | ℱ s] =ᵐ[P] 0 := by - filter_upwards [hpull,hCE] with ω hp hc - change P[(fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) | ℱ s] ω = - massSeparationSlope (M s ω)*P[(fun ω => M t ω-M s ω) | ℱ s] ω at hp - change P[(fun ω => M t ω-M s ω) | ℱ s] ω = 0 at hc - simpa only [hc,mul_zero,Pi.zero_apply] using hp - rw [integral_congr_ae he] - simp - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassUniquenessSeparation +import Mathlib.Probability.Martingale.Basic +import Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut + +namespace OAI + +/-! Bounded predictable testing of the increments in the mass uniqueness proof. -/ + +noncomputable section +open MeasureTheory Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +lemma massSeparationTest_continuous : Continuous massSeparationTest := + continuous_iff_continuousAt.mpr (fun x => (massSeparationTest_hasDerivAt x).continuousAt) + +lemma massSeparationSlope_continuous : Continuous massSeparationSlope := + continuous_iff_continuousAt.mpr (fun x => (massSeparationSlope_hasDerivAt x).continuousAt) + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] + {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} {M : ℝ≥0 → Ω → ℝ} + +omit [IsFiniteMeasure P] in +lemma martingale_mass_test_integrable (hM : Martingale M ℱ P) (t : ℝ≥0) : + Integrable (fun ω => massSeparationTest (M t ω)) P := by + apply ((hM.integrable t).norm.const_mul 2).mono' + (massSeparationTest_continuous.comp_stronglyMeasurable + ((hM.stronglyMeasurable t).mono (ℱ.le t))).aestronglyMeasurable + filter_upwards [] with ω + simpa only [Real.norm_eq_abs,massSeparationTest_zero,sub_zero] using + massSeparationTest_lipschitz 0 (M t ω) + +omit [IsFiniteMeasure P] in +lemma martingale_mass_weighted_increment_integrable (hM : Martingale M ℱ P) + (s t : ℝ≥0) : + Integrable (fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) P := by + exact ((hM.integrable t).sub (hM.integrable s)).bdd_mul + (massSeparationSlope_continuous.comp_stronglyMeasurable + ((hM.stronglyMeasurable s).mono (ℱ.le s))).aestronglyMeasurable + (ae_of_all _ (fun ω => by simpa only [Real.norm_eq_abs] using + massSeparationSlope_abs_le (M s ω))) + +lemma martingale_mass_weighted_increment_mean (hM : Martingale M ℱ P) + {s t : ℝ≥0} (hst : s ≤ t) : + (∫ ω, massSeparationSlope (M s ω)*(M t ω-M s ω) ∂P) = 0 := by + have hg := (hM.integrable t).sub (hM.integrable s) + have hfg := martingale_mass_weighted_increment_integrable hM s t + have hCE : P[(fun ω => M t ω-M s ω) | ℱ s] =ᵐ[P] 0 := by + have hsub := condExp_sub (hM.integrable t) (hM.integrable s) (ℱ s) + have hs : P[M s | ℱ s] = M s := + condExp_of_stronglyMeasurable (ℱ.le s) (hM.stronglyMeasurable s) (hM.integrable s) + filter_upwards [hsub,hM.condExp_ae_eq hst] with ω hω htω + simp only [Pi.sub_apply,hs,htω,sub_self,Pi.zero_apply] at hω ⊢ + exact hω + have hpull := condExp_mul_of_stronglyMeasurable_left + (massSeparationSlope_continuous.comp_stronglyMeasurable (hM.stronglyMeasurable s)) hfg hg + calc + _ = ∫ ω, P[(fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) | ℱ s] ω ∂P := + (integral_condExp (ℱ.le s)).symm + _ = 0 := by + have he : P[(fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) | ℱ s] =ᵐ[P] 0 := by + filter_upwards [hpull,hCE] with ω hp hc + change P[(fun ω => massSeparationSlope (M s ω)*(M t ω-M s ω)) | ℱ s] ω = + massSeparationSlope (M s ω)*P[(fun ω => M t ω-M s ω) | ℱ s] ω at hp + change P[(fun ω => M t ω-M s ω) | ℱ s] ω = 0 at hc + simpa only [hc,mul_zero,Pi.zero_apply] using hp + rw [integral_congr_ae he] + simp + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassMartingaleUniqueness.lean b/lean/OAI/Probability/SLE/Mass/MassMartingaleUniqueness.lean index 440fd58ec35261f55e2ae60dc40c4d067a546eaa..ea7c1c184af639ce5badfb1bad5c977cfdeac64b 100644 --- a/lean/OAI/Probability/SLE/Mass/MassMartingaleUniqueness.lean +++ b/lean/OAI/Probability/SLE/Mass/MassMartingaleUniqueness.lean @@ -1,123 +1,123 @@ -import OAI.Probability.SLE.Mass.MassUniquenessMesh -import OAI.Probability.SLE.Mass.MassMartingaleTesting -import Mathlib.MeasureTheory.Integral.DominatedConvergence -import Mathlib.Topology.DenseEmbedding - -namespace OAI - -/-! Uniqueness of a continuous mass process from its integrable increasing -variation and martingale difference. No second moment is imposed on a competing -mass construction. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology NNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] - {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} (A B : ℝ≥0 → Ω → ℝ) - -/-- A continuous martingale which is the difference of two increasing processes -starting at zero vanishes, using only integrability of the two endpoint masses. -/ -theorem increasing_mass_martingale_eq_at - (hM : Martingale (fun t ω => A t ω-B t ω) ℱ P) - (hA : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) - (hB : ∀ᵐ ω ∂P, Monotone (fun t => B t ω)) - (hA0 : A 0 =ᵐ[P] 0) (hB0 : B 0 =ᵐ[P] 0) - (T : ℝ≥0) (hAI : Integrable (A T) P) (hBI : Integrable (B T) P) - (hcont : ∀ᵐ ω ∂P, ContinuousOn (fun t => A t ω-B t ω) (Icc 0 T)) : - A T =ᵐ[P] B T := by - let M : ℝ≥0 → Ω → ℝ := fun t ω => A t ω-B t ω - let X : ℕ → Ω → ℝ := fun n ω => - massPartitionRemainder (fun i => M (massTimeMesh T n i) ω) (n+1) - have hXint (n : ℕ) : Integrable (X n) P := by - have hsum : Integrable (fun ω => ∑ i ∈ Finset.range (n+1), - massSeparationSlope (M (massTimeMesh T n i) ω)* - (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω)) P := - integrable_finsetSum _ (fun i _ => - martingale_mass_weighted_increment_integrable hM _ _) - exact ((martingale_mass_test_integrable hM _).sub - (martingale_mass_test_integrable hM _)).sub hsum - have hXmean (n : ℕ) : (∫ ω, X n ω ∂P) = ∫ ω, massSeparationTest (M T ω) ∂P := by - have hsum (i : ℕ) := martingale_mass_weighted_increment_integrable hM - (massTimeMesh T n i) (massTimeMesh T n (i+1)) - change (∫ ω, massSeparationTest (M (massTimeMesh T n (n+1)) ω)- - massSeparationTest (M (massTimeMesh T n 0) ω)- - ∑ i ∈ Finset.range (n+1), massSeparationSlope (M (massTimeMesh T n i) ω)* - (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω) ∂P) = _ - rw [massTimeMesh_end,massTimeMesh_zero, - integral_sub (f := fun ω => massSeparationTest (M T ω)-massSeparationTest (M 0 ω)) - (g := fun ω => ∑ i ∈ Finset.range (n+1), massSeparationSlope (M (massTimeMesh T n i) ω)* - (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω)) - ((martingale_mass_test_integrable hM T).sub - (martingale_mass_test_integrable hM 0)) (integrable_finsetSum _ (fun i _ => hsum i)), - integral_sub (f := fun ω => massSeparationTest (M T ω)) - (g := fun ω => massSeparationTest (M 0 ω)) - (martingale_mass_test_integrable hM T) (martingale_mass_test_integrable hM 0), - integral_finsetSum _ (fun i _ => hsum i)] - have hzero : (fun ω => massSeparationTest (M 0 ω)) =ᵐ[P] 0 := by - filter_upwards [hA0,hB0] with ω ha hb - simp only [M,ha,hb,Pi.zero_apply,sub_self,massSeparationTest_zero] - have hmeans : (∑ i ∈ Finset.range (n+1), ∫ ω, - massSeparationSlope (M (massTimeMesh T n i) ω)* - (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω) ∂P) = 0 := by - apply Finset.sum_eq_zero - intro i _ - exact martingale_mass_weighted_increment_mean hM (massTimeMesh_mono T n (Nat.le_succ i)) - simp only [M] at hmeans - rw [integral_congr_ae hzero,hmeans] - simp - have hbound (n : ℕ) : ∀ᵐ ω ∂P, ‖X n ω‖ ≤ 4*(A T ω+B T ω) := by - filter_upwards [hA,hB,hA0,hB0] with ω ha hb ha0 hb0 - calc - _ ≤ 4*∑ i ∈ Finset.range (n+1), - |M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω| := by - simpa only [X,Real.norm_eq_abs] using massPartitionRemainder_linear - (fun i => M (massTimeMesh T n i) ω) (n+1) - _ ≤ 4*((A T ω-A 0 ω)+(B T ω-B 0 ω)) := - mul_le_mul_of_nonneg_left (massDifference_mesh_variation - (fun t => A t ω) (fun t => B t ω) ha hb T n) (by norm_num) - _ = _ := by simp only [ha0,hb0,Pi.zero_apply,sub_zero] - have hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 0) := by - filter_upwards [hA,hB,hcont] with ω ha hb hc - exact massDifference_mesh_remainder_tendsto (fun t => A t ω) (fun t => B t ω) ha hb T hc - have hI := tendsto_integral_of_dominated_convergence (fun ω => 4*(A T ω+B T ω)) - (fun n => (hXint n).aestronglyMeasurable) ((hAI.add hBI).const_mul 4) hbound hlim - simp only [integral_zero,hXmean] at hI - have htest0 : (∫ ω, massSeparationTest (M T ω) ∂P) = 0 := - tendsto_nhds_unique tendsto_const_nhds hI - have hae := (integral_eq_zero_iff_of_nonneg - (fun ω => massSeparationTest_nonneg (M T ω)) - (martingale_mass_test_integrable hM T)).mp htest0 - filter_upwards [hae] with ω hω - exact sub_eq_zero.mp ((massSeparationTest_eq_zero (M T ω)).mp hω) - -/-- The fixed-time result holds simultaneously at every time by continuity and -a countable dense family of deterministic times. -/ -theorem increasing_mass_martingale_indistinguishable - (hM : Martingale (fun t ω => A t ω-B t ω) ℱ P) - (hA : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) - (hB : ∀ᵐ ω ∂P, Monotone (fun t => B t ω)) - (hA0 : A 0 =ᵐ[P] 0) (hB0 : B 0 =ᵐ[P] 0) - (hAI : ∀ T, Integrable (A T) P) (hBI : ∀ T, Integrable (B T) P) - (hcont : ∀ᵐ ω ∂P, Continuous (fun t => A t ω-B t ω)) : - ∀ᵐ ω ∂P, ∀ t, A t ω = B t ω := by - have heq (t : ℝ≥0) : A t =ᵐ[P] B t := - increasing_mass_martingale_eq_at A B hM hA hB hA0 hB0 t (hAI t) (hBI t) - (hcont.mono (fun _ h => h.continuousOn)) - have hcount : ∀ᵐ ω ∂P, ∀ n : ℕ, - A (TopologicalSpace.denseSeq ℝ≥0 n) ω = B (TopologicalSpace.denseSeq ℝ≥0 n) ω := - ae_all_iff.mpr (fun n => heq _) - filter_upwards [hcount,hcont] with ω he hc - have hfun : (fun t => A t ω-B t ω) = fun _ => 0 := - (TopologicalSpace.denseRange_denseSeq ℝ≥0).equalizer hc continuous_const - (funext (fun n => sub_eq_zero.mpr (he n))) - intro t - exact sub_eq_zero.mp (congr_fun hfun t) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassUniquenessMesh +import OAI.Probability.SLE.Mass.MassMartingaleTesting +import Mathlib.MeasureTheory.Integral.DominatedConvergence +import Mathlib.Topology.DenseEmbedding + +namespace OAI + +/-! Uniqueness of a continuous mass process from its integrable increasing +variation and martingale difference. No second moment is imposed on a competing +mass construction. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology NNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] + {ℱ : Filtration ℝ≥0 ‹MeasurableSpace Ω›} (A B : ℝ≥0 → Ω → ℝ) + +/-- A continuous martingale which is the difference of two increasing processes +starting at zero vanishes, using only integrability of the two endpoint masses. -/ +theorem increasing_mass_martingale_eq_at + (hM : Martingale (fun t ω => A t ω-B t ω) ℱ P) + (hA : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) + (hB : ∀ᵐ ω ∂P, Monotone (fun t => B t ω)) + (hA0 : A 0 =ᵐ[P] 0) (hB0 : B 0 =ᵐ[P] 0) + (T : ℝ≥0) (hAI : Integrable (A T) P) (hBI : Integrable (B T) P) + (hcont : ∀ᵐ ω ∂P, ContinuousOn (fun t => A t ω-B t ω) (Icc 0 T)) : + A T =ᵐ[P] B T := by + let M : ℝ≥0 → Ω → ℝ := fun t ω => A t ω-B t ω + let X : ℕ → Ω → ℝ := fun n ω => + massPartitionRemainder (fun i => M (massTimeMesh T n i) ω) (n+1) + have hXint (n : ℕ) : Integrable (X n) P := by + have hsum : Integrable (fun ω => ∑ i ∈ Finset.range (n+1), + massSeparationSlope (M (massTimeMesh T n i) ω)* + (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω)) P := + integrable_finsetSum _ (fun i _ => + martingale_mass_weighted_increment_integrable hM _ _) + exact ((martingale_mass_test_integrable hM _).sub + (martingale_mass_test_integrable hM _)).sub hsum + have hXmean (n : ℕ) : (∫ ω, X n ω ∂P) = ∫ ω, massSeparationTest (M T ω) ∂P := by + have hsum (i : ℕ) := martingale_mass_weighted_increment_integrable hM + (massTimeMesh T n i) (massTimeMesh T n (i+1)) + change (∫ ω, massSeparationTest (M (massTimeMesh T n (n+1)) ω)- + massSeparationTest (M (massTimeMesh T n 0) ω)- + ∑ i ∈ Finset.range (n+1), massSeparationSlope (M (massTimeMesh T n i) ω)* + (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω) ∂P) = _ + rw [massTimeMesh_end,massTimeMesh_zero, + integral_sub (f := fun ω => massSeparationTest (M T ω)-massSeparationTest (M 0 ω)) + (g := fun ω => ∑ i ∈ Finset.range (n+1), massSeparationSlope (M (massTimeMesh T n i) ω)* + (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω)) + ((martingale_mass_test_integrable hM T).sub + (martingale_mass_test_integrable hM 0)) (integrable_finsetSum _ (fun i _ => hsum i)), + integral_sub (f := fun ω => massSeparationTest (M T ω)) + (g := fun ω => massSeparationTest (M 0 ω)) + (martingale_mass_test_integrable hM T) (martingale_mass_test_integrable hM 0), + integral_finsetSum _ (fun i _ => hsum i)] + have hzero : (fun ω => massSeparationTest (M 0 ω)) =ᵐ[P] 0 := by + filter_upwards [hA0,hB0] with ω ha hb + simp only [M,ha,hb,Pi.zero_apply,sub_self,massSeparationTest_zero] + have hmeans : (∑ i ∈ Finset.range (n+1), ∫ ω, + massSeparationSlope (M (massTimeMesh T n i) ω)* + (M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω) ∂P) = 0 := by + apply Finset.sum_eq_zero + intro i _ + exact martingale_mass_weighted_increment_mean hM (massTimeMesh_mono T n (Nat.le_succ i)) + simp only [M] at hmeans + rw [integral_congr_ae hzero,hmeans] + simp + have hbound (n : ℕ) : ∀ᵐ ω ∂P, ‖X n ω‖ ≤ 4*(A T ω+B T ω) := by + filter_upwards [hA,hB,hA0,hB0] with ω ha hb ha0 hb0 + calc + _ ≤ 4*∑ i ∈ Finset.range (n+1), + |M (massTimeMesh T n (i+1)) ω-M (massTimeMesh T n i) ω| := by + simpa only [X,Real.norm_eq_abs] using massPartitionRemainder_linear + (fun i => M (massTimeMesh T n i) ω) (n+1) + _ ≤ 4*((A T ω-A 0 ω)+(B T ω-B 0 ω)) := + mul_le_mul_of_nonneg_left (massDifference_mesh_variation + (fun t => A t ω) (fun t => B t ω) ha hb T n) (by norm_num) + _ = _ := by simp only [ha0,hb0,Pi.zero_apply,sub_zero] + have hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 0) := by + filter_upwards [hA,hB,hcont] with ω ha hb hc + exact massDifference_mesh_remainder_tendsto (fun t => A t ω) (fun t => B t ω) ha hb T hc + have hI := tendsto_integral_of_dominated_convergence (fun ω => 4*(A T ω+B T ω)) + (fun n => (hXint n).aestronglyMeasurable) ((hAI.add hBI).const_mul 4) hbound hlim + simp only [integral_zero,hXmean] at hI + have htest0 : (∫ ω, massSeparationTest (M T ω) ∂P) = 0 := + tendsto_nhds_unique tendsto_const_nhds hI + have hae := (integral_eq_zero_iff_of_nonneg + (fun ω => massSeparationTest_nonneg (M T ω)) + (martingale_mass_test_integrable hM T)).mp htest0 + filter_upwards [hae] with ω hω + exact sub_eq_zero.mp ((massSeparationTest_eq_zero (M T ω)).mp hω) + +/-- The fixed-time result holds simultaneously at every time by continuity and +a countable dense family of deterministic times. -/ +theorem increasing_mass_martingale_indistinguishable + (hM : Martingale (fun t ω => A t ω-B t ω) ℱ P) + (hA : ∀ᵐ ω ∂P, Monotone (fun t => A t ω)) + (hB : ∀ᵐ ω ∂P, Monotone (fun t => B t ω)) + (hA0 : A 0 =ᵐ[P] 0) (hB0 : B 0 =ᵐ[P] 0) + (hAI : ∀ T, Integrable (A T) P) (hBI : ∀ T, Integrable (B T) P) + (hcont : ∀ᵐ ω ∂P, Continuous (fun t => A t ω-B t ω)) : + ∀ᵐ ω ∂P, ∀ t, A t ω = B t ω := by + have heq (t : ℝ≥0) : A t =ᵐ[P] B t := + increasing_mass_martingale_eq_at A B hM hA hB hA0 hB0 t (hAI t) (hBI t) + (hcont.mono (fun _ h => h.continuousOn)) + have hcount : ∀ᵐ ω ∂P, ∀ n : ℕ, + A (TopologicalSpace.denseSeq ℝ≥0 n) ω = B (TopologicalSpace.denseSeq ℝ≥0 n) ω := + ae_all_iff.mpr (fun n => heq _) + filter_upwards [hcount,hcont] with ω he hc + have hfun : (fun t => A t ω-B t ω) = fun _ => 0 := + (TopologicalSpace.denseRange_denseSeq ℝ≥0).equalizer hc continuous_const + (funext (fun n => sub_eq_zero.mpr (he n))) + intro t + exact sub_eq_zero.mp (congr_fun hfun t) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassPotentialCompensator.lean b/lean/OAI/Probability/SLE/Mass/MassPotentialCompensator.lean index f7570827c5c8ee745b512f641d6361dd365fbc37..802bc5ce976c150f503e34630ea73ebfeedd8989 100644 --- a/lean/OAI/Probability/SLE/Mass/MassPotentialCompensator.lean +++ b/lean/OAI/Probability/SLE/Mass/MassPotentialCompensator.lean @@ -1,57 +1,57 @@ -import OAI.Probability.SLE.Mass.MassCompensatorVariance - -namespace OAI - -/-! A remaining-mass potential determines the predictable mass increments. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] - -/-- The difference of the current potential and the one-step averaged -future potential is the conditional expectation of the mass increment. -/ -theorem remainingPotential_increment - (ℱ : Filtration ℕ mΩ) (A V : ℕ → Ω → ℝ) (L : Ω → ℝ) - (hA : StronglyAdapted ℱ A) (hAI : ∀ i, Integrable (A i) P) (hLI : Integrable L P) - (hV : ∀ i, V i =ᵐ[P] P[L-A i|ℱ i]) (i : ℕ) : - V i-P[V (i+1)|ℱ i] =ᵐ[P] P[A (i+1)-A i|ℱ i] := by - have hnext := (condExp_congr_ae (m := ℱ i) (hV (i+1))).trans - (condExp_condExp_of_le (ℱ.mono (Nat.le_succ i)) (ℱ.le (i+1))) - have hcur := (hV i).trans (condExp_sub hLI (hAI i) (ℱ i)) - have hnext' := hnext.trans (condExp_sub hLI (hAI (i+1)) (ℱ i)) - have hinc := condExp_sub (hAI (i+1)) (hAI i) (ℱ i) - have hz := condExp_of_stronglyMeasurable (ℱ.le i) (hA i) (hAI i) - filter_upwards [hcur,hnext',hinc] with ω hc hn hi - simp only [Pi.sub_apply,hz] at hc hn hi ⊢ - linarith - -omit [IsProbabilityMeasure P] in -/-- Pathwise telescoping after identifying each predictable increment. -/ -lemma predictable_mass_sum_eq - (ℱ : Filtration ℕ mΩ) (A : ℕ → Ω → ℝ) - (q : ℕ → Ω → ℝ) - (hq : ∀ i, q i =ᵐ[P] P[A (i+1)-A i|ℱ i]) (N : ℕ) : - (fun ω => ∑ i ∈ Finset.range N, q i ω) =ᵐ[P] - fun ω => A N ω-A 0 ω-centeredMassPartialSum P ℱ - (fun i => A (i+1)-A i) N ω := by - filter_upwards [ae_all_iff.mpr hq] with ω hω - simp only [centeredMassPartialSum,Pi.sub_apply] - have hs := Finset.sum_sub_distrib (s := Finset.range N) - (f := fun i => A (i+1) ω-A i ω) - (g := fun i => condExp (ℱ i) P (A (i+1)-A i) ω) - rw [Finset.sum_range_sub (fun i => A i ω) N] at hs - have hq' : (∑ i ∈ Finset.range N, q i ω) = - ∑ i ∈ Finset.range N, condExp (ℱ i) P (A (i+1)-A i) ω := - Finset.sum_congr rfl (fun i _ => hω i) - rw [hq'] - linarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassCompensatorVariance + +namespace OAI + +/-! A remaining-mass potential determines the predictable mass increments. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] + +/-- The difference of the current potential and the one-step averaged +future potential is the conditional expectation of the mass increment. -/ +theorem remainingPotential_increment + (ℱ : Filtration ℕ mΩ) (A V : ℕ → Ω → ℝ) (L : Ω → ℝ) + (hA : StronglyAdapted ℱ A) (hAI : ∀ i, Integrable (A i) P) (hLI : Integrable L P) + (hV : ∀ i, V i =ᵐ[P] P[L-A i|ℱ i]) (i : ℕ) : + V i-P[V (i+1)|ℱ i] =ᵐ[P] P[A (i+1)-A i|ℱ i] := by + have hnext := (condExp_congr_ae (m := ℱ i) (hV (i+1))).trans + (condExp_condExp_of_le (ℱ.mono (Nat.le_succ i)) (ℱ.le (i+1))) + have hcur := (hV i).trans (condExp_sub hLI (hAI i) (ℱ i)) + have hnext' := hnext.trans (condExp_sub hLI (hAI (i+1)) (ℱ i)) + have hinc := condExp_sub (hAI (i+1)) (hAI i) (ℱ i) + have hz := condExp_of_stronglyMeasurable (ℱ.le i) (hA i) (hAI i) + filter_upwards [hcur,hnext',hinc] with ω hc hn hi + simp only [Pi.sub_apply,hz] at hc hn hi ⊢ + linarith + +omit [IsProbabilityMeasure P] in +/-- Pathwise telescoping after identifying each predictable increment. -/ +lemma predictable_mass_sum_eq + (ℱ : Filtration ℕ mΩ) (A : ℕ → Ω → ℝ) + (q : ℕ → Ω → ℝ) + (hq : ∀ i, q i =ᵐ[P] P[A (i+1)-A i|ℱ i]) (N : ℕ) : + (fun ω => ∑ i ∈ Finset.range N, q i ω) =ᵐ[P] + fun ω => A N ω-A 0 ω-centeredMassPartialSum P ℱ + (fun i => A (i+1)-A i) N ω := by + filter_upwards [ae_all_iff.mpr hq] with ω hω + simp only [centeredMassPartialSum,Pi.sub_apply] + have hs := Finset.sum_sub_distrib (s := Finset.range N) + (f := fun i => A (i+1) ω-A i ω) + (g := fun i => condExp (ℱ i) P (A (i+1)-A i) ω) + rw [Finset.sum_range_sub (fun i => A i ω) N] at hs + have hq' : (∑ i ∈ Finset.range N, q i ω) = + ∑ i ∈ Finset.range N, condExp (ℱ i) P (A (i+1)-A i) ω := + Finset.sum_congr rfl (fun i _ => hω i) + rw [hq'] + linarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassUniquenessMesh.lean b/lean/OAI/Probability/SLE/Mass/MassUniquenessMesh.lean index 140fee9dd44795b2834210e4469e49c9b33ff3d4..e4c7bed746290f16d56c9e96ee4b411b940eec35 100644 --- a/lean/OAI/Probability/SLE/Mass/MassUniquenessMesh.lean +++ b/lean/OAI/Probability/SLE/Mass/MassUniquenessMesh.lean @@ -1,81 +1,81 @@ -import OAI.Probability.SLE.Mass.MassUniquenessPartitions -import Mathlib.Topology.Instances.NNReal.Lemmas -import Mathlib.Topology.UniformSpace.HeineCantor - -namespace OAI - -/-! Uniform finite time partitions used in the continuous-mass uniqueness proof. -/ - -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -def massTimeMesh (T : ℝ≥0) (n i : ℕ) : ℝ≥0 := T*(i:ℝ≥0)/(n+1:ℝ≥0) - -@[simp] lemma massTimeMesh_zero (T : ℝ≥0) (n : ℕ) : massTimeMesh T n 0 = 0 := by - simp [massTimeMesh] - -@[simp] lemma massTimeMesh_end (T : ℝ≥0) (n : ℕ) : massTimeMesh T n (n+1) = T := by - simp [massTimeMesh] - -lemma massTimeMesh_mono (T : ℝ≥0) (n : ℕ) : Monotone (massTimeMesh T n) := by - intro i j hij - exact div_le_div_of_nonneg_right - (mul_le_mul_of_nonneg_left (by exact_mod_cast hij) (show 0 ≤ T from zero_le)) - (by positivity) - -lemma massTimeMesh_mem (T : ℝ≥0) (n i : ℕ) (hi : i ≤ n+1) : - massTimeMesh T n i ∈ Icc 0 T := by - exact ⟨zero_le,(massTimeMesh_mono T n hi).trans_eq (massTimeMesh_end T n)⟩ - -lemma massTimeMesh_dist (T : ℝ≥0) (n i : ℕ) : - dist (massTimeMesh T n (i+1)) (massTimeMesh T n i) = (T:ℝ)/(n+1:ℝ) := by - rw [NNReal.dist_eq] - simp only [massTimeMesh,NNReal.coe_div,NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_add, - NNReal.coe_one,Nat.cast_add,Nat.cast_one] - rw [show (T:ℝ)*(i+1:ℝ)/(n+1)-(T:ℝ)*(i:ℝ)/(n+1) = T/(n+1) by ring, - abs_of_nonneg (by positivity)] - -lemma massTimeMesh_small_increments (T : ℝ≥0) (f : ℝ≥0 → ℝ) - (hf : ContinuousOn f (Icc 0 T)) : - ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, - ∀ i < n+1, |f (massTimeMesh T n (i+1))-f (massTimeMesh T n i)| ≤ ε := by - intro ε hε - obtain ⟨δ,hδ,hfδ⟩ := Metric.uniformContinuousOn_iff.mp - (isCompact_Icc.uniformContinuousOn_of_continuous hf) ε hε - have hlim : Tendsto (fun n : ℕ => (T:ℝ)/(n+1:ℝ)) atTop (𝓝 0) := by - simpa only [mul_one_div,mul_zero] using - (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (T:ℝ) - filter_upwards [hlim.eventually (gt_mem_nhds hδ)] with n hn i hi - have h := hfδ (massTimeMesh T n (i+1)) (massTimeMesh_mem T n (i+1) (by omega)) - (massTimeMesh T n i) (massTimeMesh_mem T n i (by omega)) - (by simpa only [massTimeMesh_dist] using hn) - exact le_of_lt (by simpa only [Real.dist_eq] using h) - -lemma massDifference_mesh_variation (A B : ℝ≥0 → ℝ) - (hA : Monotone A) (hB : Monotone B) (T : ℝ≥0) (n : ℕ) : - (∑ i ∈ Finset.range (n+1), - |(A (massTimeMesh T n (i+1))-B (massTimeMesh T n (i+1)))- - (A (massTimeMesh T n i)-B (massTimeMesh T n i))|) ≤ - (A T-A 0)+(B T-B 0) := by - simpa only [Function.comp_def,massTimeMesh_zero,massTimeMesh_end] using - massDifference_partition_variation (A ∘ massTimeMesh T n) (B ∘ massTimeMesh T n) - (hA.comp (massTimeMesh_mono T n)) (hB.comp (massTimeMesh_mono T n)) (n+1) - -lemma massDifference_mesh_remainder_tendsto (A B : ℝ≥0 → ℝ) - (hA : Monotone A) (hB : Monotone B) (T : ℝ≥0) - (hcont : ContinuousOn (fun t => A t-B t) (Icc 0 T)) : - Tendsto (fun n => massPartitionRemainder - (fun i => A (massTimeMesh T n i)-B (massTimeMesh T n i)) (n+1)) atTop (𝓝 0) := by - exact massPartitionRemainder_tendsto_zero _ _ ((A T-A 0)+(B T-B 0)) - (add_nonneg (sub_nonneg.mpr (hA zero_le)) (sub_nonneg.mpr (hB zero_le))) - (massDifference_mesh_variation A B hA hB T) - (massTimeMesh_small_increments T _ hcont) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassUniquenessPartitions +import Mathlib.Topology.Instances.NNReal.Lemmas +import Mathlib.Topology.UniformSpace.HeineCantor + +namespace OAI + +/-! Uniform finite time partitions used in the continuous-mass uniqueness proof. -/ + +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +def massTimeMesh (T : ℝ≥0) (n i : ℕ) : ℝ≥0 := T*(i:ℝ≥0)/(n+1:ℝ≥0) + +@[simp] lemma massTimeMesh_zero (T : ℝ≥0) (n : ℕ) : massTimeMesh T n 0 = 0 := by + simp [massTimeMesh] + +@[simp] lemma massTimeMesh_end (T : ℝ≥0) (n : ℕ) : massTimeMesh T n (n+1) = T := by + simp [massTimeMesh] + +lemma massTimeMesh_mono (T : ℝ≥0) (n : ℕ) : Monotone (massTimeMesh T n) := by + intro i j hij + exact div_le_div_of_nonneg_right + (mul_le_mul_of_nonneg_left (by exact_mod_cast hij) (show 0 ≤ T from zero_le)) + (by positivity) + +lemma massTimeMesh_mem (T : ℝ≥0) (n i : ℕ) (hi : i ≤ n+1) : + massTimeMesh T n i ∈ Icc 0 T := by + exact ⟨zero_le,(massTimeMesh_mono T n hi).trans_eq (massTimeMesh_end T n)⟩ + +lemma massTimeMesh_dist (T : ℝ≥0) (n i : ℕ) : + dist (massTimeMesh T n (i+1)) (massTimeMesh T n i) = (T:ℝ)/(n+1:ℝ) := by + rw [NNReal.dist_eq] + simp only [massTimeMesh,NNReal.coe_div,NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_add, + NNReal.coe_one,Nat.cast_add,Nat.cast_one] + rw [show (T:ℝ)*(i+1:ℝ)/(n+1)-(T:ℝ)*(i:ℝ)/(n+1) = T/(n+1) by ring, + abs_of_nonneg (by positivity)] + +lemma massTimeMesh_small_increments (T : ℝ≥0) (f : ℝ≥0 → ℝ) + (hf : ContinuousOn f (Icc 0 T)) : + ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, + ∀ i < n+1, |f (massTimeMesh T n (i+1))-f (massTimeMesh T n i)| ≤ ε := by + intro ε hε + obtain ⟨δ,hδ,hfδ⟩ := Metric.uniformContinuousOn_iff.mp + (isCompact_Icc.uniformContinuousOn_of_continuous hf) ε hε + have hlim : Tendsto (fun n : ℕ => (T:ℝ)/(n+1:ℝ)) atTop (𝓝 0) := by + simpa only [mul_one_div,mul_zero] using + (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (T:ℝ) + filter_upwards [hlim.eventually (gt_mem_nhds hδ)] with n hn i hi + have h := hfδ (massTimeMesh T n (i+1)) (massTimeMesh_mem T n (i+1) (by omega)) + (massTimeMesh T n i) (massTimeMesh_mem T n i (by omega)) + (by simpa only [massTimeMesh_dist] using hn) + exact le_of_lt (by simpa only [Real.dist_eq] using h) + +lemma massDifference_mesh_variation (A B : ℝ≥0 → ℝ) + (hA : Monotone A) (hB : Monotone B) (T : ℝ≥0) (n : ℕ) : + (∑ i ∈ Finset.range (n+1), + |(A (massTimeMesh T n (i+1))-B (massTimeMesh T n (i+1)))- + (A (massTimeMesh T n i)-B (massTimeMesh T n i))|) ≤ + (A T-A 0)+(B T-B 0) := by + simpa only [Function.comp_def,massTimeMesh_zero,massTimeMesh_end] using + massDifference_partition_variation (A ∘ massTimeMesh T n) (B ∘ massTimeMesh T n) + (hA.comp (massTimeMesh_mono T n)) (hB.comp (massTimeMesh_mono T n)) (n+1) + +lemma massDifference_mesh_remainder_tendsto (A B : ℝ≥0 → ℝ) + (hA : Monotone A) (hB : Monotone B) (T : ℝ≥0) + (hcont : ContinuousOn (fun t => A t-B t) (Icc 0 T)) : + Tendsto (fun n => massPartitionRemainder + (fun i => A (massTimeMesh T n i)-B (massTimeMesh T n i)) (n+1)) atTop (𝓝 0) := by + exact massPartitionRemainder_tendsto_zero _ _ ((A T-A 0)+(B T-B 0)) + (add_nonneg (sub_nonneg.mpr (hA zero_le)) (sub_nonneg.mpr (hB zero_le))) + (massDifference_mesh_variation A B hA hB T) + (massTimeMesh_small_increments T _ hcont) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassUniquenessPartitions.lean b/lean/OAI/Probability/SLE/Mass/MassUniquenessPartitions.lean index 50d56c511aa1e4a1b85880d66657abd8d1bf023f..8c842684e7cfac7961e69856a3d19b2f7d455c4e 100644 --- a/lean/OAI/Probability/SLE/Mass/MassUniquenessPartitions.lean +++ b/lean/OAI/Probability/SLE/Mass/MassUniquenessPartitions.lean @@ -1,87 +1,87 @@ -import OAI.Probability.SLE.Mass.MassUniquenessSeparation -import Mathlib.Analysis.SpecificLimits.Basic - -namespace OAI - -/-! Finite-variation estimates for the smooth separator on time partitions. -/ - -noncomputable section -open Set Filter -open scoped Topology NNReal - -namespace SLEExactGauge - -def massPartitionRemainder (x : ℕ → ℝ) (N : ℕ) : ℝ := - massSeparationTest (x N)-massSeparationTest (x 0) - - ∑ i ∈ Finset.range N, massSeparationSlope (x i)*(x (i+1)-x i) - -lemma massPartitionRemainder_eq_sum (x : ℕ → ℝ) (N : ℕ) : - massPartitionRemainder x N = ∑ i ∈ Finset.range N, - (massSeparationTest (x (i+1))-massSeparationTest (x i)- - massSeparationSlope (x i)*(x (i+1)-x i)) := by - rw [Finset.sum_sub_distrib, - Finset.sum_range_sub (fun i => massSeparationTest (x i)) N] - rfl - -lemma massPartitionRemainder_linear (x : ℕ → ℝ) (N : ℕ) : - |massPartitionRemainder x N| ≤ 4*∑ i ∈ Finset.range N, |x (i+1)-x i| := by - rw [massPartitionRemainder_eq_sum,Finset.mul_sum] - exact (Finset.abs_sum_le_sum_abs _ _).trans - (Finset.sum_le_sum (fun i _ => massSeparationTest_remainder_linear (x i) (x (i+1)))) - -lemma massPartitionRemainder_small (x : ℕ → ℝ) (N : ℕ) {ε : ℝ} - (hx : ∀ i < N, |x (i+1)-x i| ≤ ε) : - |massPartitionRemainder x N| ≤ (8*ε)*∑ i ∈ Finset.range N, |x (i+1)-x i| := by - rw [massPartitionRemainder_eq_sum,Finset.mul_sum] - apply (Finset.abs_sum_le_sum_abs _ _).trans - apply Finset.sum_le_sum - intro i hi - exact (massSeparationTest_remainder_le (x i) (x (i+1))).trans (by - rw [pow_two,mul_assoc] - exact mul_le_mul_of_nonneg_left - (mul_le_mul_of_nonneg_right (hx i (Finset.mem_range.mp hi)) (abs_nonneg _)) (by norm_num)) - -lemma massDifference_partition_variation (A B : ℕ → ℝ) - (hA : Monotone A) (hB : Monotone B) (N : ℕ) : - (∑ i ∈ Finset.range N, |(A (i+1)-B (i+1))-(A i-B i)|) ≤ - (A N-A 0)+(B N-B 0) := by - calc - _ ≤ ∑ i ∈ Finset.range N, ((A (i+1)-A i)+(B (i+1)-B i)) := by - apply Finset.sum_le_sum - intro i _ - have hAi : 0 ≤ A (i+1)-A i := sub_nonneg.mpr (hA (Nat.le_succ i)) - have hBi : 0 ≤ B (i+1)-B i := sub_nonneg.mpr (hB (Nat.le_succ i)) - calc - _ = |(A (i+1)-A i)-(B (i+1)-B i)| := by congr 1; ring - _ ≤ |A (i+1)-A i|+|B (i+1)-B i| := abs_sub _ _ - _ = _ := by rw [abs_of_nonneg hAi,abs_of_nonneg hBi] - _ = _ := by rw [Finset.sum_add_distrib,Finset.sum_range_sub,Finset.sum_range_sub] - -lemma massPartitionRemainder_tendsto_zero - (x : ℕ → ℕ → ℝ) (N : ℕ → ℕ) (D : ℝ) (hD : 0 ≤ D) - (hvar : ∀ n, (∑ i ∈ Finset.range (N n), |x n (i+1)-x n i|) ≤ D) - (hsmall : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ i < N n, |x n (i+1)-x n i| ≤ ε) : - Tendsto (fun n => massPartitionRemainder (x n) (N n)) atTop (𝓝 0) := by - apply Metric.tendsto_atTop.mpr - intro ε hε - let η := ε/(16*(D+1)) - have hη : 0 < η := div_pos hε (by positivity) - obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp (hsmall η hη) - refine ⟨n₀,fun n hn => ?_⟩ - rw [Real.dist_eq,sub_zero] - calc - _ ≤ (8*η)*∑ i ∈ Finset.range (N n), |x n (i+1)-x n i| := - massPartitionRemainder_small (x n) (N n) (hn₀ n hn) - _ ≤ (8*η)*D := mul_le_mul_of_nonneg_left (hvar n) (by positivity) - _ < ε := by - dsimp only [η] - have hpos : 0 < D+1 := by linarith - rw [show 8*(ε/(16*(D+1)))*D = (8*ε*D)/(16*(D+1)) by ring] - apply (div_lt_iff₀ (by positivity : 0 < 16*(D+1))).mpr - nlinarith - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassUniquenessSeparation +import Mathlib.Analysis.SpecificLimits.Basic + +namespace OAI + +/-! Finite-variation estimates for the smooth separator on time partitions. -/ + +noncomputable section +open Set Filter +open scoped Topology NNReal + +namespace SLEExactGauge + +def massPartitionRemainder (x : ℕ → ℝ) (N : ℕ) : ℝ := + massSeparationTest (x N)-massSeparationTest (x 0) - + ∑ i ∈ Finset.range N, massSeparationSlope (x i)*(x (i+1)-x i) + +lemma massPartitionRemainder_eq_sum (x : ℕ → ℝ) (N : ℕ) : + massPartitionRemainder x N = ∑ i ∈ Finset.range N, + (massSeparationTest (x (i+1))-massSeparationTest (x i)- + massSeparationSlope (x i)*(x (i+1)-x i)) := by + rw [Finset.sum_sub_distrib, + Finset.sum_range_sub (fun i => massSeparationTest (x i)) N] + rfl + +lemma massPartitionRemainder_linear (x : ℕ → ℝ) (N : ℕ) : + |massPartitionRemainder x N| ≤ 4*∑ i ∈ Finset.range N, |x (i+1)-x i| := by + rw [massPartitionRemainder_eq_sum,Finset.mul_sum] + exact (Finset.abs_sum_le_sum_abs _ _).trans + (Finset.sum_le_sum (fun i _ => massSeparationTest_remainder_linear (x i) (x (i+1)))) + +lemma massPartitionRemainder_small (x : ℕ → ℝ) (N : ℕ) {ε : ℝ} + (hx : ∀ i < N, |x (i+1)-x i| ≤ ε) : + |massPartitionRemainder x N| ≤ (8*ε)*∑ i ∈ Finset.range N, |x (i+1)-x i| := by + rw [massPartitionRemainder_eq_sum,Finset.mul_sum] + apply (Finset.abs_sum_le_sum_abs _ _).trans + apply Finset.sum_le_sum + intro i hi + exact (massSeparationTest_remainder_le (x i) (x (i+1))).trans (by + rw [pow_two,mul_assoc] + exact mul_le_mul_of_nonneg_left + (mul_le_mul_of_nonneg_right (hx i (Finset.mem_range.mp hi)) (abs_nonneg _)) (by norm_num)) + +lemma massDifference_partition_variation (A B : ℕ → ℝ) + (hA : Monotone A) (hB : Monotone B) (N : ℕ) : + (∑ i ∈ Finset.range N, |(A (i+1)-B (i+1))-(A i-B i)|) ≤ + (A N-A 0)+(B N-B 0) := by + calc + _ ≤ ∑ i ∈ Finset.range N, ((A (i+1)-A i)+(B (i+1)-B i)) := by + apply Finset.sum_le_sum + intro i _ + have hAi : 0 ≤ A (i+1)-A i := sub_nonneg.mpr (hA (Nat.le_succ i)) + have hBi : 0 ≤ B (i+1)-B i := sub_nonneg.mpr (hB (Nat.le_succ i)) + calc + _ = |(A (i+1)-A i)-(B (i+1)-B i)| := by congr 1; ring + _ ≤ |A (i+1)-A i|+|B (i+1)-B i| := abs_sub _ _ + _ = _ := by rw [abs_of_nonneg hAi,abs_of_nonneg hBi] + _ = _ := by rw [Finset.sum_add_distrib,Finset.sum_range_sub,Finset.sum_range_sub] + +lemma massPartitionRemainder_tendsto_zero + (x : ℕ → ℕ → ℝ) (N : ℕ → ℕ) (D : ℝ) (hD : 0 ≤ D) + (hvar : ∀ n, (∑ i ∈ Finset.range (N n), |x n (i+1)-x n i|) ≤ D) + (hsmall : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop, ∀ i < N n, |x n (i+1)-x n i| ≤ ε) : + Tendsto (fun n => massPartitionRemainder (x n) (N n)) atTop (𝓝 0) := by + apply Metric.tendsto_atTop.mpr + intro ε hε + let η := ε/(16*(D+1)) + have hη : 0 < η := div_pos hε (by positivity) + obtain ⟨n₀,hn₀⟩ := eventually_atTop.mp (hsmall η hη) + refine ⟨n₀,fun n hn => ?_⟩ + rw [Real.dist_eq,sub_zero] + calc + _ ≤ (8*η)*∑ i ∈ Finset.range (N n), |x n (i+1)-x n i| := + massPartitionRemainder_small (x n) (N n) (hn₀ n hn) + _ ≤ (8*η)*D := mul_le_mul_of_nonneg_left (hvar n) (by positivity) + _ < ε := by + dsimp only [η] + have hpos : 0 < D+1 := by linarith + rw [show 8*(ε/(16*(D+1)))*D = (8*ε*D)/(16*(D+1)) by ring] + apply (div_lt_iff₀ (by positivity : 0 < 16*(D+1))).mpr + nlinarith + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MassUniquenessSeparation.lean b/lean/OAI/Probability/SLE/Mass/MassUniquenessSeparation.lean index 06adc2c0210cb35a0eeeb9d6be1166304b4be97f..1cfe9a26a3ad469e97aae6cae1a2c0399c6c678c 100644 --- a/lean/OAI/Probability/SLE/Mass/MassUniquenessSeparation.lean +++ b/lean/OAI/Probability/SLE/Mass/MassUniquenessSeparation.lean @@ -1,124 +1,124 @@ -import Mathlib.Analysis.Calculus.MeanValue -import Mathlib.Analysis.Calculus.Deriv.Pow -import Mathlib.Analysis.Calculus.Deriv.Inv -import Mathlib.Analysis.SpecialFunctions.Pow.Real - -namespace OAI - -/-! A bounded smooth test detecting a nonzero continuous finite-variation -martingale. Its bounded derivatives permit an integrable-variation argument. -/ - -noncomputable section -open Set -open scoped Topology - -namespace SLEExactGauge - -def massSeparationTest (x : ℝ) : ℝ := x^2 / (1+x^2) -def massSeparationSlope (x : ℝ) : ℝ := 2*x / (1+x^2)^2 - -theorem massSeparationTest_nonneg (x : ℝ) : 0 ≤ massSeparationTest x := by - unfold massSeparationTest - positivity - -@[simp] theorem massSeparationTest_zero : massSeparationTest 0 = 0 := by - norm_num [massSeparationTest] - -@[simp] theorem massSeparationTest_eq_zero (x : ℝ) : massSeparationTest x = 0 ↔ x = 0 := by - have hp : 0 < 1+x^2 := by positivity - simp only [massSeparationTest,div_eq_zero_iff,hp.ne',or_false,pow_eq_zero_iff (by decide : (2:ℕ) ≠ 0)] - -theorem massSeparationTest_hasDerivAt (x : ℝ) : - HasDerivAt massSeparationTest (massSeparationSlope x) x := by - have hd := ((hasDerivAt_id x).pow 2).div - ((hasDerivAt_const x 1).add ((hasDerivAt_id x).pow 2)) (by positivity : 1+x^2 ≠ 0) - convert hd using 1 - · rfl - · dsimp only [massSeparationSlope,id_eq,Pi.pow_apply,Pi.add_apply,Pi.div_apply] - field_simp - ring - -theorem massSeparationSlope_hasDerivAt (x : ℝ) : - HasDerivAt massSeparationSlope ((2-6*x^2)/(1+x^2)^3) x := by - have hd := ((hasDerivAt_id x).const_mul 2).div - (((hasDerivAt_const x 1).add ((hasDerivAt_id x).pow 2)).pow 2) - (by positivity : (1+x^2)^2 ≠ 0) - convert hd using 1 - · rfl - · dsimp only [id_eq,Pi.pow_apply,Pi.add_apply,Pi.div_apply] - field_simp - ring - -theorem massSeparationSlope_abs_le (x : ℝ) : |massSeparationSlope x| ≤ 2 := by - have hy : 0 ≤ x^2 := sq_nonneg x - have hx : |x| ≤ 1+x^2 := by - nlinarith [sq_nonneg (|x|-1),sq_abs x] - have hp : 0 < 1+x^2 := by positivity - rw [massSeparationSlope,abs_div,abs_mul,abs_of_pos (sq_pos_of_pos hp)] - norm_num only [abs_of_pos (by norm_num : (0:ℝ)<2)] - apply (div_le_iff₀ (sq_pos_of_pos hp)).mpr - nlinarith [sq_nonneg (x^2)] - -theorem massSeparationSlope_deriv_abs_le (x : ℝ) : - |(2-6*x^2)/(1+x^2)^3| ≤ 8 := by - have hp : 0 < 1+x^2 := by positivity - rw [abs_div,abs_of_pos (pow_pos hp 3)] - apply (div_le_iff₀ (pow_pos hp 3)).mpr - have hn : |2-6*x^2| ≤ 2+6*x^2 := by - have hh := abs_sub (2:ℝ) (6*x^2) - norm_num [abs_mul,abs_of_nonneg (sq_nonneg x)] at hh ⊢ - exact hh - have hb : 2+6*x^2 ≤ 8*(1+x^2)^3 := by - nlinarith [sq_nonneg (x^2),pow_nonneg (sq_nonneg x) 3] - exact hn.trans hb - -theorem massSeparationTest_lipschitz (x y : ℝ) : - |massSeparationTest y-massSeparationTest x| ≤ 2*|y-x| := by - have h := Convex.norm_image_sub_le_of_norm_deriv_le - (s := (univ:Set ℝ)) (f := massSeparationTest) (C := 2) (x := x) (y := y) - (fun z _ => (massSeparationTest_hasDerivAt z).differentiableAt) - (fun z _ => by simpa only [(massSeparationTest_hasDerivAt z).deriv,Real.norm_eq_abs] - using massSeparationSlope_abs_le z) convex_univ (mem_univ x) (mem_univ y) - simpa only [Real.norm_eq_abs] using h - -theorem massSeparationSlope_lipschitz (x y : ℝ) : - |massSeparationSlope y-massSeparationSlope x| ≤ 8*|y-x| := by - have h := Convex.norm_image_sub_le_of_norm_deriv_le - (s := (univ:Set ℝ)) (f := massSeparationSlope) (C := 8) (x := x) (y := y) - (fun z _ => (massSeparationSlope_hasDerivAt z).differentiableAt) - (fun z _ => by simpa only [(massSeparationSlope_hasDerivAt z).deriv,Real.norm_eq_abs] - using massSeparationSlope_deriv_abs_le z) convex_univ (mem_univ x) (mem_univ y) - simpa only [Real.norm_eq_abs] using h - -theorem massSeparationTest_remainder_le (x y : ℝ) : - |massSeparationTest y-massSeparationTest x-massSeparationSlope x*(y-x)| ≤ 8*|y-x|^2 := by - let f (z : ℝ) := massSeparationTest z-massSeparationTest x-massSeparationSlope x*(z-x) - have hd (z : ℝ) : HasDerivAt f (massSeparationSlope z-massSeparationSlope x) z := by - convert ((massSeparationTest_hasDerivAt z).sub_const - (massSeparationTest x)).sub (((hasDerivAt_id z).sub_const x).const_mul (massSeparationSlope x)) using 1 - · rfl - · ring - have h := Convex.norm_image_sub_le_of_norm_deriv_le - (s := Metric.closedBall x |y-x|) (f := f) (C := 8*|y-x|) (x := x) (y := y) - (fun z _ => (hd z).differentiableAt) - (fun z hz => by - rw [(hd z).deriv,Real.norm_eq_abs] - exact (massSeparationSlope_lipschitz x z).trans - (mul_le_mul_of_nonneg_left (by simpa only [Metric.mem_closedBall,Real.dist_eq] using hz) (by norm_num))) - (convex_closedBall x _) (Metric.mem_closedBall_self (abs_nonneg _)) - (by simp only [Metric.mem_closedBall,Real.dist_eq,le_refl]) - simpa only [f,sub_self,mul_zero,sub_zero,Real.norm_eq_abs,pow_two,mul_assoc] using h - -theorem massSeparationTest_remainder_linear (x y : ℝ) : - |massSeparationTest y-massSeparationTest x-massSeparationSlope x*(y-x)| ≤ 4*|y-x| := by - calc - _ ≤ |massSeparationTest y-massSeparationTest x|+|massSeparationSlope x*(y-x)| := abs_sub _ _ - _ ≤ 2*|y-x|+2*|y-x| := add_le_add (massSeparationTest_lipschitz x y) - (by rw [abs_mul]; exact mul_le_mul_of_nonneg_right (massSeparationSlope_abs_le x) (abs_nonneg _)) - _ = _ := by ring - -end SLEExactGauge - -end - -end OAI +import Mathlib.Analysis.Calculus.MeanValue +import Mathlib.Analysis.Calculus.Deriv.Pow +import Mathlib.Analysis.Calculus.Deriv.Inv +import Mathlib.Analysis.SpecialFunctions.Pow.Real + +namespace OAI + +/-! A bounded smooth test detecting a nonzero continuous finite-variation +martingale. Its bounded derivatives permit an integrable-variation argument. -/ + +noncomputable section +open Set +open scoped Topology + +namespace SLEExactGauge + +def massSeparationTest (x : ℝ) : ℝ := x^2 / (1+x^2) +def massSeparationSlope (x : ℝ) : ℝ := 2*x / (1+x^2)^2 + +theorem massSeparationTest_nonneg (x : ℝ) : 0 ≤ massSeparationTest x := by + unfold massSeparationTest + positivity + +@[simp] theorem massSeparationTest_zero : massSeparationTest 0 = 0 := by + norm_num [massSeparationTest] + +@[simp] theorem massSeparationTest_eq_zero (x : ℝ) : massSeparationTest x = 0 ↔ x = 0 := by + have hp : 0 < 1+x^2 := by positivity + simp only [massSeparationTest,div_eq_zero_iff,hp.ne',or_false,pow_eq_zero_iff (by decide : (2:ℕ) ≠ 0)] + +theorem massSeparationTest_hasDerivAt (x : ℝ) : + HasDerivAt massSeparationTest (massSeparationSlope x) x := by + have hd := ((hasDerivAt_id x).pow 2).div + ((hasDerivAt_const x 1).add ((hasDerivAt_id x).pow 2)) (by positivity : 1+x^2 ≠ 0) + convert hd using 1 + · rfl + · dsimp only [massSeparationSlope,id_eq,Pi.pow_apply,Pi.add_apply,Pi.div_apply] + field_simp + ring + +theorem massSeparationSlope_hasDerivAt (x : ℝ) : + HasDerivAt massSeparationSlope ((2-6*x^2)/(1+x^2)^3) x := by + have hd := ((hasDerivAt_id x).const_mul 2).div + (((hasDerivAt_const x 1).add ((hasDerivAt_id x).pow 2)).pow 2) + (by positivity : (1+x^2)^2 ≠ 0) + convert hd using 1 + · rfl + · dsimp only [id_eq,Pi.pow_apply,Pi.add_apply,Pi.div_apply] + field_simp + ring + +theorem massSeparationSlope_abs_le (x : ℝ) : |massSeparationSlope x| ≤ 2 := by + have hy : 0 ≤ x^2 := sq_nonneg x + have hx : |x| ≤ 1+x^2 := by + nlinarith [sq_nonneg (|x|-1),sq_abs x] + have hp : 0 < 1+x^2 := by positivity + rw [massSeparationSlope,abs_div,abs_mul,abs_of_pos (sq_pos_of_pos hp)] + norm_num only [abs_of_pos (by norm_num : (0:ℝ)<2)] + apply (div_le_iff₀ (sq_pos_of_pos hp)).mpr + nlinarith [sq_nonneg (x^2)] + +theorem massSeparationSlope_deriv_abs_le (x : ℝ) : + |(2-6*x^2)/(1+x^2)^3| ≤ 8 := by + have hp : 0 < 1+x^2 := by positivity + rw [abs_div,abs_of_pos (pow_pos hp 3)] + apply (div_le_iff₀ (pow_pos hp 3)).mpr + have hn : |2-6*x^2| ≤ 2+6*x^2 := by + have hh := abs_sub (2:ℝ) (6*x^2) + norm_num [abs_mul,abs_of_nonneg (sq_nonneg x)] at hh ⊢ + exact hh + have hb : 2+6*x^2 ≤ 8*(1+x^2)^3 := by + nlinarith [sq_nonneg (x^2),pow_nonneg (sq_nonneg x) 3] + exact hn.trans hb + +theorem massSeparationTest_lipschitz (x y : ℝ) : + |massSeparationTest y-massSeparationTest x| ≤ 2*|y-x| := by + have h := Convex.norm_image_sub_le_of_norm_deriv_le + (s := (univ:Set ℝ)) (f := massSeparationTest) (C := 2) (x := x) (y := y) + (fun z _ => (massSeparationTest_hasDerivAt z).differentiableAt) + (fun z _ => by simpa only [(massSeparationTest_hasDerivAt z).deriv,Real.norm_eq_abs] + using massSeparationSlope_abs_le z) convex_univ (mem_univ x) (mem_univ y) + simpa only [Real.norm_eq_abs] using h + +theorem massSeparationSlope_lipschitz (x y : ℝ) : + |massSeparationSlope y-massSeparationSlope x| ≤ 8*|y-x| := by + have h := Convex.norm_image_sub_le_of_norm_deriv_le + (s := (univ:Set ℝ)) (f := massSeparationSlope) (C := 8) (x := x) (y := y) + (fun z _ => (massSeparationSlope_hasDerivAt z).differentiableAt) + (fun z _ => by simpa only [(massSeparationSlope_hasDerivAt z).deriv,Real.norm_eq_abs] + using massSeparationSlope_deriv_abs_le z) convex_univ (mem_univ x) (mem_univ y) + simpa only [Real.norm_eq_abs] using h + +theorem massSeparationTest_remainder_le (x y : ℝ) : + |massSeparationTest y-massSeparationTest x-massSeparationSlope x*(y-x)| ≤ 8*|y-x|^2 := by + let f (z : ℝ) := massSeparationTest z-massSeparationTest x-massSeparationSlope x*(z-x) + have hd (z : ℝ) : HasDerivAt f (massSeparationSlope z-massSeparationSlope x) z := by + convert ((massSeparationTest_hasDerivAt z).sub_const + (massSeparationTest x)).sub (((hasDerivAt_id z).sub_const x).const_mul (massSeparationSlope x)) using 1 + · rfl + · ring + have h := Convex.norm_image_sub_le_of_norm_deriv_le + (s := Metric.closedBall x |y-x|) (f := f) (C := 8*|y-x|) (x := x) (y := y) + (fun z _ => (hd z).differentiableAt) + (fun z hz => by + rw [(hd z).deriv,Real.norm_eq_abs] + exact (massSeparationSlope_lipschitz x z).trans + (mul_le_mul_of_nonneg_left (by simpa only [Metric.mem_closedBall,Real.dist_eq] using hz) (by norm_num))) + (convex_closedBall x _) (Metric.mem_closedBall_self (abs_nonneg _)) + (by simp only [Metric.mem_closedBall,Real.dist_eq,le_refl]) + simpa only [f,sub_self,mul_zero,sub_zero,Real.norm_eq_abs,pow_two,mul_assoc] using h + +theorem massSeparationTest_remainder_linear (x y : ℝ) : + |massSeparationTest y-massSeparationTest x-massSeparationSlope x*(y-x)| ≤ 4*|y-x| := by + calc + _ ≤ |massSeparationTest y-massSeparationTest x|+|massSeparationSlope x*(y-x)| := abs_sub _ _ + _ ≤ 2*|y-x|+2*|y-x| := add_le_add (massSeparationTest_lipschitz x y) + (by rw [abs_mul]; exact mul_le_mul_of_nonneg_right (massSeparationSlope_abs_le x) (abs_nonneg _)) + _ = _ := by ring + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MomentLevels.lean b/lean/OAI/Probability/SLE/Mass/MomentLevels.lean index 1c44d6f5156901b6e8f582fa923f7ecc88b5f005..1e7375de459df209971538cb596fd6729aa22109 100644 --- a/lean/OAI/Probability/SLE/Mass/MomentLevels.lean +++ b/lean/OAI/Probability/SLE/Mass/MomentLevels.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.MomentGauge - -namespace OAI - -/-! -The prescribed retained level in the first paper's moment-batch construction. -Its definition counts bad positive integer levels. Monotonicity makes those -levels an initial segment, so one plus their cardinal is a good level. -The moment convergence hypotheses here are explicit internal proof inputs. --/ - -noncomputable section -open Set Filter -open scoped Topology - -namespace SLEExactGauge.Target - -theorem batchLevel_ge_one (κ : ℝ) (W : WienerLaw) (n : ℕ) : - 1 ≤ batchLevel κ W n := by - unfold batchLevel - omega - -theorem finite_bad_moment_levels (κ : ℝ) (W : WienerLaw) (n : ℕ) - (hm : 0 < moment κ W n) - (hlim : Tendsto (fun j : ℕ => finiteMoment κ W n j) atTop (𝓝 (moment κ W n))) : - {j : ℕ | 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4}.Finite := by - have hlt : 3 * moment κ W n / 4 < moment κ W n := by linarith - have hevent : ∀ᶠ j : ℕ in atTop, 3 * moment κ W n / 4 < finiteMoment κ W n j := - (tendsto_order.mp hlim).1 _ hlt - obtain ⟨N, hN⟩ := eventually_atTop.mp hevent - apply (finite_Iio N).subset - intro j hj - show j < N - by_contra hjN - exact (not_le_of_gt (hN j (le_of_not_gt hjN))) hj.2 - -/-- The actual `Target.batchLevel`, not a separately chosen large level, -retains more than three quarters of the limiting moment. -/ -theorem finiteMoment_batchLevel_gt (κ : ℝ) (W : WienerLaw) (n : ℕ) - (hmono : Monotone (fun j : ℕ => finiteMoment κ W n j)) - (hm : 0 < moment κ W n) - (hlim : Tendsto (fun j : ℕ => finiteMoment κ W n j) atTop (𝓝 (moment κ W n))) : - 3 * moment κ W n / 4 < finiteMoment κ W n (batchLevel κ W n) := by - classical - let B := {j : ℕ // 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4} - let : Finite B := (finite_bad_moment_levels κ W n hm hlim).to_subtype - by_contra hbad - have hbad' : finiteMoment κ W n (batchLevel κ W n) ≤ 3 * moment κ W n / 4 := - le_of_not_gt hbad - let f : Fin (batchLevel κ W n) → B := fun j => - ⟨(j : ℕ) + 1, by - refine ⟨by omega, ?_⟩ - exact (hmono (Nat.succ_le_of_lt j.isLt)).trans hbad'⟩ - have hinj : Function.Injective f := by - intro i j hij - apply Fin.ext - have hval := congrArg Subtype.val hij - dsimp only [f] at hval - omega - have hcard := Nat.card_le_card_of_injective f hinj - rw [Nat.card_fin] at hcard - change 1 + Nat.card B ≤ Nat.card B at hcard - omega - -theorem batchLevel_retains_moment (κ : ℝ) (W : WienerLaw) (n : ℕ) - (hmono : Monotone (fun j : ℕ => finiteMoment κ W n j)) - (hm : 0 < moment κ W n) - (hlim : Tendsto (fun j : ℕ => finiteMoment κ W n j) atTop (𝓝 (moment κ W n))) : - 1 ≤ batchLevel κ W n ∧ - 3 * moment κ W n / 4 < finiteMoment κ W n (batchLevel κ W n) := - ⟨batchLevel_ge_one κ W n, finiteMoment_batchLevel_gt κ W n hmono hm hlim⟩ - -end SLEExactGauge.Target - -end - -end OAI +import OAI.Probability.SLE.MomentGauge + +namespace OAI + +/-! +The prescribed retained level in the first paper's moment-batch construction. +Its definition counts bad positive integer levels. Monotonicity makes those +levels an initial segment, so one plus their cardinal is a good level. +The moment convergence hypotheses here are explicit internal proof inputs. +-/ + +noncomputable section +open Set Filter +open scoped Topology + +namespace SLEExactGauge.Target + +theorem batchLevel_ge_one (κ : ℝ) (W : WienerLaw) (n : ℕ) : + 1 ≤ batchLevel κ W n := by + unfold batchLevel + omega + +theorem finite_bad_moment_levels (κ : ℝ) (W : WienerLaw) (n : ℕ) + (hm : 0 < moment κ W n) + (hlim : Tendsto (fun j : ℕ => finiteMoment κ W n j) atTop (𝓝 (moment κ W n))) : + {j : ℕ | 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4}.Finite := by + have hlt : 3 * moment κ W n / 4 < moment κ W n := by linarith + have hevent : ∀ᶠ j : ℕ in atTop, 3 * moment κ W n / 4 < finiteMoment κ W n j := + (tendsto_order.mp hlim).1 _ hlt + obtain ⟨N, hN⟩ := eventually_atTop.mp hevent + apply (finite_Iio N).subset + intro j hj + show j < N + by_contra hjN + exact (not_le_of_gt (hN j (le_of_not_gt hjN))) hj.2 + +/-- The actual `Target.batchLevel`, not a separately chosen large level, +retains more than three quarters of the limiting moment. -/ +theorem finiteMoment_batchLevel_gt (κ : ℝ) (W : WienerLaw) (n : ℕ) + (hmono : Monotone (fun j : ℕ => finiteMoment κ W n j)) + (hm : 0 < moment κ W n) + (hlim : Tendsto (fun j : ℕ => finiteMoment κ W n j) atTop (𝓝 (moment κ W n))) : + 3 * moment κ W n / 4 < finiteMoment κ W n (batchLevel κ W n) := by + classical + let B := {j : ℕ // 1 ≤ j ∧ finiteMoment κ W n j ≤ 3 * moment κ W n / 4} + let : Finite B := (finite_bad_moment_levels κ W n hm hlim).to_subtype + by_contra hbad + have hbad' : finiteMoment κ W n (batchLevel κ W n) ≤ 3 * moment κ W n / 4 := + le_of_not_gt hbad + let f : Fin (batchLevel κ W n) → B := fun j => + ⟨(j : ℕ) + 1, by + refine ⟨by omega, ?_⟩ + exact (hmono (Nat.succ_le_of_lt j.isLt)).trans hbad'⟩ + have hinj : Function.Injective f := by + intro i j hij + apply Fin.ext + have hval := congrArg Subtype.val hij + dsimp only [f] at hval + omega + have hcard := Nat.card_le_card_of_injective f hinj + rw [Nat.card_fin] at hcard + change 1 + Nat.card B ≤ Nat.card B at hcard + omega + +theorem batchLevel_retains_moment (κ : ℝ) (W : WienerLaw) (n : ℕ) + (hmono : Monotone (fun j : ℕ => finiteMoment κ W n j)) + (hm : 0 < moment κ W n) + (hlim : Tendsto (fun j : ℕ => finiteMoment κ W n j) atTop (𝓝 (moment κ W n))) : + 1 ≤ batchLevel κ W n ∧ + 3 * moment κ W n / 4 < finiteMoment κ W n (batchLevel κ W n) := + ⟨batchLevel_ge_one κ W n, finiteMoment_batchLevel_gt κ W n hmono hm hlim⟩ + +end SLEExactGauge.Target + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/MonotoneMassExhaustion.lean b/lean/OAI/Probability/SLE/Mass/MonotoneMassExhaustion.lean index a0085bbbc4da7988eb1d5e5a8252c30de74d5382..f6d1b9694a697df364e1bf7a4a09e9fbac958a2c 100644 --- a/lean/OAI/Probability/SLE/Mass/MonotoneMassExhaustion.lean +++ b/lean/OAI/Probability/SLE/Mass/MonotoneMassExhaustion.lean @@ -1,33 +1,33 @@ -import Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence -import Mathlib.Topology.Order.MonotoneConvergence - -namespace OAI - -/-! Nonnegative decreasing remaining masses vanish almost surely when -their expectations tend to zero. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped ENNReal Topology - -namespace SLEExactGauge - -theorem ae_tendsto_zero_of_antitone_lintegral - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (f : ℕ → Ω → ℝ≥0∞) - (hf : ∀ n, Measurable (f n)) (hanti : ∀ᵐ ω ∂P, Antitone (fun n => f n ω)) - (hmean : Tendsto (fun n => ∫⁻ ω, f n ω ∂P) atTop (𝓝 0)) : - ∀ᵐ ω ∂P, Tendsto (fun n => f n ω) atTop (𝓝 0) := by - have hi : (∫⁻ ω, liminf (fun n => f n ω) atTop ∂P) = 0 := by - apply le_zero_iff.mp - exact (lintegral_liminf_le hf).trans_eq hmean.liminf_eq - have hz := (lintegral_eq_zero_iff (Measurable.liminf hf)).mp hi - filter_upwards [hanti,hz] with ω ha hz - have ht := tendsto_atTop_iInf ha - have he : (⨅ n, f n ω) = 0 := ht.liminf_eq.symm.trans hz - rwa [he] at ht - -end SLEExactGauge - -end - -end OAI +import Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence +import Mathlib.Topology.Order.MonotoneConvergence + +namespace OAI + +/-! Nonnegative decreasing remaining masses vanish almost surely when +their expectations tend to zero. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped ENNReal Topology + +namespace SLEExactGauge + +theorem ae_tendsto_zero_of_antitone_lintegral + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (f : ℕ → Ω → ℝ≥0∞) + (hf : ∀ n, Measurable (f n)) (hanti : ∀ᵐ ω ∂P, Antitone (fun n => f n ω)) + (hmean : Tendsto (fun n => ∫⁻ ω, f n ω ∂P) atTop (𝓝 0)) : + ∀ᵐ ω ∂P, Tendsto (fun n => f n ω) atTop (𝓝 0) := by + have hi : (∫⁻ ω, liminf (fun n => f n ω) atTop ∂P) = 0 := by + apply le_zero_iff.mp + exact (lintegral_liminf_le hf).trans_eq hmean.liminf_eq + have hz := (lintegral_eq_zero_iff (Measurable.liminf hf)).mp hi + filter_upwards [hanti,hz] with ω ha hz + have ht := tendsto_atTop_iInf ha + have he : (⨅ n, f n ω) = 0 := ht.liminf_eq.symm.trans hz + rwa [he] at ht + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/NonnegativeL2Passage.lean b/lean/OAI/Probability/SLE/Mass/NonnegativeL2Passage.lean index eb89ec6a62d1409302d5e5f51af74b64d4952715..b2940443784a031233f4ea75497c0e0c67f20430 100644 --- a/lean/OAI/Probability/SLE/Mass/NonnegativeL2Passage.lean +++ b/lean/OAI/Probability/SLE/Mass/NonnegativeL2Passage.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.ExponentialMartingales - -namespace OAI - -/-! Uniform second moments of terminal tests also control every smaller -nonnegative test, including tests evaluated at random times. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped ENNReal Topology - -namespace SLEExactGauge - -lemma memLp_two_of_nonnegative_square_bound - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (X : Ω → ℝ) - (hm : Measurable X) (hX : ∀ ω, 0 ≤ X ω) - (M : ℝ≥0∞) (hM : M < ∞) - (hb : (∫⁻ ω, ENNReal.ofReal (X ω)^2 ∂P) ≤ M) : - MemLp X 2 P ∧ (∫ ω, X ω^2 ∂P) ≤ M.toReal := by - have hi : Integrable (fun ω => X ω^2) P := by - refine ⟨(hm.pow_const 2).aestronglyMeasurable,?_⟩ - apply (hasFiniteIntegral_iff_ofReal (ae_of_all _ (fun ω => sq_nonneg (X ω)))).mpr - simpa only [ENNReal.ofReal_pow (hX _)] using hb.trans_lt hM - have hmem := (memLp_two_iff_integrable_sq hm.aestronglyMeasurable).mpr hi - refine ⟨hmem,?_⟩ - have he : ENNReal.ofReal (∫ ω, X ω^2 ∂P) ≤ M := by - rw [ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun ω => sq_nonneg (X ω)))] - simpa only [ENNReal.ofReal_pow (hX _)] using hb - have ht := ENNReal.toReal_mono hM.ne he - rwa [ENNReal.toReal_ofReal (integral_nonneg (fun ω => sq_nonneg (X ω)))] at ht - -theorem integral_tendsto_of_nonnegative_L2_domination - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] - (X Z : ℕ → Ω → ℝ) (Y : Ω → ℝ) - (hXm : ∀ n, Measurable (X n)) (hX0 : ∀ n ω, 0 ≤ X n ω) - (hXZ : ∀ n ω, X n ω ≤ Z n ω) (hZ : ∀ n, MemLp (Z n) 2 P) - (A : ℝ) (hA : ∀ n, (∫ ω, Z n ω^2 ∂P) ≤ A) - (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) : - Integrable Y P ∧ - Tendsto (fun n => ∫ ω, X n ω ∂P) atTop (𝓝 (∫ ω, Y ω ∂P)) := by - have hX (n : ℕ) : MemLp (X n) 2 P := - (hZ n).mono' (hXm n).aestronglyMeasurable - (ae_of_all _ (fun ω => by simpa only [Real.norm_of_nonneg (hX0 n ω)] using hXZ n ω)) - have hsq (n : ℕ) : (∫ ω, X n ω^2 ∂P) ≤ A := by - apply (integral_mono (hX n).integrable_sq (hZ n).integrable_sq ?_).trans (hA n) - intro ω - exact pow_le_pow_left₀ (hX0 n ω) (hXZ n ω) 2 - obtain ⟨hY,hL1⟩ := finite_L1_tendsto_of_square_bound hX A hsq hlim - exact ⟨hY,tendsto_integral_of_L1' Y - (Eventually.of_forall (fun n => (hX n).integrable (by norm_num))) hL1⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.ExponentialMartingales + +namespace OAI + +/-! Uniform second moments of terminal tests also control every smaller +nonnegative test, including tests evaluated at random times. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped ENNReal Topology + +namespace SLEExactGauge + +lemma memLp_two_of_nonnegative_square_bound + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (X : Ω → ℝ) + (hm : Measurable X) (hX : ∀ ω, 0 ≤ X ω) + (M : ℝ≥0∞) (hM : M < ∞) + (hb : (∫⁻ ω, ENNReal.ofReal (X ω)^2 ∂P) ≤ M) : + MemLp X 2 P ∧ (∫ ω, X ω^2 ∂P) ≤ M.toReal := by + have hi : Integrable (fun ω => X ω^2) P := by + refine ⟨(hm.pow_const 2).aestronglyMeasurable,?_⟩ + apply (hasFiniteIntegral_iff_ofReal (ae_of_all _ (fun ω => sq_nonneg (X ω)))).mpr + simpa only [ENNReal.ofReal_pow (hX _)] using hb.trans_lt hM + have hmem := (memLp_two_iff_integrable_sq hm.aestronglyMeasurable).mpr hi + refine ⟨hmem,?_⟩ + have he : ENNReal.ofReal (∫ ω, X ω^2 ∂P) ≤ M := by + rw [ofReal_integral_eq_lintegral_ofReal hi (ae_of_all _ (fun ω => sq_nonneg (X ω)))] + simpa only [ENNReal.ofReal_pow (hX _)] using hb + have ht := ENNReal.toReal_mono hM.ne he + rwa [ENNReal.toReal_ofReal (integral_nonneg (fun ω => sq_nonneg (X ω)))] at ht + +theorem integral_tendsto_of_nonnegative_L2_domination + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsFiniteMeasure P] + (X Z : ℕ → Ω → ℝ) (Y : Ω → ℝ) + (hXm : ∀ n, Measurable (X n)) (hX0 : ∀ n ω, 0 ≤ X n ω) + (hXZ : ∀ n ω, X n ω ≤ Z n ω) (hZ : ∀ n, MemLp (Z n) 2 P) + (A : ℝ) (hA : ∀ n, (∫ ω, Z n ω^2 ∂P) ≤ A) + (hlim : ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 (Y ω))) : + Integrable Y P ∧ + Tendsto (fun n => ∫ ω, X n ω ∂P) atTop (𝓝 (∫ ω, Y ω ∂P)) := by + have hX (n : ℕ) : MemLp (X n) 2 P := + (hZ n).mono' (hXm n).aestronglyMeasurable + (ae_of_all _ (fun ω => by simpa only [Real.norm_of_nonneg (hX0 n ω)] using hXZ n ω)) + have hsq (n : ℕ) : (∫ ω, X n ω^2 ∂P) ≤ A := by + apply (integral_mono (hX n).integrable_sq (hZ n).integrable_sq ?_).trans (hA n) + intro ω + exact pow_le_pow_left₀ (hX0 n ω) (hXZ n ω) 2 + obtain ⟨hY,hL1⟩ := finite_L1_tendsto_of_square_bound hX A hsq hlim + exact ⟨hY,tendsto_integral_of_L1' Y + (Eventually.of_forall (fun n => (hX n).integrable (by norm_num))) hL1⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/NormalizedDiskCompensator.lean b/lean/OAI/Probability/SLE/Mass/NormalizedDiskCompensator.lean index ff7d5699ec06f6824c83eea0e515783477c6c20f..40b15bb3e7465b7780d7c53927b69a1869331d6f 100644 --- a/lean/OAI/Probability/SLE/Mass/NormalizedDiskCompensator.lean +++ b/lean/OAI/Probability/SLE/Mass/NormalizedDiskCompensator.lean @@ -1,92 +1,92 @@ -import OAI.Probability.SLE.Geometry.CanonicalNormalizedPotential -import OAI.Probability.SLE.Mass.CanonicalMassCompensator - -namespace OAI - -/-! Recovery of the actual canonical disk mass on the exact target -Brownian-time dyadic grid. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint.CanonicalMass - -variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} - {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) - -def normalizedDiskScalar (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (t : ℝ≥0) (ω : Ω) : ℝ := C.diskScalar θ hθ (t/κ.toNNReal) ω - -lemma normalizedDiskScalar_measurable (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (t : ℝ≥0) : Measurable (C.normalizedDiskScalar θ hθ t) := - (compactWeightedMass_measurable _ _ - (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp - C.progressive.adapted.measurable - -lemma normalizedDiskScalar_integrable (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (t : ℝ≥0) : Integrable (C.normalizedDiskScalar θ hθ t) P := by - exact (compactWeightedMass_kernel_integrable P κ (C.process (t/κ.toNNReal)) - C.progressive.adapted.measurable - (by simpa only [Measure.restrict_univ] using - C.uniqueness_process_mean_le (t/κ.toNNReal) univ) - _ _ (Target.diskSourceHalfPlane_compact hθ)).1 - -lemma normalizedDiskScalar_stronglyAdapted (hκ : 0<κ) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) : - StronglyAdapted (Target.normalizedFiltration κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) - (C.normalizedDiskScalar θ hθ) := by - intro t - erw [Target.normalizedFiltration_at κ hκ] - exact ((compactWeightedMass_measurable _ _ - (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp - (C.progressive.adapted (t/κ.toNNReal))).stronglyMeasurable - -theorem normalizedDiskScalar_paths (hκ : 0<κ) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) : - ∀ᵐ ω ∂P, Monotone (fun t => C.normalizedDiskScalar θ hθ t ω) ∧ - C.normalizedDiskScalar θ hθ 0 ω=0 ∧ - Continuous (fun t => C.normalizedDiskScalar θ hθ t ω) ∧ - Tendsto (fun t => C.normalizedDiskScalar θ hθ t ω) atTop - (𝓝 (C.diskTerminalScalar θ hθ ω)) ∧ - ∀ t, C.normalizedDiskScalar θ hθ t ω ≤ C.diskTerminalScalar θ hθ ω := by - filter_upwards [C.compactWeightedMass_paths (Target.diskMassWeight κ θ hθ) - (Target.diskSourceHalfPlane θ) (Target.diskSourceHalfPlane_compact hθ)] with ω hω - refine ⟨?_,?_,hω.2.2.1.comp (continuous_id.div_const _),?_,fun t => hω.2.2.2.2 _⟩ - · intro s t hst - exact hω.1 (div_le_div_of_nonneg_right hst zero_le) - · simpa only [normalizedDiskScalar,diskScalar,zero_div] using hω.2.1 - · exact hω.2.2.2.1.comp - ((tendsto_div_const_atTop_of_pos (Real.toNNReal_pos.mpr hκ)).mpr tendsto_id) - -theorem normalized_disk_compensator_recovery [IsProbabilityMeasure P] - (hκ : 0<κ) (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : Ω → ℝ≥0∞) (hT : Measurable T) : - TendstoInMeasure P - (fun n ω => meshMassCompensator P - (compensatorMeshFiltration (Target.normalizedFiltration κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0)) - (C.normalizedDiskScalar θ hθ) n (compensatorStopIndex n (T ω)) ω) atTop - (fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) - (C.diskTerminalScalar θ hθ ω) (T ω)) := by - apply meshMassCompensator_stopped_tendstoInMeasure P _ _ _ _ _ - (C.compactWeightedMass_terminal_memLp _ _ (Target.diskSourceHalfPlane_compact hθ)) - (C.normalizedDiskScalar_paths hκ θ hθ) T hT - · intro t - exact C.normalizedDiskScalar_measurable θ hθ t - · intro n i - change StronglyMeasurable[Target.normalizedFiltration κ B - (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) - hc h0 (compensatorMeshTime n i)] _ - exact C.normalizedDiskScalar_stronglyAdapted hκ hc h0 θ hθ _ - -end SLEExactGauge.Multipoint.CanonicalMass - -end - -end OAI +import OAI.Probability.SLE.Geometry.CanonicalNormalizedPotential +import OAI.Probability.SLE.Mass.CanonicalMassCompensator + +namespace OAI + +/-! Recovery of the actual canonical disk mass on the exact target +Brownian-time dyadic grid. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint.CanonicalMass + +variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} + {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) + +def normalizedDiskScalar (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (t : ℝ≥0) (ω : Ω) : ℝ := C.diskScalar θ hθ (t/κ.toNNReal) ω + +lemma normalizedDiskScalar_measurable (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (t : ℝ≥0) : Measurable (C.normalizedDiskScalar θ hθ t) := + (compactWeightedMass_measurable _ _ + (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp + C.progressive.adapted.measurable + +lemma normalizedDiskScalar_integrable (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (t : ℝ≥0) : Integrable (C.normalizedDiskScalar θ hθ t) P := by + exact (compactWeightedMass_kernel_integrable P κ (C.process (t/κ.toNNReal)) + C.progressive.adapted.measurable + (by simpa only [Measure.restrict_univ] using + C.uniqueness_process_mean_le (t/κ.toNNReal) univ) + _ _ (Target.diskSourceHalfPlane_compact hθ)).1 + +lemma normalizedDiskScalar_stronglyAdapted (hκ : 0<κ) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) : + StronglyAdapted (Target.normalizedFiltration κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0) + (C.normalizedDiskScalar θ hθ) := by + intro t + erw [Target.normalizedFiltration_at κ hκ] + exact ((compactWeightedMass_measurable _ _ + (Target.diskSourceHalfPlane_compact hθ).measurableSet).comp + (C.progressive.adapted (t/κ.toNNReal))).stronglyMeasurable + +theorem normalizedDiskScalar_paths (hκ : 0<κ) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) : + ∀ᵐ ω ∂P, Monotone (fun t => C.normalizedDiskScalar θ hθ t ω) ∧ + C.normalizedDiskScalar θ hθ 0 ω=0 ∧ + Continuous (fun t => C.normalizedDiskScalar θ hθ t ω) ∧ + Tendsto (fun t => C.normalizedDiskScalar θ hθ t ω) atTop + (𝓝 (C.diskTerminalScalar θ hθ ω)) ∧ + ∀ t, C.normalizedDiskScalar θ hθ t ω ≤ C.diskTerminalScalar θ hθ ω := by + filter_upwards [C.compactWeightedMass_paths (Target.diskMassWeight κ θ hθ) + (Target.diskSourceHalfPlane θ) (Target.diskSourceHalfPlane_compact hθ)] with ω hω + refine ⟨?_,?_,hω.2.2.1.comp (continuous_id.div_const _),?_,fun t => hω.2.2.2.2 _⟩ + · intro s t hst + exact hω.1 (div_le_div_of_nonneg_right hst zero_le) + · simpa only [normalizedDiskScalar,diskScalar,zero_div] using hω.2.1 + · exact hω.2.2.2.1.comp + ((tendsto_div_const_atTop_of_pos (Real.toNNReal_pos.mpr hκ)).mpr tendsto_id) + +theorem normalized_disk_compensator_recovery [IsProbabilityMeasure P] + (hκ : 0<κ) (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) (T : Ω → ℝ≥0∞) (hT : Measurable T) : + TendstoInMeasure P + (fun n ω => meshMassCompensator P + (compensatorMeshFiltration (Target.normalizedFiltration κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) hc h0)) + (C.normalizedDiskScalar θ hθ) n (compensatorStopIndex n (T ω)) ω) atTop + (fun ω => massAtExtendedTime (fun t => C.normalizedDiskScalar θ hθ t ω) + (C.diskTerminalScalar θ hθ ω) (T ω)) := by + apply meshMassCompensator_stopped_tendstoInMeasure P _ _ _ _ _ + (C.compactWeightedMass_terminal_memLp _ _ (Target.diskSourceHalfPlane_compact hθ)) + (C.normalizedDiskScalar_paths hκ θ hθ) T hT + · intro t + exact C.normalizedDiskScalar_measurable θ hθ t + · intro n i + change StronglyMeasurable[Target.normalizedFiltration κ B + (fun r => (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r))) + hc h0 (compensatorMeshTime n i)] _ + exact C.normalizedDiskScalar_stronglyAdapted hκ hc h0 θ hθ _ + +end SLEExactGauge.Multipoint.CanonicalMass + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/OriginalShiftedMass.lean b/lean/OAI/Probability/SLE/Mass/OriginalShiftedMass.lean index 5d36dcf1fc2162d97449859bee51b17dba93932d..403d0d72c3be906e928328fc5c933df46c0f8b03 100644 --- a/lean/OAI/Probability/SLE/Mass/OriginalShiftedMass.lean +++ b/lean/OAI/Probability/SLE/Mass/OriginalShiftedMass.lean @@ -1,58 +1,58 @@ -import OAI.Probability.SLE.Mass.CanonicalMassRemaining -import OAI.Probability.SLE.Stochastic.ProgressiveStoppedTime - -namespace OAI - -/-! The original canonical mass evaluated after a finite stopping time. -Its remaining Green kernel is stated in the full shifted filtration. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Classical -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge.Multipoint - -lemma extendedCanonicalMass_finite_time - {Ω : Type*} (N : ℝ≥0 → Ω → Measure upperHalfPlane) - (ν : Ω → Measure upperHalfPlane) (τ : Ω → ℝ≥0) (ω : Ω) : - extendedCanonicalMass N ν (τ ω:ℝ≥0∞) ω=N (τ ω) ω := by - simp only [extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} - {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - (C : CanonicalMass P κ B hBm γ G) (τ : Ω → ℝ≥0) - (hτ : IsStoppingTime (Filtration.natural B (fun s => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - -include C τ hτ in -lemma CanonicalMass.original_shifted_adapted : - Adapted (shiftedStoppingFiltration hτ) (fun t ω => C.process (τ ω+t) ω) := - _root_.OAI.MeasureTheory.IsProgressive.shifted_adapted C.progressive τ hτ - -include C τ hτ in -lemma CanonicalMass.original_shifted_green_kernel - (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : - (P.restrict E).bind C.rational.terminal = - (P.restrict E).bind (fun ω => C.process (τ ω+t) ω)+ - (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) := by - have h := C.kernel (fun ω => ((τ ω+t:ℝ≥0):WithTop ℝ≥0)) - (stopping_time_add_nnreal hτ t) E hE - change (P.restrict E).bind C.rational.terminal = - (P.restrict E).bind (fun ω => C.process (τ ω+t) ω)+ - (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) at h - exact h - -include C τ hτ in -lemma CanonicalMass.original_shifted_mean_le (t : ℝ≥0) (E : Set Ω) : - (P.restrict E).bind (fun ω => C.process (τ ω+t) ω) ≤ initialGreenMeasure κ := by - have h := C.atTime_mean_le (fun ω => ((τ ω+t:ℝ≥0):ℝ≥0∞)) - (stopping_time_add_nnreal hτ t).measurable' E - exact h - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassRemaining +import OAI.Probability.SLE.Stochastic.ProgressiveStoppedTime + +namespace OAI + +/-! The original canonical mass evaluated after a finite stopping time. +Its remaining Green kernel is stated in the full shifted filtration. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Classical +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge.Multipoint + +lemma extendedCanonicalMass_finite_time + {Ω : Type*} (N : ℝ≥0 → Ω → Measure upperHalfPlane) + (ν : Ω → Measure upperHalfPlane) (τ : Ω → ℝ≥0) (ω : Ω) : + extendedCanonicalMass N ν (τ ω:ℝ≥0∞) ω=N (τ ω) ω := by + simp only [extendedCanonicalMass,ENNReal.coe_lt_top,ite_true,ENNReal.toNNReal_coe] + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {κ : ℝ} + {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + (C : CanonicalMass P κ B hBm γ G) (τ : Ω → ℝ≥0) + (hτ : IsStoppingTime (Filtration.natural B (fun s => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + +include C τ hτ in +lemma CanonicalMass.original_shifted_adapted : + Adapted (shiftedStoppingFiltration hτ) (fun t ω => C.process (τ ω+t) ω) := + _root_.OAI.MeasureTheory.IsProgressive.shifted_adapted C.progressive τ hτ + +include C τ hτ in +lemma CanonicalMass.original_shifted_green_kernel + (t : ℝ≥0) (E : Set Ω) (hE : MeasurableSet[shiftedStoppingFiltration hτ t] E) : + (P.restrict E).bind C.rational.terminal = + (P.restrict E).bind (fun ω => C.process (τ ω+t) ω)+ + (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) := by + have h := C.kernel (fun ω => ((τ ω+t:ℝ≥0):WithTop ℝ≥0)) + (stopping_time_add_nnreal hτ t) E hE + change (P.restrict E).bind C.rational.terminal = + (P.restrict E).bind (fun ω => C.process (τ ω+t) ω)+ + (P.restrict E).bind (fun ω => halfPlaneGreenMeasure κ B γ G (τ ω+t) ω) at h + exact h + +include C τ hτ in +lemma CanonicalMass.original_shifted_mean_le (t : ℝ≥0) (E : Set Ω) : + (P.restrict E).bind (fun ω => C.process (τ ω+t) ω) ≤ initialGreenMeasure κ := by + have h := C.atTime_mean_le (fun ω => ((τ ω+t:ℝ≥0):ℝ≥0∞)) + (stopping_time_add_nnreal hτ t).measurable' E + exact h + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/PastThresholdMass.lean b/lean/OAI/Probability/SLE/Mass/PastThresholdMass.lean index 1b017d4d9d99a6c6534bba601fbdda7166167d0f..b69f77d45a91ba176202ab92851bc3387ceafbab 100644 --- a/lean/OAI/Probability/SLE/Mass/PastThresholdMass.lean +++ b/lean/OAI/Probability/SLE/Mass/PastThresholdMass.lean @@ -1,63 +1,63 @@ -import OAI.Probability.SLE.Gauge.StrictPastCountable -import OAI.Probability.SLE.Geometry.WeightedCompactTerminal - -namespace OAI - -/-! Continuous mass crosses an intermediate level at a dense test time. -Consequently a cell with twice the threshold has at least half its mass -strictly after a successful past test. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal -attribute [local instance] Classical.propDecidable - -namespace SLEExactGauge.Multipoint - -lemma dense_time_between_mass_levels (q : ℕ → ℝ≥0) (hq : DenseRange q) - (f : ℝ≥0 → ℝ) (hf : Continuous f) (h0 : f 0=0) - (M H L : ℝ) (hH : 0 linarith) - have hopen : IsOpen (f ⁻¹' Ioo H L) := isOpen_Ioo.preimage hf - have hne : (f ⁻¹' Ioo H L).Nonempty := ⟨t,by change H linarith⟩ - exact hq.exists_mem_open hopen hne - -lemma past_threshold_detects_half - {X : Type*} [MeasurableSpace X] (μ : Measure X) (K : Set X) - (hfin : μ K≠∞) - (A : ℝ≥0 → Set X) (hA : ∀ t, MeasurableSet (A t)) - (q : ℕ → ℝ≥0) (hq : DenseRange q) (f : ℝ≥0 → ℝ) - (hf : Continuous f) (h0 : f 0=0) - (hlim : Tendsto f atTop (𝓝 (μ K).toReal)) - (hpast : ∀ n, μ (K ∩ A (q n))=ENNReal.ofReal (f (q n))) - (H : ℝ) (hH : 0 linarith) + have hopen : IsOpen (f ⁻¹' Ioo H L) := isOpen_Ioo.preimage hf + have hne : (f ⁻¹' Ioo H L).Nonempty := ⟨t,by change H linarith⟩ + exact hq.exists_mem_open hopen hne + +lemma past_threshold_detects_half + {X : Type*} [MeasurableSpace X] (μ : Measure X) (K : Set X) + (hfin : μ K≠∞) + (A : ℝ≥0 → Set X) (hA : ∀ t, MeasurableSet (A t)) + (q : ℕ → ℝ≥0) (hq : DenseRange q) (f : ℝ≥0 → ℝ) + (hf : Continuous f) (h0 : f 0=0) + (hlim : Tendsto f atTop (𝓝 (μ K).toReal)) + (hpast : ∀ n, μ (K ∩ A (q n))=ENNReal.ofReal (f (q n))) + (H : ℝ) (hH : 0 H z) - (fun z : upperHalfPlane => ENNReal.ofReal (‖deriv H z‖^dimension κ)) μ := by - have hmap : (fun z : upperHalfPlane => f (Target.diskMap θ z))= - (fun z : upperHalfPlane => H z) := funext (fun z => he z z.property) - have hder (z : upperHalfPlane) : deriv (f ∘ Target.diskMap θ) z=deriv H z := by - apply Filter.EventuallyEq.deriv_eq - filter_upwards [isOpen_upperHalfPlane.mem_nhds z.property] with w hw - exact he w hw - unfold diskChartMass - rw [hmap] - congr 1 - funext z - rw [hder] - -lemma diskToHalfPlane_real_affine (c a : ℝ) (z ξ : ℂ) - (hξ : ξ ∈ ball (0:ℂ) 1) : - diskToHalfPlane ((c:ℂ)+(a:ℂ)*z) ξ = - (c:ℂ)+(a:ℂ)*diskToHalfPlane z ξ := by - simp only [diskToHalfPlane,star_add,star_mul,Complex.star_def,Complex.conj_ofReal] - field_simp [disk_denominator_ne hξ] - ring - -lemma physicalDiskChart_normalized - (F G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) (c a θ : ℝ) - (he : G t z=(c:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I)) - {ξ : ℂ} (hξ : ξ ∈ ball (0:ℂ) 1) : - physicalDiskChart id F G t z ξ = - F t ((c:ℂ)+(a:ℂ)*Target.diskMapInv θ ξ) := by - change F t (diskToHalfPlane (G t z) ξ)=_ - rw [he,diskToHalfPlane_real_affine c a _ ξ hξ,← Target.diskMapInv_eq_diskToHalfPlane] - -lemma physicalDiskChart_mass_eq - (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (F G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) (c a : ℝ) - (he : G t z=(c:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I)) - (μ : Measure upperHalfPlane) : - diskChartMass κ θ (physicalDiskChart id F G t z) μ = - weightedImageMass (fun w : upperHalfPlane => F t ((c:ℂ)+(a:ℂ)*w)) - (fun w : upperHalfPlane => ENNReal.ofReal - (‖deriv (fun v : ℂ => F t ((c:ℂ)+(a:ℂ)*v)) w‖^dimension κ)) μ := by - apply diskChartMass_eq_of_coordinates κ θ (physicalDiskChart id F G t z) - (fun v : ℂ => F t ((c:ℂ)+(a:ℂ)*v)) - intro w hw - rw [physicalDiskChart_normalized F G t z c a θ he (Target.diskMap_maps hθ hw), - Target.diskMapInv_diskMap hθ hw] - -lemma IsCapacityTwoWitness.physical_quarter_mass_bounds - {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} - (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) - (κ : ℝ) (hκ : 0 < κ) (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) - (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) (a : ℝ) - (he : G t z=(U t:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I)) - (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : - let f := physicalDiskChart id F G t z - let ν := weightedImageMass (fun w : upperHalfPlane => F t ((U t:ℂ)+(a:ℂ)*w)) - (fun w : upperHalfPlane => ENNReal.ofReal - (‖deriv (fun v : ℂ => F t ((U t:ℂ)+(a:ℂ)*v)) w‖^dimension κ)) μ - let L := ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) - (Target.diskSourceHalfPlane θ) μ) - ENNReal.ofReal ((fullRadius G t z/16)^dimension κ)*L ≤ ν (f '' Target.diskK) ∧ - ν (f '' Target.diskK) ≤ ENNReal.ofReal ((16*fullRadius G t z)^dimension κ)*L := by - obtain ⟨hd,hi,_,hder,_⟩ := hw.physical_disk_chart id differentiableOn_id - (fun _ _ _ _ h => h) t z hz - have hder' : ‖deriv (physicalDiskChart id F G t z) 0‖=fullRadius G t z := by - simpa [physicalConformalRadius] using hder - have hb := diskChartMass_quarter_bounds hgrowth κ hκ θ hθ - (physicalDiskChart id F G t z) hd hi μ - rw [hder',physicalDiskChart_mass_eq κ θ hθ F G t z (U t) a he μ] at hb - exact hb - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.DiskChartMass +import OAI.Probability.SLE.Geometry.PhysicalCellDisk + +namespace OAI + +/-! The disk scalar controls the actual centered and rescaled inverse +Loewner pushforward. The marked angle is retained exactly. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory Metric +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge +open Multipoint + +lemma diskChartMass_eq_of_coordinates + (κ θ : ℝ) (f H : ℂ → ℂ) + (he : ∀ z ∈ upperHalfPlane, f (Target.diskMap θ z)=H z) + (μ : Measure upperHalfPlane) : + diskChartMass κ θ f μ = weightedImageMass (fun z : upperHalfPlane => H z) + (fun z : upperHalfPlane => ENNReal.ofReal (‖deriv H z‖^dimension κ)) μ := by + have hmap : (fun z : upperHalfPlane => f (Target.diskMap θ z))= + (fun z : upperHalfPlane => H z) := funext (fun z => he z z.property) + have hder (z : upperHalfPlane) : deriv (f ∘ Target.diskMap θ) z=deriv H z := by + apply Filter.EventuallyEq.deriv_eq + filter_upwards [isOpen_upperHalfPlane.mem_nhds z.property] with w hw + exact he w hw + unfold diskChartMass + rw [hmap] + congr 1 + funext z + rw [hder] + +lemma diskToHalfPlane_real_affine (c a : ℝ) (z ξ : ℂ) + (hξ : ξ ∈ ball (0:ℂ) 1) : + diskToHalfPlane ((c:ℂ)+(a:ℂ)*z) ξ = + (c:ℂ)+(a:ℂ)*diskToHalfPlane z ξ := by + simp only [diskToHalfPlane,star_add,star_mul,Complex.star_def,Complex.conj_ofReal] + field_simp [disk_denominator_ne hξ] + ring + +lemma physicalDiskChart_normalized + (F G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) (c a θ : ℝ) + (he : G t z=(c:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I)) + {ξ : ℂ} (hξ : ξ ∈ ball (0:ℂ) 1) : + physicalDiskChart id F G t z ξ = + F t ((c:ℂ)+(a:ℂ)*Target.diskMapInv θ ξ) := by + change F t (diskToHalfPlane (G t z) ξ)=_ + rw [he,diskToHalfPlane_real_affine c a _ ξ hξ,← Target.diskMapInv_eq_diskToHalfPlane] + +lemma physicalDiskChart_mass_eq + (κ θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (F G : ℝ≥0 → ℂ → ℂ) (t : ℝ≥0) (z : ℂ) (c a : ℝ) + (he : G t z=(c:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I)) + (μ : Measure upperHalfPlane) : + diskChartMass κ θ (physicalDiskChart id F G t z) μ = + weightedImageMass (fun w : upperHalfPlane => F t ((c:ℂ)+(a:ℂ)*w)) + (fun w : upperHalfPlane => ENNReal.ofReal + (‖deriv (fun v : ℂ => F t ((c:ℂ)+(a:ℂ)*v)) w‖^dimension κ)) μ := by + apply diskChartMass_eq_of_coordinates κ θ (physicalDiskChart id F G t z) + (fun v : ℂ => F t ((c:ℂ)+(a:ℂ)*v)) + intro w hw + rw [physicalDiskChart_normalized F G t z c a θ he (Target.diskMap_maps hθ hw), + Target.diskMapInv_diskMap hθ hw] + +lemma IsCapacityTwoWitness.physical_quarter_mass_bounds + {U : ℝ≥0 → ℝ} {γ : ℝ≥0 → ℂ} {G F : ℝ≥0 → ℂ → ℂ} + (hw : IsCapacityTwoWitness U γ G F) (hgrowth : PublishedDiskUnivalentGrowth) + (κ : ℝ) (hκ : 0 < κ) (θ : ℝ) (hθ : θ ∈ Ioo (0:ℝ) Real.pi) + (t : ℝ≥0) (z : ℂ) (hz : z ∈ survivingDomain γ t) (a : ℝ) + (he : G t z=(U t:ℂ)+(a:ℂ)*Complex.exp (θ*Complex.I)) + (μ : Measure upperHalfPlane) [IsFiniteMeasureOnCompacts μ] : + let f := physicalDiskChart id F G t z + let ν := weightedImageMass (fun w : upperHalfPlane => F t ((U t:ℂ)+(a:ℂ)*w)) + (fun w : upperHalfPlane => ENNReal.ofReal + (‖deriv (fun v : ℂ => F t ((U t:ℂ)+(a:ℂ)*v)) w‖^dimension κ)) μ + let L := ENNReal.ofReal (compactWeightedMass (Target.diskMassWeight κ θ hθ) + (Target.diskSourceHalfPlane θ) μ) + ENNReal.ofReal ((fullRadius G t z/16)^dimension κ)*L ≤ ν (f '' Target.diskK) ∧ + ν (f '' Target.diskK) ≤ ENNReal.ofReal ((16*fullRadius G t z)^dimension κ)*L := by + obtain ⟨hd,hi,_,hder,_⟩ := hw.physical_disk_chart id differentiableOn_id + (fun _ _ _ _ h => h) t z hz + have hder' : ‖deriv (physicalDiskChart id F G t z) 0‖=fullRadius G t z := by + simpa [physicalConformalRadius] using hder + have hb := diskChartMass_quarter_bounds hgrowth κ hκ θ hθ + (physicalDiskChart id F G t z) hd hi μ + rw [hder',physicalDiskChart_mass_eq κ θ hθ F G t z (U t) a he μ] at hb + exact hb + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/PositiveMassContinuity.lean b/lean/OAI/Probability/SLE/Mass/PositiveMassContinuity.lean index 6a742be6b94b733317363ddfdace983c2ce2a82a..757bdeb7adb1c63818e26f84bee4a0deae5debc2 100644 --- a/lean/OAI/Probability/SLE/Mass/PositiveMassContinuity.lean +++ b/lean/OAI/Probability/SLE/Mass/PositiveMassContinuity.lean @@ -1,37 +1,37 @@ -import OAI.Probability.SLE.Loewner.StoppedInversePaths - -namespace OAI - -/-! Positive compact tests determine continuity of every real compact test. -/ -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal CompactlySupported - -namespace SLEExactGauge - -lemma PositiveMassPaths.continuous_real_tests - {N : ℝ≥0 → Measure upperHalfPlane} {ν : Measure upperHalfPlane} - (h : PositiveMassPaths N ν) (f : C_c(upperHalfPlane,ℝ)) : - Continuous (fun t => ∫ z, f z ∂N t) := by - let fp : C_c(upperHalfPlane,ℝ) := f ⊔ 0 - let fm : C_c(upperHalfPlane,ℝ) := -f ⊔ 0 - have hp : ∀ z, 0 ≤ fp z := fun z => le_max_right _ _ - have hm : ∀ z, 0 ≤ fm z := fun z => le_max_right _ _ - have he (t : ℝ≥0) : (∫ z, f z ∂N t) = (∫ z, fp z ∂N t)-(∫ z, fm z ∂N t) := by - let := (h.radon t).2 - rw [← integral_sub fp.integrable fm.integrable] - apply integral_congr_ae - exact ae_of_all _ (fun z => by - change f z = max (f z) 0-max (-f z) 0 - rcases le_total 0 (f z) with hz | hz - · rw [max_eq_left hz,max_eq_right (neg_nonpos.mpr hz)]; ring - · rw [max_eq_right hz,max_eq_left (neg_nonneg.mpr hz)]; ring) - have hcont : Continuous (fun t => (∫ z, fp z ∂N t)-(∫ z, fm z ∂N t)) := - (h.continuous_tests fp hp).sub (h.continuous_tests fm hm) - exact hcont.congr (fun t => (he t).symm) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Loewner.StoppedInversePaths + +namespace OAI + +/-! Positive compact tests determine continuity of every real compact test. -/ +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal CompactlySupported + +namespace SLEExactGauge + +lemma PositiveMassPaths.continuous_real_tests + {N : ℝ≥0 → Measure upperHalfPlane} {ν : Measure upperHalfPlane} + (h : PositiveMassPaths N ν) (f : C_c(upperHalfPlane,ℝ)) : + Continuous (fun t => ∫ z, f z ∂N t) := by + let fp : C_c(upperHalfPlane,ℝ) := f ⊔ 0 + let fm : C_c(upperHalfPlane,ℝ) := -f ⊔ 0 + have hp : ∀ z, 0 ≤ fp z := fun z => le_max_right _ _ + have hm : ∀ z, 0 ≤ fm z := fun z => le_max_right _ _ + have he (t : ℝ≥0) : (∫ z, f z ∂N t) = (∫ z, fp z ∂N t)-(∫ z, fm z ∂N t) := by + let := (h.radon t).2 + rw [← integral_sub fp.integrable fm.integrable] + apply integral_congr_ae + exact ae_of_all _ (fun z => by + change f z = max (f z) 0-max (-f z) 0 + rcases le_total 0 (f z) with hz | hz + · rw [max_eq_left hz,max_eq_right (neg_nonpos.mpr hz)]; ring + · rw [max_eq_right hz,max_eq_left (neg_nonneg.mpr hz)]; ring) + have hcont : Continuous (fun t => (∫ z, fp z ∂N t)-(∫ z, fm z ∂N t)) := + (h.continuous_tests fp hp).sub (h.continuous_tests fm hm) + exact hcont.congr (fun t => (he t).symm) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/PositiveMassGerm.lean b/lean/OAI/Probability/SLE/Mass/PositiveMassGerm.lean index 6b2eaec736f589f6f36c050794d1a373d763ae14..4b839163c68781ab25fc193f70aebec3a70f40ea 100644 --- a/lean/OAI/Probability/SLE/Mass/PositiveMassGerm.lean +++ b/lean/OAI/Probability/SLE/Mass/PositiveMassGerm.lean @@ -1,98 +1,98 @@ -import OAI.Probability.SLE.Stochastic.BrownianGerm -import OAI.Probability.SLE.Mass.CanonicalMassConstruction - -namespace OAI - -/-! A uniform positive chance of early mass, together with the Brownian -germ law, gives mass before every positive time. The scaling estimate -needed to apply this lemma is kept explicit. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge - -lemma earlyMassGerm_tail {Ω : Type*} (E : ℕ → Set Ω) (k : ℕ) : - (⋂ N : ℕ, ⋃ j ≥ N, E j) = (⋂ N : ℕ, ⋃ j ≥ max N k, E j) := by - ext ω - simp only [mem_iInter,mem_iUnion] - constructor - · intro h N - exact h (max N k) - · intro h N - obtain ⟨j,hj,hE⟩ := h N - exact ⟨j,(le_max_left N k).trans hj,hE⟩ - -/-- This is the zero-one step of the canonical mass positivity argument. -The lower bound will come from scaling a single nonzero compact mean. -/ -theorem positive_mass_all_times_of_uniform_probability - {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t)) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) - (Z : ℝ≥0 → Ω → ℝ≥0∞) - (hZ : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) Z) - (hmono : ∀ᵐ ω ∂P, Monotone (fun t => Z t ω)) - (ε : ℕ → ℝ≥0) (hεanti : Antitone ε) (hε : Tendsto ε atTop (𝓝 0)) - (p : ℝ≥0∞) (hp : 0 {ω | 0 ⋃ j ≥ N, E j - let G : Set Ω := ⋂ N, A N - have hEp (j k : ℕ) (hkj : k≤j) : - MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] (E j) := by - have hz : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] - (Z (ε j)) := (hZ (ε j)).mono - ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono (hεanti hkj)) le_rfl - exact measurableSet_lt measurable_const hz - have hGp (k : ℕ) : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] G := by - change MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] - (⋂ N : ℕ, ⋃ j ≥ N, E j) - rw [earlyMassGerm_tail E k] - apply MeasurableSet.iInter - intro N - apply MeasurableSet.iUnion - intro j - apply MeasurableSet.iUnion - intro hj - exact hEp j k ((le_max_right N k).trans hj) - have hEm (j : ℕ) : MeasurableSet (E j) := - (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le _ _ (hEp j j le_rfl) - have hAm (N : ℕ) : MeasurableSet (A N) := - MeasurableSet.iUnion (fun j => MeasurableSet.iUnion (fun _ => hEm j)) - have hGm : MeasurableSet G := MeasurableSet.iInter hAm - have hAa : Antitone A := by - intro N M hNM ω hω - obtain ⟨j,hj⟩ := mem_iUnion.mp hω - obtain ⟨hjM,hωE⟩ := mem_iUnion.mp hj - exact mem_iUnion.mpr ⟨j,mem_iUnion.mpr ⟨hNM.trans hjM,hωE⟩⟩ - have hpG : p≤P G := by - change p≤P (⋂ N, A N) - rw [hAa.measure_iInter (fun N => (hAm N).nullMeasurableSet) ⟨0,measure_ne_top P _⟩] - apply le_iInf - intro N - apply (hprob N).trans (measure_mono ?_) - intro ω hω - exact mem_iUnion.mpr ⟨N,mem_iUnion.mpr ⟨le_rfl,hω⟩⟩ - have hGone : P G=1 := by - rcases brownian_germ_zero_one hB hm hc h0 ε hε G hGp with hG | hG - · exact ((ne_of_gt hp) (le_zero_iff.mp (hpG.trans hG.le))).elim - · exact hG - have hGa : ∀ᵐ ω ∂P, ω∈G := (mem_ae_iff_prob_eq_one hGm).mpr hGone - filter_upwards [hGa,hmono] with ω hω hωmono - intro t ht - obtain ⟨N,hN⟩ := (hε.eventually (eventually_lt_nhds ht)).exists - have hg : ∀ N : ℕ, ∃ j, N≤j ∧ ω∈E j := by - intro N - change ω∈⋂ N, ⋃ j ≥ N, E j at hω - obtain ⟨j,hj⟩ := mem_iUnion.mp (mem_iInter.mp hω N) - obtain ⟨hNj,hmem⟩ := mem_iUnion.mp hj - exact ⟨j,hNj,hmem⟩ - obtain ⟨j,hj,hωj⟩ := hg N - exact lt_of_lt_of_le hωj (hωmono ((hεanti hj).trans hN.le)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Stochastic.BrownianGerm +import OAI.Probability.SLE.Mass.CanonicalMassConstruction + +namespace OAI + +/-! A uniform positive chance of early mass, together with the Brownian +germ law, gives mass before every positive time. The scaling estimate +needed to apply this lemma is kept explicit. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge + +lemma earlyMassGerm_tail {Ω : Type*} (E : ℕ → Set Ω) (k : ℕ) : + (⋂ N : ℕ, ⋃ j ≥ N, E j) = (⋂ N : ℕ, ⋃ j ≥ max N k, E j) := by + ext ω + simp only [mem_iInter,mem_iUnion] + constructor + · intro h N + exact h (max N k) + · intro h N + obtain ⟨j,hj,hE⟩ := h N + exact ⟨j,(le_max_left N k).trans hj,hE⟩ + +/-- This is the zero-one step of the canonical mass positivity argument. +The lower bound will come from scaling a single nonzero compact mean. -/ +theorem positive_mass_all_times_of_uniform_probability + {Ω : Type*} [mΩ : MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {B : ℝ≥0 → Ω → ℝ} (hB : IsBrownianReal B P) (hm : ∀ t, Measurable (B t)) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) + (Z : ℝ≥0 → Ω → ℝ≥0∞) + (hZ : Adapted (Filtration.natural B (fun t => (hm t).stronglyMeasurable)) Z) + (hmono : ∀ᵐ ω ∂P, Monotone (fun t => Z t ω)) + (ε : ℕ → ℝ≥0) (hεanti : Antitone ε) (hε : Tendsto ε atTop (𝓝 0)) + (p : ℝ≥0∞) (hp : 0 {ω | 0 ⋃ j ≥ N, E j + let G : Set Ω := ⋂ N, A N + have hEp (j k : ℕ) (hkj : k≤j) : + MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] (E j) := by + have hz : Measurable[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] + (Z (ε j)) := (hZ (ε j)).mono + ((Filtration.natural B (fun t => (hm t).stronglyMeasurable)).mono (hεanti hkj)) le_rfl + exact measurableSet_lt measurable_const hz + have hGp (k : ℕ) : MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] G := by + change MeasurableSet[Filtration.natural B (fun t => (hm t).stronglyMeasurable) (ε k)] + (⋂ N : ℕ, ⋃ j ≥ N, E j) + rw [earlyMassGerm_tail E k] + apply MeasurableSet.iInter + intro N + apply MeasurableSet.iUnion + intro j + apply MeasurableSet.iUnion + intro hj + exact hEp j k ((le_max_right N k).trans hj) + have hEm (j : ℕ) : MeasurableSet (E j) := + (Filtration.natural B (fun t => (hm t).stronglyMeasurable)).le _ _ (hEp j j le_rfl) + have hAm (N : ℕ) : MeasurableSet (A N) := + MeasurableSet.iUnion (fun j => MeasurableSet.iUnion (fun _ => hEm j)) + have hGm : MeasurableSet G := MeasurableSet.iInter hAm + have hAa : Antitone A := by + intro N M hNM ω hω + obtain ⟨j,hj⟩ := mem_iUnion.mp hω + obtain ⟨hjM,hωE⟩ := mem_iUnion.mp hj + exact mem_iUnion.mpr ⟨j,mem_iUnion.mpr ⟨hNM.trans hjM,hωE⟩⟩ + have hpG : p≤P G := by + change p≤P (⋂ N, A N) + rw [hAa.measure_iInter (fun N => (hAm N).nullMeasurableSet) ⟨0,measure_ne_top P _⟩] + apply le_iInf + intro N + apply (hprob N).trans (measure_mono ?_) + intro ω hω + exact mem_iUnion.mpr ⟨N,mem_iUnion.mpr ⟨le_rfl,hω⟩⟩ + have hGone : P G=1 := by + rcases brownian_germ_zero_one hB hm hc h0 ε hε G hGp with hG | hG + · exact ((ne_of_gt hp) (le_zero_iff.mp (hpG.trans hG.le))).elim + · exact hG + have hGa : ∀ᵐ ω ∂P, ω∈G := (mem_ae_iff_prob_eq_one hGm).mpr hGone + filter_upwards [hGa,hmono] with ω hω hωmono + intro t ht + obtain ⟨N,hN⟩ := (hε.eventually (eventually_lt_nhds ht)).exists + have hg : ∀ N : ℕ, ∃ j, N≤j ∧ ω∈E j := by + intro N + change ω∈⋂ N, ⋃ j ≥ N, E j at hω + obtain ⟨j,hj⟩ := mem_iUnion.mp (mem_iInter.mp hω N) + obtain ⟨hNj,hmem⟩ := mem_iUnion.mp hj + exact ⟨j,hNj,hmem⟩ + obtain ⟨j,hj,hωj⟩ := hg N + exact lt_of_lt_of_le hωj (hωmono ((hεanti hj).trans hN.le)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/PositiveRestartMass.lean b/lean/OAI/Probability/SLE/Mass/PositiveRestartMass.lean index 5d822fa3fb52ff334199d958bf55535c25868305..4852deca5a68e8b6da865d4793f20e28bf9e9633 100644 --- a/lean/OAI/Probability/SLE/Mass/PositiveRestartMass.lean +++ b/lean/OAI/Probability/SLE/Mass/PositiveRestartMass.lean @@ -1,167 +1,167 @@ -import OAI.Probability.SLE.Mass.CanonicalMassRestart -import OAI.Probability.SLE.Geometry.RestartGreenKernel - -namespace OAI - -/-! Positive mass survives the actual inverse-Loewner transport and -passes from initial intervals to every finite Brownian restart. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped NNReal ENNReal Topology - -namespace SLEExactGauge - -lemma weightedImageMass_ne_zero {X : Type*} [MeasurableSpace X] - (J : upperHalfPlane → X) (hJ : Measurable J) - (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) (hp : ∀ z, 0 B t ω)) - (R : RawCanonicalMass) - (he : ∀ᵐ ω ∂P, ∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) - (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (t : ℝ≥0) (ht : 0 B t ω)) (hκ : 0<κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) - (fun ω => (τ ω:WithTop ℝ≥0))) - (t : ℝ≥0) (ht : 0 (ω,z)) := - @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω - have hJsec (ω : Ω) : Measurable (stoppedMassPoint B κ γ G F hw τ ω) := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ (hpair ω) - have hWsec (ω : Ω) : Measurable (stoppedMassWeight B κ F τ ω) := - @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ - inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance - (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW (hpair ω) - filter_upwards [hr τ hτ,C.raw_restart_positive_time hB hc R (he.mono fun _ h => h.1) - hpos τ hτ t ht,fresh_capacity_witness_exists hB hm hc hpublished τ hτ] with ω hω hp hf - rw [hω.1 t] - obtain ⟨β,G',F',hf⟩ := hf - apply weightedImageMass_ne_zero (stoppedMassPoint B κ γ G F hw τ ω) - (hJsec ω) - (stoppedMassWeight B κ F τ ω) - (hWsec ω) - (fun z => ?_) _ hp - exact ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos - ((hw ω).restart_inverse_derivative_norm_pos ((hc ω).const_mul _) (τ ω) hf z z.property) _) - -lemma positive_increment_fixed_times - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 < t → C.process t ω ≠ 0) - (s t : ℝ≥0) (hst : s < t) : - ∀ᵐ ω ∂P, C.process t ω-C.process s ω ≠ 0 := by - have hτ := isStoppingTime_const (Filtration.natural B (fun r => - (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) s - have h := C.positive_increment_after_stop hB hc hκ F hw hpublished hpos - (fun _ => s) hτ (t-s) (tsub_pos_iff_lt.mpr hst) - simpa only [add_tsub_cancel_of_le hst.le] using h - -lemma positive_increment_all_times - (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) - (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 < t → C.process t ω ≠ 0) : - ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, s < t → C.process t ω-C.process s ω ≠ 0 := by - have hq (p q : ℚ) : ∀ᵐ ω ∂P, - (p:ℝ).toNNReal < (q:ℝ).toNNReal → - C.process (q:ℝ).toNNReal ω-C.process (p:ℝ).toNNReal ω ≠ 0 := by - by_cases hpq : (p:ℝ).toNNReal < (q:ℝ).toNNReal - · exact (C.positive_increment_fixed_times hB hc hκ F hw hpublished hpos _ _ hpq).mono - (fun _ h _ => h) - · exact ae_of_all _ (fun _ h => (hpq h).elim) - filter_upwards [C.paths,ae_all_iff.mpr (fun p => ae_all_iff.mpr (hq p))] with ω hp hqω - intro s t hst hz - obtain ⟨p,hsp,hpt⟩ := exists_rat_btwn (show (s:ℝ) < (t:ℝ) from hst) - obtain ⟨q,hpq,hqt⟩ := exists_rat_btwn hpt - have hp0 : (0:ℝ) ≤ p := s.coe_nonneg.trans hsp.le - have hq0 : (0:ℝ) ≤ q := hp0.trans hpq.le - have hsp' : s ≤ (p:ℝ).toNNReal := by - exact_mod_cast (show (s:ℝ) ≤ ((p:ℝ).toNNReal:ℝ) by rw [Real.coe_toNNReal _ hp0]; exact hsp.le) - have hpq' : (p:ℝ).toNNReal < (q:ℝ).toNNReal := by - exact_mod_cast (show ((p:ℝ).toNNReal:ℝ) < ((q:ℝ).toNNReal:ℝ) by - rw [Real.coe_toNNReal _ hp0,Real.coe_toNNReal _ hq0]; exact hpq) - have hqt' : (q:ℝ).toNNReal ≤ t := by - exact_mod_cast (show ((q:ℝ).toNNReal:ℝ) ≤ (t:ℝ) by rw [Real.coe_toNNReal _ hq0]; exact hqt.le) - let : IsFiniteMeasureOnCompacts (C.process s ω) := (C.radon s ω).2 - have hts : C.process t ω=C.process s ω := by - have h := sigmaFinite_sub_add_cancel_of_le (hp.monotone hst.le) - rw [hz,zero_add] at h - exact h.symm - have hps : C.process (p:ℝ).toNNReal ω=C.process s ω := by - apply le_antisymm - · exact (hp.monotone (hpq'.le.trans hqt')).trans_eq hts - · exact hp.monotone hsp' - have hqs : C.process (q:ℝ).toNNReal ω=C.process s ω := by - apply le_antisymm - · exact (hp.monotone hqt').trans_eq hts - · exact hp.monotone (hsp'.trans hpq'.le) - apply hqω p q hpq' - rw [hps,hqs,Measure.sub_self] - -end Multipoint.CanonicalMass -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.CanonicalMassRestart +import OAI.Probability.SLE.Geometry.RestartGreenKernel + +namespace OAI + +/-! Positive mass survives the actual inverse-Loewner transport and +passes from initial intervals to every finite Brownian restart. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped NNReal ENNReal Topology + +namespace SLEExactGauge + +lemma weightedImageMass_ne_zero {X : Type*} [MeasurableSpace X] + (J : upperHalfPlane → X) (hJ : Measurable J) + (w : upperHalfPlane → ℝ≥0∞) (hw : Measurable w) (hp : ∀ z, 0 B t ω)) + (R : RawCanonicalMass) + (he : ∀ᵐ ω ∂P, ∀ t, R.process t (sourceDriverPath B hc ω)=C.process t ω) + (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (t : ℝ≥0) (ht : 0 B t ω)) (hκ : 0<κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B s)).stronglyMeasurable)) + (fun ω => (τ ω:WithTop ℝ≥0))) + (t : ℝ≥0) (ht : 0 (ω,z)) := + @measurable_prodMk_left Ω upperHalfPlane hτ.measurableSpace inferInstance ω + have hJsec (ω : Ω) : Measurable (stoppedMassPoint B κ γ G F hw τ ω) := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) upperHalfPlane + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassPoint B κ γ G F hw τ p.1 p.2) (fun z => (ω,z)) hJ (hpair ω) + have hWsec (ω : Ω) : Measurable (stoppedMassWeight B κ F τ ω) := + @Measurable.comp upperHalfPlane (Ω × upperHalfPlane) ℝ≥0∞ + inferInstance (hτ.measurableSpace.prod inferInstance) inferInstance + (fun p => stoppedMassWeight B κ F τ p.1 p.2) (fun z => (ω,z)) hW (hpair ω) + filter_upwards [hr τ hτ,C.raw_restart_positive_time hB hc R (he.mono fun _ h => h.1) + hpos τ hτ t ht,fresh_capacity_witness_exists hB hm hc hpublished τ hτ] with ω hω hp hf + rw [hω.1 t] + obtain ⟨β,G',F',hf⟩ := hf + apply weightedImageMass_ne_zero (stoppedMassPoint B κ γ G F hw τ ω) + (hJsec ω) + (stoppedMassWeight B κ F τ ω) + (hWsec ω) + (fun z => ?_) _ hp + exact ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos + ((hw ω).restart_inverse_derivative_norm_pos ((hc ω).const_mul _) (τ ω) hf z z.property) _) + +lemma positive_increment_fixed_times + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 < t → C.process t ω ≠ 0) + (s t : ℝ≥0) (hst : s < t) : + ∀ᵐ ω ∂P, C.process t ω-C.process s ω ≠ 0 := by + have hτ := isStoppingTime_const (Filtration.natural B (fun r => + (hBm.comp (measurable_const.prodMk measurable_id) : Measurable (B r)).stronglyMeasurable)) s + have h := C.positive_increment_after_stop hB hc hκ F hw hpublished hpos + (fun _ => s) hτ (t-s) (tsub_pos_iff_lt.mpr hst) + simpa only [add_tsub_cancel_of_le hst.le] using h + +lemma positive_increment_all_times + (hB : IsBrownianReal B P) (hc : ∀ ω, Continuous (fun t => B t ω)) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) + (hpos : ∀ᵐ ω ∂P, ∀ t : ℝ≥0, 0 < t → C.process t ω ≠ 0) : + ∀ᵐ ω ∂P, ∀ s t : ℝ≥0, s < t → C.process t ω-C.process s ω ≠ 0 := by + have hq (p q : ℚ) : ∀ᵐ ω ∂P, + (p:ℝ).toNNReal < (q:ℝ).toNNReal → + C.process (q:ℝ).toNNReal ω-C.process (p:ℝ).toNNReal ω ≠ 0 := by + by_cases hpq : (p:ℝ).toNNReal < (q:ℝ).toNNReal + · exact (C.positive_increment_fixed_times hB hc hκ F hw hpublished hpos _ _ hpq).mono + (fun _ h _ => h) + · exact ae_of_all _ (fun _ h => (hpq h).elim) + filter_upwards [C.paths,ae_all_iff.mpr (fun p => ae_all_iff.mpr (hq p))] with ω hp hqω + intro s t hst hz + obtain ⟨p,hsp,hpt⟩ := exists_rat_btwn (show (s:ℝ) < (t:ℝ) from hst) + obtain ⟨q,hpq,hqt⟩ := exists_rat_btwn hpt + have hp0 : (0:ℝ) ≤ p := s.coe_nonneg.trans hsp.le + have hq0 : (0:ℝ) ≤ q := hp0.trans hpq.le + have hsp' : s ≤ (p:ℝ).toNNReal := by + exact_mod_cast (show (s:ℝ) ≤ ((p:ℝ).toNNReal:ℝ) by rw [Real.coe_toNNReal _ hp0]; exact hsp.le) + have hpq' : (p:ℝ).toNNReal < (q:ℝ).toNNReal := by + exact_mod_cast (show ((p:ℝ).toNNReal:ℝ) < ((q:ℝ).toNNReal:ℝ) by + rw [Real.coe_toNNReal _ hp0,Real.coe_toNNReal _ hq0]; exact hpq) + have hqt' : (q:ℝ).toNNReal ≤ t := by + exact_mod_cast (show ((q:ℝ).toNNReal:ℝ) ≤ (t:ℝ) by rw [Real.coe_toNNReal _ hq0]; exact hqt.le) + let : IsFiniteMeasureOnCompacts (C.process s ω) := (C.radon s ω).2 + have hts : C.process t ω=C.process s ω := by + have h := sigmaFinite_sub_add_cancel_of_le (hp.monotone hst.le) + rw [hz,zero_add] at h + exact h.symm + have hps : C.process (p:ℝ).toNNReal ω=C.process s ω := by + apply le_antisymm + · exact (hp.monotone (hpq'.le.trans hqt')).trans_eq hts + · exact hp.monotone hsp' + have hqs : C.process (q:ℝ).toNNReal ω=C.process s ω := by + apply le_antisymm + · exact (hp.monotone hqt').trans_eq hts + · exact hp.monotone (hsp'.trans hpq'.le) + apply hqω p q hpq' + rw [hps,hqs,Measure.sub_self] + +end Multipoint.CanonicalMass +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/PositiveTraceMass.lean b/lean/OAI/Probability/SLE/Mass/PositiveTraceMass.lean index c34be2fa3d32166f5828e767ab6c4fdbd24515c9..85f3b77ae3ad90f16689ae38fb9abd0c2e9a009e 100644 --- a/lean/OAI/Probability/SLE/Mass/PositiveTraceMass.lean +++ b/lean/OAI/Probability/SLE/Mass/PositiveTraceMass.lean @@ -1,72 +1,72 @@ -import OAI.Probability.SLE.Mass.PositiveRestartMass -import OAI.Probability.SLE.Mass.CanonicalMassScaling - -namespace OAI - -/-! Positive increments and trace support force positive canonical mass -in each open set visited at a positive time. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -lemma positive_trace_mass_of_positive_increments - (γ : ℝ≥0 → ℂ) (hγ : Continuous γ) (N : ℝ≥0 → Measure upperHalfPlane) - (hmono : Monotone N) - (hinc : ∀ s t, s ≤ t → (N t-N s) - {z : upperHalfPlane | (z:ℂ) ∉ (fun r : ℝ => γ r.toNNReal) '' Icc (s:ℝ) t}=0) - (hpos : ∀ s t, s < t → N t-N s ≠ 0) - (V : Set ℂ) (hV : IsOpen V) (u T : ℝ≥0) (hu : 0 < u) (huT : u ≤ T) - (hvisit : γ u ∈ V) : 0 < N T (Subtype.val ⁻¹' V) := by - have hpre : γ ⁻¹' V ∈ 𝓝 u := hγ.continuousAt.preimage_mem_nhds (hV.mem_nhds hvisit) - obtain ⟨a,ha,haV⟩ := exists_Ioc_subset_of_mem_nhds' hpre hu - obtain ⟨b,hab,hbu⟩ := exists_between ha.2 - have hseg : (fun r : ℝ => γ r.toNNReal) '' Icc (b:ℝ) u ⊆ V := by - rintro _ ⟨r,hr,rfl⟩ - apply haV - have hbr : b ≤ r.toNNReal := by - simpa only [Real.toNNReal_coe] using Real.toNNReal_mono hr.1 - have hru : r.toNNReal ≤ u := by - simpa only [Real.toNNReal_coe] using Real.toNNReal_mono hr.2 - exact ⟨hab.trans_le hbr,hru⟩ - have hnull : (N u-N b) (Subtype.val ⁻¹' V)ᶜ=0 := by - apply le_antisymm ?_ bot_le - apply (measure_mono (show (Subtype.val ⁻¹' V)ᶜ ⊆ - {z : upperHalfPlane | (z:ℂ) ∉ (fun r : ℝ => γ r.toNNReal) '' Icc (b:ℝ) u} from ?_)).trans_eq - (hinc b u hbu.le) - intro z hz hs - exact hz (hseg hs) - have hVmeas : MeasurableSet ((Subtype.val : upperHalfPlane → ℂ) ⁻¹' V) := - (hV.preimage (continuous_subtype_val : Continuous (Subtype.val : upperHalfPlane → ℂ))).measurableSet - have he := measure_add_measure_compl (μ:=N u-N b) hVmeas - rw [hnull,add_zero] at he - have hp : 0 < (N u-N b) (Subtype.val ⁻¹' V) := by - rw [he] - exact Measure.measure_univ_pos.mpr (hpos b u hbu) - exact hp.trans_le (((Measure.sub_le : N u-N b ≤ N u).trans (hmono huT)) _) - -variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] - {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} - {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} - -lemma CanonicalMass.positive_open_visited - (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) - (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) - (F : Ω → ℝ≥0 → ℂ → ℂ) - (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) - (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : - ∀ᵐ ω ∂P, ∀ V : Set ℂ, IsOpen V → ∀ u T : ℝ≥0, - 0 < u → u ≤ T → γ ω u ∈ V → 0 < C.process T ω (Subtype.val ⁻¹' V) := by - have hp := C.positive_all_times hB hc h0 F hw - have hi := C.positive_increment_all_times hB hc hκ F hw hpublished hp - filter_upwards [C.paths,hi] with ω hω hωi - exact positive_trace_mass_of_positive_increments (γ ω) (hw ω).1 (fun t => C.process t ω) - hω.monotone hω.increment_support hωi - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.PositiveRestartMass +import OAI.Probability.SLE.Mass.CanonicalMassScaling + +namespace OAI + +/-! Positive increments and trace support force positive canonical mass +in each open set visited at a positive time. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +lemma positive_trace_mass_of_positive_increments + (γ : ℝ≥0 → ℂ) (hγ : Continuous γ) (N : ℝ≥0 → Measure upperHalfPlane) + (hmono : Monotone N) + (hinc : ∀ s t, s ≤ t → (N t-N s) + {z : upperHalfPlane | (z:ℂ) ∉ (fun r : ℝ => γ r.toNNReal) '' Icc (s:ℝ) t}=0) + (hpos : ∀ s t, s < t → N t-N s ≠ 0) + (V : Set ℂ) (hV : IsOpen V) (u T : ℝ≥0) (hu : 0 < u) (huT : u ≤ T) + (hvisit : γ u ∈ V) : 0 < N T (Subtype.val ⁻¹' V) := by + have hpre : γ ⁻¹' V ∈ 𝓝 u := hγ.continuousAt.preimage_mem_nhds (hV.mem_nhds hvisit) + obtain ⟨a,ha,haV⟩ := exists_Ioc_subset_of_mem_nhds' hpre hu + obtain ⟨b,hab,hbu⟩ := exists_between ha.2 + have hseg : (fun r : ℝ => γ r.toNNReal) '' Icc (b:ℝ) u ⊆ V := by + rintro _ ⟨r,hr,rfl⟩ + apply haV + have hbr : b ≤ r.toNNReal := by + simpa only [Real.toNNReal_coe] using Real.toNNReal_mono hr.1 + have hru : r.toNNReal ≤ u := by + simpa only [Real.toNNReal_coe] using Real.toNNReal_mono hr.2 + exact ⟨hab.trans_le hbr,hru⟩ + have hnull : (N u-N b) (Subtype.val ⁻¹' V)ᶜ=0 := by + apply le_antisymm ?_ bot_le + apply (measure_mono (show (Subtype.val ⁻¹' V)ᶜ ⊆ + {z : upperHalfPlane | (z:ℂ) ∉ (fun r : ℝ => γ r.toNNReal) '' Icc (b:ℝ) u} from ?_)).trans_eq + (hinc b u hbu.le) + intro z hz hs + exact hz (hseg hs) + have hVmeas : MeasurableSet ((Subtype.val : upperHalfPlane → ℂ) ⁻¹' V) := + (hV.preimage (continuous_subtype_val : Continuous (Subtype.val : upperHalfPlane → ℂ))).measurableSet + have he := measure_add_measure_compl (μ:=N u-N b) hVmeas + rw [hnull,add_zero] at he + have hp : 0 < (N u-N b) (Subtype.val ⁻¹' V) := by + rw [he] + exact Measure.measure_univ_pos.mpr (hpos b u hbu) + exact hp.trans_le (((Measure.sub_le : N u-N b ≤ N u).trans (hmono huT)) _) + +variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsProbabilityMeasure P] + {κ : ℝ} {B : ℝ≥0 → Ω → ℝ} {hBm : Measurable (Function.uncurry B)} + {γ : Ω → ℝ≥0 → ℂ} {G : Ω → ℝ≥0 → ℂ → ℂ} + +lemma CanonicalMass.positive_open_visited + (C : CanonicalMass P κ B hBm γ G) (hB : IsBrownianReal B P) + (hc : ∀ ω, Continuous (fun t => B t ω)) (h0 : ∀ ω, B 0 ω=0) (hκ : 0 < κ) + (F : Ω → ℝ≥0 → ℂ → ℂ) + (hw : ∀ ω, IsCapacityTwoWitness (fun t => Real.sqrt κ*B t ω) (γ ω) (G ω) (F ω)) + (hpublished : PublishedTraceGeneration (P.map (sourceDriverPath B hc)) κ (fun t p => p t)) : + ∀ᵐ ω ∂P, ∀ V : Set ℂ, IsOpen V → ∀ u T : ℝ≥0, + 0 < u → u ≤ T → γ ω u ∈ V → 0 < C.process T ω (Subtype.val ⁻¹' V) := by + have hp := C.positive_all_times hB hc h0 F hw + have hi := C.positive_increment_all_times hB hc hκ F hw hpublished hp + filter_upwards [C.paths,hi] with ω hω hωi + exact positive_trace_mass_of_positive_increments (γ ω) (hw ω).1 (fun t => C.process t ω) + hω.monotone hω.increment_support hωi + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonApproximationList.lean b/lean/OAI/Probability/SLE/Mass/RadonApproximationList.lean index f1fab3ca0e1aa343890668ef2b45b77416bc040a..7000a03fe44da72330d18469b2b0953e70705bd4 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonApproximationList.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonApproximationList.lean @@ -1,56 +1,56 @@ -import OAI.Probability.SLE.Mass.RadonTestLimit -import Mathlib.Topology.Bases -import Mathlib.Topology.Instances.Real.Lemmas - -namespace OAI - -/-! A countable list of positive Radon measures approximates every finite -list of compact-test integrals. This uses separability of the image in -real sequence space, without a topology on the space of measures. -/ - -noncomputable section -open Set Filter MeasureTheory Classical -open scoped Topology CompactlySupported - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - -omit [T2Space X] [LocallyCompactSpace X] [BorelSpace X] in -lemma LocalTestFamily.exists_countable_measure_approximants (D : LocalTestFamily X) : - ∃ μ : ℕ → Measure X, (∀ k, (μ k).Regular) ∧ - ∀ (ν : Measure X), ν.Regular → ∀ n : ℕ, ∀ ε : ℝ, 0 < ε → - ∃ k, ∀ j < n+1, |(∫ x, D.test j x ∂μ k)-(∫ x, D.test j x ∂ν)| < ε := by - let R := {ν : Measure X // ν.Regular} - let V : R → ℕ → ℝ := fun ν j => ∫ x, D.test j x ∂ν.val - let S := range V - let : Nonempty S := ⟨⟨V ⟨0,inferInstance⟩,⟨⟨0,inferInstance⟩,rfl⟩⟩⟩ - let q : ℕ → S := TopologicalSpace.denseSeq S - have hq : DenseRange q := TopologicalSpace.denseRange_denseSeq S - choose μ hμ using fun k => (q k).property - refine ⟨fun k => (μ k).val,fun k => (μ k).property,?_⟩ - intro ν hν n ε hε - let v : S := ⟨V ⟨ν,hν⟩,⟨⟨ν,hν⟩,rfl⟩⟩ - let U : Set S := {w | ∀ j : Fin (n+1), |w.val j.val-v.val j.val| < ε} - have hU : IsOpen U := by - simp only [U,Set.ofPred_forall] - apply isOpen_iInter_of_finite - intro j - exact isOpen_lt (((continuous_apply j.val).comp continuous_subtype_val).sub continuous_const).abs - continuous_const - have hv : v ∈ U := by - intro j - simpa only [sub_self,abs_zero] using hε - obtain ⟨k,hk⟩ := hq.exists_mem_open hU ⟨v,hv⟩ - refine ⟨k,fun j hj => ?_⟩ - have hh := hk (⟨j,hj⟩ : Fin (n+1)) - change |(V (μ k)) j-(V ⟨ν,hν⟩) j| < ε - rw [hμ k] - exact hh - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonTestLimit +import Mathlib.Topology.Bases +import Mathlib.Topology.Instances.Real.Lemmas + +namespace OAI + +/-! A countable list of positive Radon measures approximates every finite +list of compact-test integrals. This uses separability of the image in +real sequence space, without a topology on the space of measures. -/ + +noncomputable section +open Set Filter MeasureTheory Classical +open scoped Topology CompactlySupported + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + +omit [T2Space X] [LocallyCompactSpace X] [BorelSpace X] in +lemma LocalTestFamily.exists_countable_measure_approximants (D : LocalTestFamily X) : + ∃ μ : ℕ → Measure X, (∀ k, (μ k).Regular) ∧ + ∀ (ν : Measure X), ν.Regular → ∀ n : ℕ, ∀ ε : ℝ, 0 < ε → + ∃ k, ∀ j < n+1, |(∫ x, D.test j x ∂μ k)-(∫ x, D.test j x ∂ν)| < ε := by + let R := {ν : Measure X // ν.Regular} + let V : R → ℕ → ℝ := fun ν j => ∫ x, D.test j x ∂ν.val + let S := range V + let : Nonempty S := ⟨⟨V ⟨0,inferInstance⟩,⟨⟨0,inferInstance⟩,rfl⟩⟩⟩ + let q : ℕ → S := TopologicalSpace.denseSeq S + have hq : DenseRange q := TopologicalSpace.denseRange_denseSeq S + choose μ hμ using fun k => (q k).property + refine ⟨fun k => (μ k).val,fun k => (μ k).property,?_⟩ + intro ν hν n ε hε + let v : S := ⟨V ⟨ν,hν⟩,⟨⟨ν,hν⟩,rfl⟩⟩ + let U : Set S := {w | ∀ j : Fin (n+1), |w.val j.val-v.val j.val| < ε} + have hU : IsOpen U := by + simp only [U,Set.ofPred_forall] + apply isOpen_iInter_of_finite + intro j + exact isOpen_lt (((continuous_apply j.val).comp continuous_subtype_val).sub continuous_const).abs + continuous_const + have hv : v ∈ U := by + intro j + simpa only [sub_self,abs_zero] using hε + obtain ⟨k,hk⟩ := hq.exists_mem_open hU ⟨v,hv⟩ + refine ⟨k,fun j hj => ?_⟩ + have hh := hk (⟨j,hj⟩ : Fin (n+1)) + change |(V (μ k)) j-(V ⟨ν,hν⟩) j| < ε + rw [hμ k] + exact hh + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonFactorization.lean b/lean/OAI/Probability/SLE/Mass/RadonFactorization.lean index 530985e1384039d818dc6553abf502fbc68028b1..2d9c90a0d5071cfe5db2be769592005bb311b2aa 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonFactorization.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonFactorization.lean @@ -1,97 +1,97 @@ -import OAI.Probability.SLE.Mass.RadonApproximationList -import OAI.Probability.SLE.Mass.RawRadonLimit -import OAI.Probability.SLE.Mass.RadonTestDetermines -import Mathlib.MeasureTheory.Function.FactorsThrough - -namespace OAI - -/-! Doob--Dynkin factorization of a locally finite Radon kernel through -countably many real compact tests. The factor is regular on every input and -is reconstructed in the target data space, without ambient null-set changes. -/ - -noncomputable section -open Set Filter MeasureTheory Classical -open scoped Topology CompactlySupported - -namespace SLEExactGauge - -lemma exists_measurable_nat_choice - {Y : Type*} [MeasurableSpace Y] (p : Y → ℕ → Prop) - (hm : ∀ k, MeasurableSet {y | p y k}) : - ∃ i : Y → ℕ, Measurable i ∧ ∀ y, (∃ k, p y k) → p y (i y) := by - let q (y : Y) (k : ℕ) := (∃ j, p y j) → p y k - have hq (y : Y) : ∃ k, q y k := by - by_cases h : ∃ j, p y j - · obtain ⟨k,hk⟩ := h - exact ⟨k,fun _ => hk⟩ - · exact ⟨0,fun h' => (h h').elim⟩ - have hqm (k : ℕ) : MeasurableSet {y | q y k} := by - have he : {y | q y k} = (⋃ j, {y | p y j})ᶜ ∪ {y | p y k} := by - ext y - simp only [q,mem_ofPred_eq,mem_union,mem_compl_iff,mem_iUnion] - tauto - rw [he] - exact (MeasurableSet.iUnion hm).compl.union (hm k) - exact ⟨fun y => Nat.find (hq y),measurable_find hq hqm, - fun y hy => Nat.find_spec (hq y) hy⟩ - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] - -/-- Countable compact tests suffice for a raw measurable Radon factor. -The factorization equality holds at every original input. -/ -theorem LocalTestFamily.measurable_radon_factor - (D : LocalTestFamily X) {Ω Y : Type*} [MeasurableSpace Y] - (F : Ω → Y) (μ : Ω → Measure X) (hμ : ∀ ω, (μ ω).Regular) - (htest : ∀ j, Measurable[MeasurableSpace.comap F inferInstance] - (fun ω => ∫ x, D.test j x ∂μ ω)) : - ∃ ν : Y → Measure X, Measurable ν ∧ (∀ y, (ν y).Regular) ∧ ∀ ω, ν (F ω)=μ ω := by - choose g hgm hg using fun j => (htest j).exists_eq_measurable_comp - obtain ⟨a,ha,happrox⟩ := D.exists_countable_measure_approximants - let p (n : ℕ) (y : Y) (k : ℕ) : Prop := - ∀ j < n+1, |(∫ x, D.test j x ∂a k)-g j y| < 1/(n+1:ℝ) - have hpm (n k : ℕ) : MeasurableSet {y | p n y k} := by - simp only [p,Set.ofPred_forall] - apply MeasurableSet.iInter - intro j - apply MeasurableSet.iInter - intro _ - simpa only [Real.norm_eq_abs,Pi.sub_apply] using - measurableSet_lt (measurable_const.sub (hgm j)).norm measurable_const - choose i him hi using fun n => exists_measurable_nat_choice (p n) (hpm n) - let μn (n : ℕ) (y : Y) := a (i n y) - have hμnm (n : ℕ) (f : C_c(X,ℝ)) : Measurable (fun y => ∫ x, f x ∂μn n y) := - (measurable_of_countable (fun k => ∫ x, f x ∂a k)).comp (him n) - have hvalid (n : ℕ) (ω : Ω) : p n (F ω) (i n (F ω)) := by - apply hi n - obtain ⟨k,hk⟩ := happrox (μ ω) (hμ ω) n (1/(n+1:ℝ)) (by positivity) - refine ⟨k,fun j hj => ?_⟩ - have he : g j (F ω) = ∫ x, D.test j x ∂μ ω := congr_fun (hg j) ω |>.symm - rw [he] - exact hk j hj - have hlim (ω : Ω) (j : ℕ) : Tendsto (fun n => ∫ x, D.test j x ∂μn n (F ω)) atTop - (𝓝 (∫ x, D.test j x ∂μ ω)) := by - apply Metric.tendsto_atTop.mpr - intro ε hε - obtain ⟨N,hN⟩ := eventually_atTop.mp - ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually (gt_mem_nhds hε)) - refine ⟨max j N,fun n hn => ?_⟩ - rw [Real.dist_eq] - have hjn : j < n+1 := by omega - have hv := hvalid n ω j hjn - have he : g j (F ω) = ∫ x, D.test j x ∂μ ω := congr_fun (hg j) ω |>.symm - rw [he] at hv - exact hv.trans (hN n (le_trans (le_max_right _ _) hn)) - obtain ⟨ν,hνm,hν,hνlim⟩ := D.exists_raw_radon_limit μn (fun n y => ha (i n y)) hμnm - refine ⟨ν,hνm,hν,fun ω => ?_⟩ - let := hν (F ω) - let := hμ ω - apply D.measure_eq_of_test_integrals - intro j - exact tendsto_nhds_unique (hνlim (F ω) (fun j => (hlim ω j).cauchySeq) (D.test j)) (hlim ω j) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonApproximationList +import OAI.Probability.SLE.Mass.RawRadonLimit +import OAI.Probability.SLE.Mass.RadonTestDetermines +import Mathlib.MeasureTheory.Function.FactorsThrough + +namespace OAI + +/-! Doob--Dynkin factorization of a locally finite Radon kernel through +countably many real compact tests. The factor is regular on every input and +is reconstructed in the target data space, without ambient null-set changes. -/ + +noncomputable section +open Set Filter MeasureTheory Classical +open scoped Topology CompactlySupported + +namespace SLEExactGauge + +lemma exists_measurable_nat_choice + {Y : Type*} [MeasurableSpace Y] (p : Y → ℕ → Prop) + (hm : ∀ k, MeasurableSet {y | p y k}) : + ∃ i : Y → ℕ, Measurable i ∧ ∀ y, (∃ k, p y k) → p y (i y) := by + let q (y : Y) (k : ℕ) := (∃ j, p y j) → p y k + have hq (y : Y) : ∃ k, q y k := by + by_cases h : ∃ j, p y j + · obtain ⟨k,hk⟩ := h + exact ⟨k,fun _ => hk⟩ + · exact ⟨0,fun h' => (h h').elim⟩ + have hqm (k : ℕ) : MeasurableSet {y | q y k} := by + have he : {y | q y k} = (⋃ j, {y | p y j})ᶜ ∪ {y | p y k} := by + ext y + simp only [q,mem_ofPred_eq,mem_union,mem_compl_iff,mem_iUnion] + tauto + rw [he] + exact (MeasurableSet.iUnion hm).compl.union (hm k) + exact ⟨fun y => Nat.find (hq y),measurable_find hq hqm, + fun y hy => Nat.find_spec (hq y) hy⟩ + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] + +/-- Countable compact tests suffice for a raw measurable Radon factor. +The factorization equality holds at every original input. -/ +theorem LocalTestFamily.measurable_radon_factor + (D : LocalTestFamily X) {Ω Y : Type*} [MeasurableSpace Y] + (F : Ω → Y) (μ : Ω → Measure X) (hμ : ∀ ω, (μ ω).Regular) + (htest : ∀ j, Measurable[MeasurableSpace.comap F inferInstance] + (fun ω => ∫ x, D.test j x ∂μ ω)) : + ∃ ν : Y → Measure X, Measurable ν ∧ (∀ y, (ν y).Regular) ∧ ∀ ω, ν (F ω)=μ ω := by + choose g hgm hg using fun j => (htest j).exists_eq_measurable_comp + obtain ⟨a,ha,happrox⟩ := D.exists_countable_measure_approximants + let p (n : ℕ) (y : Y) (k : ℕ) : Prop := + ∀ j < n+1, |(∫ x, D.test j x ∂a k)-g j y| < 1/(n+1:ℝ) + have hpm (n k : ℕ) : MeasurableSet {y | p n y k} := by + simp only [p,Set.ofPred_forall] + apply MeasurableSet.iInter + intro j + apply MeasurableSet.iInter + intro _ + simpa only [Real.norm_eq_abs,Pi.sub_apply] using + measurableSet_lt (measurable_const.sub (hgm j)).norm measurable_const + choose i him hi using fun n => exists_measurable_nat_choice (p n) (hpm n) + let μn (n : ℕ) (y : Y) := a (i n y) + have hμnm (n : ℕ) (f : C_c(X,ℝ)) : Measurable (fun y => ∫ x, f x ∂μn n y) := + (measurable_of_countable (fun k => ∫ x, f x ∂a k)).comp (him n) + have hvalid (n : ℕ) (ω : Ω) : p n (F ω) (i n (F ω)) := by + apply hi n + obtain ⟨k,hk⟩ := happrox (μ ω) (hμ ω) n (1/(n+1:ℝ)) (by positivity) + refine ⟨k,fun j hj => ?_⟩ + have he : g j (F ω) = ∫ x, D.test j x ∂μ ω := congr_fun (hg j) ω |>.symm + rw [he] + exact hk j hj + have hlim (ω : Ω) (j : ℕ) : Tendsto (fun n => ∫ x, D.test j x ∂μn n (F ω)) atTop + (𝓝 (∫ x, D.test j x ∂μ ω)) := by + apply Metric.tendsto_atTop.mpr + intro ε hε + obtain ⟨N,hN⟩ := eventually_atTop.mp + ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).eventually (gt_mem_nhds hε)) + refine ⟨max j N,fun n hn => ?_⟩ + rw [Real.dist_eq] + have hjn : j < n+1 := by omega + have hv := hvalid n ω j hjn + have he : g j (F ω) = ∫ x, D.test j x ∂μ ω := congr_fun (hg j) ω |>.symm + rw [he] at hv + exact hv.trans (hN n (le_trans (le_max_right _ _) hn)) + obtain ⟨ν,hνm,hν,hνlim⟩ := D.exists_raw_radon_limit μn (fun n y => ha (i n y)) hμnm + refine ⟨ν,hνm,hν,fun ω => ?_⟩ + let := hν (F ω) + let := hμ ω + apply D.measure_eq_of_test_integrals + intro j + exact tendsto_nhds_unique (hνlim (F ω) (fun j => (hlim ω j).cauchySeq) (D.test j)) (hlim ω j) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonKernelIntegral.lean b/lean/OAI/Probability/SLE/Mass/RadonKernelIntegral.lean index 1e4cfbfc5007aaf3f965d1f34a27b4780468f504..652bb357d24a5977ca8d4d305b23637983fd2edb 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonKernelIntegral.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonKernelIntegral.lean @@ -1,67 +1,67 @@ -import Mathlib.MeasureTheory.Measure.GiryMonad -import Mathlib.MeasureTheory.Integral.Bochner.Basic - -namespace OAI - -/-! Signed integrals of a random Radon measure can be computed from its -mean measure. The positive and negative parts use Tonelli separately. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped ENNReal - -namespace SLEExactGauge - -lemma lintegral_ofReal_lt_top_of_integrable - {X : Type*} [MeasurableSpace X] {μ : Measure X} {f : X → ℝ} - (hf : Integrable f μ) : (∫⁻ x, ENNReal.ofReal (f x) ∂μ) < ⊤ := by - apply (lintegral_mono (fun x => ENNReal.ofReal_le_ofReal (le_abs_self (f x)))).trans_lt - exact (hasFiniteIntegral_iff_ofReal (Eventually.of_forall (fun x => norm_nonneg (f x)))).mp - hf.norm.hasFiniteIntegral - -theorem integrable_integral_bind_of_sections - {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] - (P : Measure Ω) (μ : Ω → Measure X) (hμ : Measurable μ) - (f : X → ℝ) (hf : Measurable f) - (hfi : ∀ ω, Integrable f (μ ω)) (hfP : Integrable f (P.bind μ)) : - Integrable (fun ω => ∫ x, f x ∂μ ω) P ∧ - (∫ ω, (∫ x, f x ∂μ ω) ∂P) = ∫ x, f x ∂P.bind μ := by - let A (ω : Ω) := ∫⁻ x, ENNReal.ofReal (f x) ∂μ ω - let B (ω : Ω) := ∫⁻ x, ENNReal.ofReal (-f x) ∂μ ω - have hAm : Measurable A := (Measure.measurable_lintegral hf.ennreal_ofReal).comp hμ - have hBm : Measurable B := (Measure.measurable_lintegral hf.neg.ennreal_ofReal).comp hμ - have hA : (∫⁻ ω, A ω ∂P) = ∫⁻ x, ENNReal.ofReal (f x) ∂P.bind μ := - (Measure.lintegral_bind hμ.aemeasurable hf.ennreal_ofReal.aemeasurable).symm - have hB : (∫⁻ ω, B ω ∂P) = ∫⁻ x, ENNReal.ofReal (-f x) ∂P.bind μ := - (Measure.lintegral_bind hμ.aemeasurable hf.neg.ennreal_ofReal.aemeasurable).symm - have hAf : (∫⁻ ω, A ω ∂P) < ⊤ := hA.trans_lt (lintegral_ofReal_lt_top_of_integrable hfP) - have hBf : (∫⁻ ω, B ω ∂P) < ⊤ := hB.trans_lt (lintegral_ofReal_lt_top_of_integrable hfP.neg) - have he (ω : Ω) : (∫ x, f x ∂μ ω) = (A ω).toReal - (B ω).toReal := - integral_eq_lintegral_pos_part_sub_lintegral_neg_part (hfi ω) - refine ⟨?_,?_⟩ - · convert (integrable_toReal_of_lintegral_ne_top hAm.aemeasurable hAf.ne).sub - (integrable_toReal_of_lintegral_ne_top hBm.aemeasurable hBf.ne) using 1 - funext ω - exact he ω - simp_rw [he] - rw [integral_sub (integrable_toReal_of_lintegral_ne_top hAm.aemeasurable hAf.ne) - (integrable_toReal_of_lintegral_ne_top hBm.aemeasurable hBf.ne), - integral_toReal hAm.aemeasurable (Eventually.of_forall (fun ω => - lintegral_ofReal_lt_top_of_integrable (hfi ω))), - integral_toReal hBm.aemeasurable (Eventually.of_forall (fun ω => - lintegral_ofReal_lt_top_of_integrable (hfi ω).neg)),hA,hB] - exact (integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfP).symm - -theorem integral_bind_of_integrable_sections - {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] - (P : Measure Ω) (μ : Ω → Measure X) (hμ : Measurable μ) - (f : X → ℝ) (hf : Measurable f) - (hfi : ∀ ω, Integrable f (μ ω)) (hfP : Integrable f (P.bind μ)) : - (∫ ω, (∫ x, f x ∂μ ω) ∂P) = ∫ x, f x ∂P.bind μ := - (integrable_integral_bind_of_sections P μ hμ f hf hfi hfP).2 - -end SLEExactGauge - -end - -end OAI +import Mathlib.MeasureTheory.Measure.GiryMonad +import Mathlib.MeasureTheory.Integral.Bochner.Basic + +namespace OAI + +/-! Signed integrals of a random Radon measure can be computed from its +mean measure. The positive and negative parts use Tonelli separately. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped ENNReal + +namespace SLEExactGauge + +lemma lintegral_ofReal_lt_top_of_integrable + {X : Type*} [MeasurableSpace X] {μ : Measure X} {f : X → ℝ} + (hf : Integrable f μ) : (∫⁻ x, ENNReal.ofReal (f x) ∂μ) < ⊤ := by + apply (lintegral_mono (fun x => ENNReal.ofReal_le_ofReal (le_abs_self (f x)))).trans_lt + exact (hasFiniteIntegral_iff_ofReal (Eventually.of_forall (fun x => norm_nonneg (f x)))).mp + hf.norm.hasFiniteIntegral + +theorem integrable_integral_bind_of_sections + {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] + (P : Measure Ω) (μ : Ω → Measure X) (hμ : Measurable μ) + (f : X → ℝ) (hf : Measurable f) + (hfi : ∀ ω, Integrable f (μ ω)) (hfP : Integrable f (P.bind μ)) : + Integrable (fun ω => ∫ x, f x ∂μ ω) P ∧ + (∫ ω, (∫ x, f x ∂μ ω) ∂P) = ∫ x, f x ∂P.bind μ := by + let A (ω : Ω) := ∫⁻ x, ENNReal.ofReal (f x) ∂μ ω + let B (ω : Ω) := ∫⁻ x, ENNReal.ofReal (-f x) ∂μ ω + have hAm : Measurable A := (Measure.measurable_lintegral hf.ennreal_ofReal).comp hμ + have hBm : Measurable B := (Measure.measurable_lintegral hf.neg.ennreal_ofReal).comp hμ + have hA : (∫⁻ ω, A ω ∂P) = ∫⁻ x, ENNReal.ofReal (f x) ∂P.bind μ := + (Measure.lintegral_bind hμ.aemeasurable hf.ennreal_ofReal.aemeasurable).symm + have hB : (∫⁻ ω, B ω ∂P) = ∫⁻ x, ENNReal.ofReal (-f x) ∂P.bind μ := + (Measure.lintegral_bind hμ.aemeasurable hf.neg.ennreal_ofReal.aemeasurable).symm + have hAf : (∫⁻ ω, A ω ∂P) < ⊤ := hA.trans_lt (lintegral_ofReal_lt_top_of_integrable hfP) + have hBf : (∫⁻ ω, B ω ∂P) < ⊤ := hB.trans_lt (lintegral_ofReal_lt_top_of_integrable hfP.neg) + have he (ω : Ω) : (∫ x, f x ∂μ ω) = (A ω).toReal - (B ω).toReal := + integral_eq_lintegral_pos_part_sub_lintegral_neg_part (hfi ω) + refine ⟨?_,?_⟩ + · convert (integrable_toReal_of_lintegral_ne_top hAm.aemeasurable hAf.ne).sub + (integrable_toReal_of_lintegral_ne_top hBm.aemeasurable hBf.ne) using 1 + funext ω + exact he ω + simp_rw [he] + rw [integral_sub (integrable_toReal_of_lintegral_ne_top hAm.aemeasurable hAf.ne) + (integrable_toReal_of_lintegral_ne_top hBm.aemeasurable hBf.ne), + integral_toReal hAm.aemeasurable (Eventually.of_forall (fun ω => + lintegral_ofReal_lt_top_of_integrable (hfi ω))), + integral_toReal hBm.aemeasurable (Eventually.of_forall (fun ω => + lintegral_ofReal_lt_top_of_integrable (hfi ω).neg)),hA,hB] + exact (integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfP).symm + +theorem integral_bind_of_integrable_sections + {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] + (P : Measure Ω) (μ : Ω → Measure X) (hμ : Measurable μ) + (f : X → ℝ) (hf : Measurable f) + (hfi : ∀ ω, Integrable f (μ ω)) (hfP : Integrable f (P.bind μ)) : + (∫ ω, (∫ x, f x ∂μ ω) ∂P) = ∫ x, f x ∂P.bind μ := + (integrable_integral_bind_of_sections P μ hμ f hf hfi hfP).2 + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonPathCertificates.lean b/lean/OAI/Probability/SLE/Mass/RadonPathCertificates.lean index b06b2877e6c26028d3bad1fd90cb2a862173db93..62338820d44c8b586b5a5457de6a387261cc488d 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonPathCertificates.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonPathCertificates.lean @@ -1,77 +1,77 @@ -import OAI.Probability.SLE.Mass.RadonTestDetermines -import OAI.Probability.SLE.Mass.RadonTestLimitProperties -import OAI.Probability.SLE.Mass.RationalTraceMassProgressiveVersion - -namespace OAI - -/-! Countable compact-test certificates for order and continuous Radon paths. -The cutoff coordinates in the determining family supply local mass bounds. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported NNReal - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - -private lemma positive_part_error {a b e : ℝ} (ha : 0 ≤ a) (h : |a-b| ≤ e) : - |a-max b 0| ≤ e := by - rcases le_total 0 b with hb | hb - · simpa only [max_eq_left hb] using h - · rw [max_eq_right hb,sub_zero,abs_of_nonneg ha] - rw [abs_of_nonneg (sub_nonneg.mpr (hb.trans ha))] at h - linarith - -theorem LocalTestFamily.measure_le_of_positive_tests (D : LocalTestFamily X) - (μ ν : Measure X) [μ.Regular] [ν.Regular] - (h : ∀ j, (∫ x, (D.test j ⊔ 0) x ∂μ) ≤ ∫ x, (D.test j ⊔ 0) x ∂ν) : μ ≤ ν := by - apply radon_measure_le_of_nonneg_tests - intro f hf - obtain ⟨k,_,happrox⟩ := D.weighted_approximation f - let A := ∫ x, f x ∂μ - let B := ∫ x, f x ∂ν - let M := (∫ x, D.test k x ∂μ)+(∫ x, D.test k x ∂ν) - have hb (n : ℕ) : A ≤ B+(1/(n+1:ℝ))*M := by - obtain ⟨j,hj⟩ := happrox (1/(n+1:ℝ)) (by positivity) - have hp (x : X) : |f x-(D.test j ⊔ 0) x| ≤ (1/(n+1:ℝ))*D.test k x := - positive_part_error (hf x) (hj x) - have hμ := integral_sub_le_cutoff μ f (D.test j ⊔ 0) (D.test k) _ hp - have hν := integral_sub_le_cutoff ν f (D.test j ⊔ 0) (D.test k) _ hp - have hh := h j - dsimp only [A,B,M] - have hμ' := le_abs_self ((∫ x, f x ∂μ)-(∫ x, (D.test j ⊔ 0) x ∂μ)) - have hν' := neg_le_abs ((∫ x, f x ∂ν)-(∫ x, (D.test j ⊔ 0) x ∂ν)) - nlinarith - have hl : Tendsto (fun n : ℕ => B+(1/(n+1:ℝ))*M) atTop (𝓝 B) := by - simpa only [zero_mul,add_zero] using tendsto_const_nhds.add - ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).mul_const M) - exact le_of_tendsto_of_tendsto' tendsto_const_nhds hl hb - -theorem LocalTestFamily.continuous_integrals_of_tests (D : LocalTestFamily X) - {T : Type*} [TopologicalSpace T] [SequentialSpace T] - (N : T → Measure X) (hr : ∀ t, (N t).Regular) - (hc : ∀ j, Continuous (fun t => ∫ x, D.test j x ∂N t)) : - ∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂N t) := by - intro f - apply SeqContinuous.continuous - intro t a hta - let (n : ℕ) : IsFiniteMeasureOnCompacts (N (t n)) := by - let := hr (t n) - infer_instance - obtain ⟨ν,hν,_,hl⟩ := D.exists_radon_limit (fun n => N (t n)) - (fun j => ⟨∫ x, D.test j x ∂N a,(hc j).continuousAt.tendsto.comp hta⟩) - have he : ν=N a := by - let := hν - let := hr a - apply D.measure_eq_of_test_integrals - intro j - exact tendsto_nhds_unique (hl (D.test j)) ((hc j).continuousAt.tendsto.comp hta) - simpa only [he,Function.comp_def] using hl f - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonTestDetermines +import OAI.Probability.SLE.Mass.RadonTestLimitProperties +import OAI.Probability.SLE.Mass.RationalTraceMassProgressiveVersion + +namespace OAI + +/-! Countable compact-test certificates for order and continuous Radon paths. +The cutoff coordinates in the determining family supply local mass bounds. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported NNReal + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + +private lemma positive_part_error {a b e : ℝ} (ha : 0 ≤ a) (h : |a-b| ≤ e) : + |a-max b 0| ≤ e := by + rcases le_total 0 b with hb | hb + · simpa only [max_eq_left hb] using h + · rw [max_eq_right hb,sub_zero,abs_of_nonneg ha] + rw [abs_of_nonneg (sub_nonneg.mpr (hb.trans ha))] at h + linarith + +theorem LocalTestFamily.measure_le_of_positive_tests (D : LocalTestFamily X) + (μ ν : Measure X) [μ.Regular] [ν.Regular] + (h : ∀ j, (∫ x, (D.test j ⊔ 0) x ∂μ) ≤ ∫ x, (D.test j ⊔ 0) x ∂ν) : μ ≤ ν := by + apply radon_measure_le_of_nonneg_tests + intro f hf + obtain ⟨k,_,happrox⟩ := D.weighted_approximation f + let A := ∫ x, f x ∂μ + let B := ∫ x, f x ∂ν + let M := (∫ x, D.test k x ∂μ)+(∫ x, D.test k x ∂ν) + have hb (n : ℕ) : A ≤ B+(1/(n+1:ℝ))*M := by + obtain ⟨j,hj⟩ := happrox (1/(n+1:ℝ)) (by positivity) + have hp (x : X) : |f x-(D.test j ⊔ 0) x| ≤ (1/(n+1:ℝ))*D.test k x := + positive_part_error (hf x) (hj x) + have hμ := integral_sub_le_cutoff μ f (D.test j ⊔ 0) (D.test k) _ hp + have hν := integral_sub_le_cutoff ν f (D.test j ⊔ 0) (D.test k) _ hp + have hh := h j + dsimp only [A,B,M] + have hμ' := le_abs_self ((∫ x, f x ∂μ)-(∫ x, (D.test j ⊔ 0) x ∂μ)) + have hν' := neg_le_abs ((∫ x, f x ∂ν)-(∫ x, (D.test j ⊔ 0) x ∂ν)) + nlinarith + have hl : Tendsto (fun n : ℕ => B+(1/(n+1:ℝ))*M) atTop (𝓝 B) := by + simpa only [zero_mul,add_zero] using tendsto_const_nhds.add + ((tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).mul_const M) + exact le_of_tendsto_of_tendsto' tendsto_const_nhds hl hb + +theorem LocalTestFamily.continuous_integrals_of_tests (D : LocalTestFamily X) + {T : Type*} [TopologicalSpace T] [SequentialSpace T] + (N : T → Measure X) (hr : ∀ t, (N t).Regular) + (hc : ∀ j, Continuous (fun t => ∫ x, D.test j x ∂N t)) : + ∀ f : C_c(X,ℝ), Continuous (fun t => ∫ x, f x ∂N t) := by + intro f + apply SeqContinuous.continuous + intro t a hta + let (n : ℕ) : IsFiniteMeasureOnCompacts (N (t n)) := by + let := hr (t n) + infer_instance + obtain ⟨ν,hν,_,hl⟩ := D.exists_radon_limit (fun n => N (t n)) + (fun j => ⟨∫ x, D.test j x ∂N a,(hc j).continuousAt.tendsto.comp hta⟩) + have he : ν=N a := by + let := hν + let := hr a + apply D.measure_eq_of_test_integrals + intro j + exact tendsto_nhds_unique (hl (D.test j)) ((hc j).continuousAt.tendsto.comp hta) + simpa only [he,Function.comp_def] using hl f + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonProcessAEUniqueness.lean b/lean/OAI/Probability/SLE/Mass/RadonProcessAEUniqueness.lean index 03f04266a254faafa22abcad587378d531421bf0..61838b5b553df66b309db23be91df0a9714f5d2e 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonProcessAEUniqueness.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonProcessAEUniqueness.lean @@ -1,39 +1,39 @@ -import OAI.Probability.SLE.Mass.RadonProcessUniqueness - -namespace OAI - -/-! A single almost-sure Radon certificate suffices when upgrading -simultaneous positive-test identities to identities of measure paths. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported NNReal - -namespace SLEExactGauge - - theorem random_radon_process_eq_of_positive_tests_ae - {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (D : LocalTestFamily X) (N M : ℝ≥0 → Ω → Measure X) - (hNr : ∀ᵐ ω ∂P, ∀ t, (N t ω).Regular) - (hMr : ∀ᵐ ω ∂P, ∀ t, (M t ω).Regular) - (htest : ∀ f : C_c(X,ℝ), (∀ x, 0 ≤ f x) → - ∀ᵐ ω ∂P, ∀ t, (∫ x, f x ∂N t ω)=∫ x, f x ∂M t ω) : - ∀ᵐ ω ∂P, ∀ t, N t ω=M t ω := by - have hp (j : ℕ) := htest (D.test j ⊔ 0) (fun x => le_max_right _ _) - have hn (j : ℕ) := htest ((-D.test j) ⊔ 0) (fun x => le_max_right _ _) - filter_upwards [hNr,hMr,ae_all_iff.mpr hp,ae_all_iff.mpr hn] with ω hN hM hωp hωn - intro t - let := hN t - let := hM t - apply D.measure_eq_of_test_integrals - intro j - rw [compact_test_integral_pos_sub_neg (N t ω) (D.test j), - compact_test_integral_pos_sub_neg (M t ω) (D.test j),hωp j t,hωn j t] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonProcessUniqueness + +namespace OAI + +/-! A single almost-sure Radon certificate suffices when upgrading +simultaneous positive-test identities to identities of measure paths. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported NNReal + +namespace SLEExactGauge + + theorem random_radon_process_eq_of_positive_tests_ae + {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (D : LocalTestFamily X) (N M : ℝ≥0 → Ω → Measure X) + (hNr : ∀ᵐ ω ∂P, ∀ t, (N t ω).Regular) + (hMr : ∀ᵐ ω ∂P, ∀ t, (M t ω).Regular) + (htest : ∀ f : C_c(X,ℝ), (∀ x, 0 ≤ f x) → + ∀ᵐ ω ∂P, ∀ t, (∫ x, f x ∂N t ω)=∫ x, f x ∂M t ω) : + ∀ᵐ ω ∂P, ∀ t, N t ω=M t ω := by + have hp (j : ℕ) := htest (D.test j ⊔ 0) (fun x => le_max_right _ _) + have hn (j : ℕ) := htest ((-D.test j) ⊔ 0) (fun x => le_max_right _ _) + filter_upwards [hNr,hMr,ae_all_iff.mpr hp,ae_all_iff.mpr hn] with ω hN hM hωp hωn + intro t + let := hN t + let := hM t + apply D.measure_eq_of_test_integrals + intro j + rw [compact_test_integral_pos_sub_neg (N t ω) (D.test j), + compact_test_integral_pos_sub_neg (M t ω) (D.test j),hωp j t,hωn j t] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonProcessUniqueness.lean b/lean/OAI/Probability/SLE/Mass/RadonProcessUniqueness.lean index 924a138f9d28f962858a22c00bf7493cc0523575..a84624335503bcb5aecd69765f4ae00520f8b62f 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonProcessUniqueness.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonProcessUniqueness.lean @@ -1,51 +1,51 @@ -import OAI.Probability.SLE.Mass.MassKernelUniqueness -import OAI.Probability.SLE.Mass.RadonTestDetermines - -namespace OAI - -/-! Countably many positive compact tests give one common full-measure event -on which all times of two random Radon processes agree. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported NNReal - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - -omit [T2Space X] [LocallyCompactSpace X] in -lemma compact_test_integral_pos_sub_neg (μ : Measure X) [μ.Regular] (f : C_c(X,ℝ)) : - (∫ x, f x ∂μ) = (∫ x, (f ⊔ 0) x ∂μ)-(∫ x, ((-f) ⊔ 0) x ∂μ) := by - rw [← integral_sub (f ⊔ 0).integrable ((-f) ⊔ 0).integrable] - apply integral_congr_ae - filter_upwards [] with x - change f x = max (f x) 0-max (-f x) 0 - rcases le_total 0 (f x) with h | h - · rw [max_eq_left h,max_eq_right (neg_nonpos.mpr h),sub_zero] - · rw [max_eq_right h,max_eq_left (neg_nonneg.mpr h),zero_sub,neg_neg] - -theorem random_radon_process_eq_of_positive_tests - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (D : LocalTestFamily X) (N M : ℝ≥0 → Ω → Measure X) - (hNr : ∀ t ω, (N t ω).Regular) (hMr : ∀ t ω, (M t ω).Regular) - (htest : ∀ f : C_c(X,ℝ), (∀ x, 0 ≤ f x) → - ∀ᵐ ω ∂P, ∀ t, (∫ x, f x ∂N t ω) = ∫ x, f x ∂M t ω) : - ∀ᵐ ω ∂P, ∀ t, N t ω = M t ω := by - have hp (j : ℕ) := htest (D.test j ⊔ 0) (fun x => le_max_right _ _) - have hn (j : ℕ) := htest ((-D.test j) ⊔ 0) (fun x => le_max_right _ _) - filter_upwards [ae_all_iff.mpr hp,ae_all_iff.mpr hn] with ω hωp hωn - intro t - let := hNr t ω - let := hMr t ω - apply D.measure_eq_of_test_integrals - intro j - rw [compact_test_integral_pos_sub_neg (N t ω) (D.test j), - compact_test_integral_pos_sub_neg (M t ω) (D.test j),hωp j t,hωn j t] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.MassKernelUniqueness +import OAI.Probability.SLE.Mass.RadonTestDetermines + +namespace OAI + +/-! Countably many positive compact tests give one common full-measure event +on which all times of two random Radon processes agree. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported NNReal + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + +omit [T2Space X] [LocallyCompactSpace X] in +lemma compact_test_integral_pos_sub_neg (μ : Measure X) [μ.Regular] (f : C_c(X,ℝ)) : + (∫ x, f x ∂μ) = (∫ x, (f ⊔ 0) x ∂μ)-(∫ x, ((-f) ⊔ 0) x ∂μ) := by + rw [← integral_sub (f ⊔ 0).integrable ((-f) ⊔ 0).integrable] + apply integral_congr_ae + filter_upwards [] with x + change f x = max (f x) 0-max (-f x) 0 + rcases le_total 0 (f x) with h | h + · rw [max_eq_left h,max_eq_right (neg_nonpos.mpr h),sub_zero] + · rw [max_eq_right h,max_eq_left (neg_nonneg.mpr h),zero_sub,neg_neg] + +theorem random_radon_process_eq_of_positive_tests + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (D : LocalTestFamily X) (N M : ℝ≥0 → Ω → Measure X) + (hNr : ∀ t ω, (N t ω).Regular) (hMr : ∀ t ω, (M t ω).Regular) + (htest : ∀ f : C_c(X,ℝ), (∀ x, 0 ≤ f x) → + ∀ᵐ ω ∂P, ∀ t, (∫ x, f x ∂N t ω) = ∫ x, f x ∂M t ω) : + ∀ᵐ ω ∂P, ∀ t, N t ω = M t ω := by + have hp (j : ℕ) := htest (D.test j ⊔ 0) (fun x => le_max_right _ _) + have hn (j : ℕ) := htest ((-D.test j) ⊔ 0) (fun x => le_max_right _ _) + filter_upwards [ae_all_iff.mpr hp,ae_all_iff.mpr hn] with ω hωp hωn + intro t + let := hNr t ω + let := hMr t ω + apply D.measure_eq_of_test_integrals + intro j + rw [compact_test_integral_pos_sub_neg (N t ω) (D.test j), + compact_test_integral_pos_sub_neg (M t ω) (D.test j),hωp j t,hωn j t] + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonRandomTesting.lean b/lean/OAI/Probability/SLE/Mass/RadonRandomTesting.lean index e199bd1227710da38b76e296bba2b8b11ee04a97..8052e4fe42f637c7014241eba5696488e6c6a629 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonRandomTesting.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonRandomTesting.lean @@ -1,64 +1,64 @@ -import OAI.Probability.SLE.Geometry.RandomSpatialTesting -import OAI.Probability.SLE.LoewnerCharts - -namespace OAI - -/-! Nonnegative random tests of locally finite half-plane measures. -Compact restrictions remove any need for a finite total mass. -/ - -noncomputable section -open Set Filter MeasureTheory ProbabilityTheory Classical -open scoped ENNReal - -namespace SLEExactGauge - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -private lemma finite_halfPlane_random_test_measurable - {Ω : Type*} [MeasurableSpace Ω] - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) - (hfin : ∀ ω, IsFiniteMeasure (μ ω)) - (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable f) : - Measurable (fun ω => ∫⁻ z, f (ω,z) ∂μ ω) := by - let K : Kernel Ω upperHalfPlane := ⟨μ,hμ⟩ - have hp : Measurable (fun ω => (μ ω).map (Prod.mk ω)) := by - apply Measure.measurable_of_measurable_coe - intro A hA - have h := Kernel.measurable_kernel_prodMk_left_of_finite (κ := K) hA hfin - change Measurable (fun a => μ a (Prod.mk a ⁻¹' A)) at h - simpa only [Measure.map_apply measurable_prodMk_left hA] using h - have h := (Measure.measurable_lintegral hf).comp hp - convert h using 1 - funext ω - exact (lintegral_map hf measurable_prodMk_left).symm - -theorem halfPlane_random_test_measurable - {Ω : Type*} [MeasurableSpace Ω] - (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) - (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω)) - (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable f) : - Measurable (fun ω => ∫⁻ z, f (ω,z) ∂μ ω) := by - have he (ω : Ω) : (∫⁻ z, f (ω,z) ∂μ ω) = - ⨆ n, ∫⁻ z in compactCovering upperHalfPlane n, f (ω,z) ∂μ ω := by - rw [← setLIntegral_iUnion_of_directed _ (Monotone.directed_le (compactCovering_subset upperHalfPlane)), - iUnion_compactCovering,Measure.restrict_univ] - simp_rw [he] - apply Measurable.iSup - intro n - have hK := (isCompact_compactCovering upperHalfPlane n).measurableSet - have hm : Measurable (fun ω => (μ ω).restrict (compactCovering upperHalfPlane n)) := by - apply Measure.measurable_of_measurable_coe - intro A hA - simp only [Measure.restrict_apply hA] - exact (Measure.measurable_coe (hA.inter hK)).comp hμ - apply finite_halfPlane_random_test_measurable _ hm _ f hf - intro ω - let := hfin ω - exact ⟨by simpa only [Measure.restrict_apply_univ] using - (isCompact_compactCovering upperHalfPlane n).measure_lt_top (μ := μ ω)⟩ - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Geometry.RandomSpatialTesting +import OAI.Probability.SLE.LoewnerCharts + +namespace OAI + +/-! Nonnegative random tests of locally finite half-plane measures. +Compact restrictions remove any need for a finite total mass. -/ + +noncomputable section +open Set Filter MeasureTheory ProbabilityTheory Classical +open scoped ENNReal + +namespace SLEExactGauge + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +private lemma finite_halfPlane_random_test_measurable + {Ω : Type*} [MeasurableSpace Ω] + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) + (hfin : ∀ ω, IsFiniteMeasure (μ ω)) + (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable f) : + Measurable (fun ω => ∫⁻ z, f (ω,z) ∂μ ω) := by + let K : Kernel Ω upperHalfPlane := ⟨μ,hμ⟩ + have hp : Measurable (fun ω => (μ ω).map (Prod.mk ω)) := by + apply Measure.measurable_of_measurable_coe + intro A hA + have h := Kernel.measurable_kernel_prodMk_left_of_finite (κ := K) hA hfin + change Measurable (fun a => μ a (Prod.mk a ⁻¹' A)) at h + simpa only [Measure.map_apply measurable_prodMk_left hA] using h + have h := (Measure.measurable_lintegral hf).comp hp + convert h using 1 + funext ω + exact (lintegral_map hf measurable_prodMk_left).symm + +theorem halfPlane_random_test_measurable + {Ω : Type*} [MeasurableSpace Ω] + (μ : Ω → Measure upperHalfPlane) (hμ : Measurable μ) + (hfin : ∀ ω, IsFiniteMeasureOnCompacts (μ ω)) + (f : Ω × upperHalfPlane → ℝ≥0∞) (hf : Measurable f) : + Measurable (fun ω => ∫⁻ z, f (ω,z) ∂μ ω) := by + have he (ω : Ω) : (∫⁻ z, f (ω,z) ∂μ ω) = + ⨆ n, ∫⁻ z in compactCovering upperHalfPlane n, f (ω,z) ∂μ ω := by + rw [← setLIntegral_iUnion_of_directed _ (Monotone.directed_le (compactCovering_subset upperHalfPlane)), + iUnion_compactCovering,Measure.restrict_univ] + simp_rw [he] + apply Measurable.iSup + intro n + have hK := (isCompact_compactCovering upperHalfPlane n).measurableSet + have hm : Measurable (fun ω => (μ ω).restrict (compactCovering upperHalfPlane n)) := by + apply Measure.measurable_of_measurable_coe + intro A hA + simp only [Measure.restrict_apply hA] + exact (Measure.measurable_coe (hA.inter hK)).comp hμ + apply finite_halfPlane_random_test_measurable _ hm _ f hf + intro ω + let := hfin ω + exact ⟨by simpa only [Measure.restrict_apply_univ] using + (isCompact_compactCovering upperHalfPlane n).measure_lt_top (μ := μ ω)⟩ + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonTerminalUniqueness.lean b/lean/OAI/Probability/SLE/Mass/RadonTerminalUniqueness.lean index 9c39003d25695ae4cbf9040ad4b66f89992b459f..dafe94c724802064a55c848dfde77d0c401f8017 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonTerminalUniqueness.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonTerminalUniqueness.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Mass.RadonProcessUniqueness -import OAI.Probability.SLE.Mass.RationalTraceMassLocalVariation -import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract - -namespace OAI - -/-! Local exhaustion determines the terminal measure once the finite-time -increasing Radon family is fixed. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped ENNReal NNReal Topology - -namespace SLEExactGauge - -lemma radon_terminal_eq_of_common_exhaustion - {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - (μ ν : Measure X) [μ.Regular] [ν.Regular] - (N : ℝ≥0 → Measure X) - (hμ : ∀ t, N t ≤ μ) (hν : ∀ t, N t ≤ ν) - (hlμ : ∀ K : Set X, IsCompact K → Tendsto (fun t : ℝ≥0 => (μ-N t) K) atTop (𝓝 0)) - (hlν : ∀ K : Set X, IsCompact K → Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) : - μ = ν := by - have hle (ρ σ : Measure X) [ρ.Regular] (hρ : ∀ t, N t ≤ ρ) (hσ : ∀ t, N t ≤ σ) - (hl : ∀ K : Set X, IsCompact K → Tendsto (fun t : ℝ≥0 => (ρ-N t) K) atTop (𝓝 0)) - (K : Set X) (hK : IsCompact K) : ρ K ≤ σ K := by - have hb (t : ℝ≥0) : ρ K ≤ σ K+(ρ-N t) K := by - let : IsFiniteMeasureOnCompacts (N t) := - ⟨fun L hL => lt_of_le_of_lt (hρ t L) hL.measure_lt_top⟩ - rw [sub_apply_of_finite_restriction (hρ t) hK.measurableSet hK.measure_lt_top.ne] - calc - ρ K = (ρ K-N t K)+N t K := (tsub_add_cancel_of_le (hρ t K)).symm - _ ≤ (ρ K-N t K)+σ K := add_le_add le_rfl (hσ t K) - _ = _ := add_comm _ _ - have hlim := (tendsto_const_nhds (x := σ K)).add (hl K hK) - simp only [add_zero] at hlim - exact le_of_tendsto_of_tendsto' tendsto_const_nhds hlim hb - apply Measure.OuterRegular.ext_isOpen - apply Measure.InnerRegularWRT.eq_of_innerRegularWRT_of_forall_eq - Measure.Regular.innerRegular Measure.Regular.innerRegular - intro K hK - exact (hle μ ν hμ hν hlμ K hK).antisymm (hle ν μ hν hμ hlν K hK) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonProcessUniqueness +import OAI.Probability.SLE.Mass.RationalTraceMassLocalVariation +import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract + +namespace OAI + +/-! Local exhaustion determines the terminal measure once the finite-time +increasing Radon family is fixed. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped ENNReal NNReal Topology + +namespace SLEExactGauge + +lemma radon_terminal_eq_of_common_exhaustion + {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + (μ ν : Measure X) [μ.Regular] [ν.Regular] + (N : ℝ≥0 → Measure X) + (hμ : ∀ t, N t ≤ μ) (hν : ∀ t, N t ≤ ν) + (hlμ : ∀ K : Set X, IsCompact K → Tendsto (fun t : ℝ≥0 => (μ-N t) K) atTop (𝓝 0)) + (hlν : ∀ K : Set X, IsCompact K → Tendsto (fun t : ℝ≥0 => (ν-N t) K) atTop (𝓝 0)) : + μ = ν := by + have hle (ρ σ : Measure X) [ρ.Regular] (hρ : ∀ t, N t ≤ ρ) (hσ : ∀ t, N t ≤ σ) + (hl : ∀ K : Set X, IsCompact K → Tendsto (fun t : ℝ≥0 => (ρ-N t) K) atTop (𝓝 0)) + (K : Set X) (hK : IsCompact K) : ρ K ≤ σ K := by + have hb (t : ℝ≥0) : ρ K ≤ σ K+(ρ-N t) K := by + let : IsFiniteMeasureOnCompacts (N t) := + ⟨fun L hL => lt_of_le_of_lt (hρ t L) hL.measure_lt_top⟩ + rw [sub_apply_of_finite_restriction (hρ t) hK.measurableSet hK.measure_lt_top.ne] + calc + ρ K = (ρ K-N t K)+N t K := (tsub_add_cancel_of_le (hρ t K)).symm + _ ≤ (ρ K-N t K)+σ K := add_le_add le_rfl (hσ t K) + _ = _ := add_comm _ _ + have hlim := (tendsto_const_nhds (x := σ K)).add (hl K hK) + simp only [add_zero] at hlim + exact le_of_tendsto_of_tendsto' tendsto_const_nhds hlim hb + apply Measure.OuterRegular.ext_isOpen + apply Measure.InnerRegularWRT.eq_of_innerRegularWRT_of_forall_eq + Measure.Regular.innerRegular Measure.Regular.innerRegular + intro K hK + exact (hle μ ν hμ hν hlμ K hK).antisymm (hle ν μ hν hμ hlν K hK) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonTestDetermines.lean b/lean/OAI/Probability/SLE/Mass/RadonTestDetermines.lean index 47041cca8db6d5d14378c29a9ec4332934ed64ad..fa80d83ea268deee524709416607a877852adee6 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonTestDetermines.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonTestDetermines.lean @@ -1,49 +1,49 @@ -import OAI.Probability.SLE.Mass.RadonTestLimit - -namespace OAI - -/-! The constructed countable local family determines a regular measure. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - -theorem LocalTestFamily.measure_eq_of_test_integrals (D : LocalTestFamily X) - (μ ν : Measure X) [μ.Regular] [ν.Regular] - (h : ∀ j, (∫ x, D.test j x ∂μ) = ∫ x, D.test j x ∂ν) : μ = ν := by - apply Measure.ext_of_integral_eq_on_compactlySupported - intro f - let M (n : ℕ) := if n % 2 = 0 then μ else ν - let (n : ℕ) : IsFiniteMeasureOnCompacts (M n) := by - dsimp only [M] - split_ifs <;> infer_instance - have hc : CauchySeq (fun n => ∫ x, f x ∂M n) := D.integrals_cauchy M (fun j => by - refine ⟨∫ x, D.test j x ∂ν,?_⟩ - have he : (fun n => ∫ x, D.test j x ∂M n) = fun _ => ∫ x, D.test j x ∂ν := by - funext n - dsimp only [M] - split_ifs - · exact h j - · rfl - rw [he] - exact tendsto_const_nhds) f - apply eq_of_dist_eq_zero - apply le_antisymm _ dist_nonneg - apply le_of_forall_pos_le_add - intro ε hε - obtain ⟨N,hN⟩ := Metric.cauchySeq_iff.mp hc ε hε - have hh := hN (2*N) (by omega) (2*N+1) (by omega) - have hm0 : (2*N)%2=0 := by omega - have hm1 : (2*N+1)%2≠0 := by omega - simpa only [M,hm0,hm1,ite_true,ite_false,zero_add] using hh.le - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonTestLimit + +namespace OAI + +/-! The constructed countable local family determines a regular measure. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + +theorem LocalTestFamily.measure_eq_of_test_integrals (D : LocalTestFamily X) + (μ ν : Measure X) [μ.Regular] [ν.Regular] + (h : ∀ j, (∫ x, D.test j x ∂μ) = ∫ x, D.test j x ∂ν) : μ = ν := by + apply Measure.ext_of_integral_eq_on_compactlySupported + intro f + let M (n : ℕ) := if n % 2 = 0 then μ else ν + let (n : ℕ) : IsFiniteMeasureOnCompacts (M n) := by + dsimp only [M] + split_ifs <;> infer_instance + have hc : CauchySeq (fun n => ∫ x, f x ∂M n) := D.integrals_cauchy M (fun j => by + refine ⟨∫ x, D.test j x ∂ν,?_⟩ + have he : (fun n => ∫ x, D.test j x ∂M n) = fun _ => ∫ x, D.test j x ∂ν := by + funext n + dsimp only [M] + split_ifs + · exact h j + · rfl + rw [he] + exact tendsto_const_nhds) f + apply eq_of_dist_eq_zero + apply le_antisymm _ dist_nonneg + apply le_of_forall_pos_le_add + intro ε hε + obtain ⟨N,hN⟩ := Metric.cauchySeq_iff.mp hc ε hε + have hh := hN (2*N) (by omega) (2*N+1) (by omega) + have hm0 : (2*N)%2=0 := by omega + have hm1 : (2*N+1)%2≠0 := by omega + simpa only [M,hm0,hm1,ite_true,ite_false,zero_add] using hh.le + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonTestFamily.lean b/lean/OAI/Probability/SLE/Mass/RadonTestFamily.lean index c3dace8daf4c7a0411ff0f9556fa227d770a4a05..11c732d6aef684eaea2322e3ca8f435a49897f24 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonTestFamily.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonTestFamily.lean @@ -1,118 +1,118 @@ -import OAI.Probability.SLE.Mass.RadonTestLimit -import Mathlib.Topology.ContinuousMap.SecondCountableSpace -import Mathlib.Topology.UrysohnsLemma - -namespace OAI - -/-! A countable determining family on a second-countable locally compact -space, constructed from compact exhaustion and separability on compacts. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [SecondCountableTopology X] - -omit [LocallyCompactSpace X] in -/-- Compact support in a fixed compact set admits a countable uniformly -dense family, with the same support constraint. -/ -theorem exists_compact_test_sequence (K : Set X) (hK : IsCompact K) : - ∃ f : ℕ → C_c(X, ℝ), (∀ j, tsupport (f j) ⊆ K) ∧ - ∀ g : C_c(X, ℝ), tsupport g ⊆ K → ∀ ε : ℝ, 0 < ε → - ∃ j, ∀ x, |g x-f j x| ≤ ε := by - classical - let : CompactSpace K := isCompact_iff_compactSpace.mp hK - let S : Set C(K, ℝ) := {g | ∃ f : C_c(X, ℝ), tsupport f ⊆ K ∧ ∀ x : K, g x=f x} - have hzero : (0 : C(K, ℝ)) ∈ S := by - refine ⟨0, ?_, fun _ => rfl⟩ - simp - let : Nonempty S := ⟨⟨0,hzero⟩⟩ - obtain ⟨d,hd⟩ := TopologicalSpace.exists_dense_seq S - have hrep (j : ℕ) : ∃ f : C_c(X, ℝ), tsupport f ⊆ K ∧ ∀ x : K, (d j).1 x=f x := - (d j).2 - choose f hf hrep using hrep - refine ⟨f,hf,fun g hg ε hε => ?_⟩ - let r : C(K, ℝ) := ⟨fun x => g x, g.continuous.comp continuous_subtype_val⟩ - have hr : r ∈ S := ⟨g,hg,fun _ => rfl⟩ - obtain ⟨j,hj⟩ := Metric.denseRange_iff.mp hd ⟨r,hr⟩ ε hε - refine ⟨j,fun x => ?_⟩ - by_cases hx : x ∈ K - · have hb := ContinuousMap.norm_coe_le_norm (r-(d j).1) ⟨x,hx⟩ - have hb' : |g x-f j x| ≤ dist (⟨r,hr⟩ : S) (d j) := by - change |g x-f j x| ≤ dist r (d j).1 - rw [dist_eq_norm] - simpa only [ContinuousMap.sub_apply, Real.norm_eq_abs, hrep j ⟨x,hx⟩, r, - ContinuousMap.coe_mk] using hb - exact hb'.trans hj.le - · have hgx : g x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hg h)) - have hfx : f j x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hf j h)) - simpa only [hgx,hfx,sub_self,abs_zero] using hε.le - -omit [SecondCountableTopology X] in -/-- One nonnegative compactly supported cutoff dominates one on a given -compact set. -/ -theorem exists_compact_test_cutoff (K : Set X) (hK : IsCompact K) : - ∃ f : C_c(X, ℝ), (∀ x, 0 ≤ f x) ∧ ∀ x ∈ K, 1 ≤ f x := by - obtain ⟨f,hf,hfc,_,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen - hK isOpen_univ (subset_univ K) - refine ⟨⟨f,hfc⟩,fun x => (h01 x).1,fun x hx => ?_⟩ - change 1 ≤ f x - rw [hf hx] - rfl - -/-- The determining family required by Radon reconstruction exists -without an additional density or cutoff hypothesis. -/ -theorem exists_localTestFamily : Nonempty (LocalTestFamily X) := by - classical - let K := CompactExhaustion.choice X - have hdense (n : ℕ) := exists_compact_test_sequence (K n) (K.isCompact n) - choose f hf hfdense using hdense - have hcut (n : ℕ) := exists_compact_test_cutoff (K n) (K.isCompact n) - choose ψ hψ0 hψ1 using hcut - let test (j : ℕ) : C_c(X, ℝ) := - match Encodable.decode (α := ℕ ⊕ (ℕ × ℕ)) j with - | some (.inl n) => ψ n - | some (.inr (n,k)) => f n k - | none => 0 - have htest_cut (n : ℕ) : test (Encodable.encode (Sum.inl n : ℕ ⊕ (ℕ × ℕ)))=ψ n := by - simp only [test, Encodable.encodek] - have htest_dense (n j : ℕ) : - test (Encodable.encode (Sum.inr (n,j) : ℕ ⊕ (ℕ × ℕ)))=f n j := by - simp only [test, Encodable.encodek] - refine ⟨{ - compactSet := K - isCompact := K.isCompact - exhaustive := fun _ h => K.exists_superset_of_isCompact h - test := test - cutoffIndex := fun n => Encodable.encode (Sum.inl n : ℕ ⊕ (ℕ × ℕ)) - cutoff_nonneg := ?_ - cutoff_one_le := ?_ - dense := ?_ }⟩ - · intro n x - rw [htest_cut] - exact hψ0 n x - · intro n x hx - rw [htest_cut] - exact hψ1 n x hx - · intro n g hg ε hε - obtain ⟨j,hj⟩ := hfdense n g hg ε hε - refine ⟨Encodable.encode (Sum.inr (n,j) : ℕ ⊕ (ℕ × ℕ)), ?_, ?_⟩ - · rw [htest_dense] - exact hf n j - · rw [htest_dense] - exact hj - -/-- A chosen countable family can be used for all times and all random -outcomes of a measure-valued construction. -/ -def localTestFamily (X : Type*) [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [SecondCountableTopology X] : LocalTestFamily X := - Classical.choice exists_localTestFamily - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonTestLimit +import Mathlib.Topology.ContinuousMap.SecondCountableSpace +import Mathlib.Topology.UrysohnsLemma + +namespace OAI + +/-! A countable determining family on a second-countable locally compact +space, constructed from compact exhaustion and separability on compacts. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [SecondCountableTopology X] + +omit [LocallyCompactSpace X] in +/-- Compact support in a fixed compact set admits a countable uniformly +dense family, with the same support constraint. -/ +theorem exists_compact_test_sequence (K : Set X) (hK : IsCompact K) : + ∃ f : ℕ → C_c(X, ℝ), (∀ j, tsupport (f j) ⊆ K) ∧ + ∀ g : C_c(X, ℝ), tsupport g ⊆ K → ∀ ε : ℝ, 0 < ε → + ∃ j, ∀ x, |g x-f j x| ≤ ε := by + classical + let : CompactSpace K := isCompact_iff_compactSpace.mp hK + let S : Set C(K, ℝ) := {g | ∃ f : C_c(X, ℝ), tsupport f ⊆ K ∧ ∀ x : K, g x=f x} + have hzero : (0 : C(K, ℝ)) ∈ S := by + refine ⟨0, ?_, fun _ => rfl⟩ + simp + let : Nonempty S := ⟨⟨0,hzero⟩⟩ + obtain ⟨d,hd⟩ := TopologicalSpace.exists_dense_seq S + have hrep (j : ℕ) : ∃ f : C_c(X, ℝ), tsupport f ⊆ K ∧ ∀ x : K, (d j).1 x=f x := + (d j).2 + choose f hf hrep using hrep + refine ⟨f,hf,fun g hg ε hε => ?_⟩ + let r : C(K, ℝ) := ⟨fun x => g x, g.continuous.comp continuous_subtype_val⟩ + have hr : r ∈ S := ⟨g,hg,fun _ => rfl⟩ + obtain ⟨j,hj⟩ := Metric.denseRange_iff.mp hd ⟨r,hr⟩ ε hε + refine ⟨j,fun x => ?_⟩ + by_cases hx : x ∈ K + · have hb := ContinuousMap.norm_coe_le_norm (r-(d j).1) ⟨x,hx⟩ + have hb' : |g x-f j x| ≤ dist (⟨r,hr⟩ : S) (d j) := by + change |g x-f j x| ≤ dist r (d j).1 + rw [dist_eq_norm] + simpa only [ContinuousMap.sub_apply, Real.norm_eq_abs, hrep j ⟨x,hx⟩, r, + ContinuousMap.coe_mk] using hb + exact hb'.trans hj.le + · have hgx : g x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hg h)) + have hfx : f j x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hf j h)) + simpa only [hgx,hfx,sub_self,abs_zero] using hε.le + +omit [SecondCountableTopology X] in +/-- One nonnegative compactly supported cutoff dominates one on a given +compact set. -/ +theorem exists_compact_test_cutoff (K : Set X) (hK : IsCompact K) : + ∃ f : C_c(X, ℝ), (∀ x, 0 ≤ f x) ∧ ∀ x ∈ K, 1 ≤ f x := by + obtain ⟨f,hf,hfc,_,h01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen + hK isOpen_univ (subset_univ K) + refine ⟨⟨f,hfc⟩,fun x => (h01 x).1,fun x hx => ?_⟩ + change 1 ≤ f x + rw [hf hx] + rfl + +/-- The determining family required by Radon reconstruction exists +without an additional density or cutoff hypothesis. -/ +theorem exists_localTestFamily : Nonempty (LocalTestFamily X) := by + classical + let K := CompactExhaustion.choice X + have hdense (n : ℕ) := exists_compact_test_sequence (K n) (K.isCompact n) + choose f hf hfdense using hdense + have hcut (n : ℕ) := exists_compact_test_cutoff (K n) (K.isCompact n) + choose ψ hψ0 hψ1 using hcut + let test (j : ℕ) : C_c(X, ℝ) := + match Encodable.decode (α := ℕ ⊕ (ℕ × ℕ)) j with + | some (.inl n) => ψ n + | some (.inr (n,k)) => f n k + | none => 0 + have htest_cut (n : ℕ) : test (Encodable.encode (Sum.inl n : ℕ ⊕ (ℕ × ℕ)))=ψ n := by + simp only [test, Encodable.encodek] + have htest_dense (n j : ℕ) : + test (Encodable.encode (Sum.inr (n,j) : ℕ ⊕ (ℕ × ℕ)))=f n j := by + simp only [test, Encodable.encodek] + refine ⟨{ + compactSet := K + isCompact := K.isCompact + exhaustive := fun _ h => K.exists_superset_of_isCompact h + test := test + cutoffIndex := fun n => Encodable.encode (Sum.inl n : ℕ ⊕ (ℕ × ℕ)) + cutoff_nonneg := ?_ + cutoff_one_le := ?_ + dense := ?_ }⟩ + · intro n x + rw [htest_cut] + exact hψ0 n x + · intro n x hx + rw [htest_cut] + exact hψ1 n x hx + · intro n g hg ε hε + obtain ⟨j,hj⟩ := hfdense n g hg ε hε + refine ⟨Encodable.encode (Sum.inr (n,j) : ℕ ⊕ (ℕ × ℕ)), ?_, ?_⟩ + · rw [htest_dense] + exact hf n j + · rw [htest_dense] + exact hj + +/-- A chosen countable family can be used for all times and all random +outcomes of a measure-valued construction. -/ +def localTestFamily (X : Type*) [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [SecondCountableTopology X] : LocalTestFamily X := + Classical.choice exists_localTestFamily + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonTestLimit.lean b/lean/OAI/Probability/SLE/Mass/RadonTestLimit.lean index a574783b1bf742227b82656c294305d542f3d4cf..c27a84d8d8ca84eeea4db66ef954933d4a618c20 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonTestLimit.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonTestLimit.lean @@ -1,138 +1,138 @@ -import Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real -import Mathlib.Topology.MetricSpace.Sequences - -namespace OAI - -/-! Deterministic reconstruction in `mass:construction`. A countable -family locally dense in compactly supported continuous functions, -including nonnegative exhaustion cutoffs, determines one Radon limit. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - -/-- A countable determining family with explicit support control. -Density is in the uniform norm within each compact exhaustion set. -/ -structure LocalTestFamily (X : Type*) [TopologicalSpace X] where - compactSet : ℕ → Set X - isCompact : ∀ n, IsCompact (compactSet n) - exhaustive : ∀ K : Set X, IsCompact K → ∃ n, K ⊆ compactSet n - test : ℕ → C_c(X, ℝ) - cutoffIndex : ℕ → ℕ - cutoff_nonneg : ∀ n x, 0 ≤ test (cutoffIndex n) x - cutoff_one_le : ∀ n x, x ∈ compactSet n → 1 ≤ test (cutoffIndex n) x - dense : ∀ n (f : C_c(X, ℝ)), tsupport f ⊆ compactSet n → - ∀ ε : ℝ, 0 < ε → ∃ j, tsupport (test j) ⊆ compactSet n ∧ - ∀ x, |f x-test j x| ≤ ε - -omit [T2Space X] [LocallyCompactSpace X] [MeasurableSpace X] [BorelSpace X] in -lemma LocalTestFamily.weighted_approximation (D : LocalTestFamily X) (f : C_c(X, ℝ)) : - ∃ k, (∀ x, 0 ≤ D.test k x) ∧ ∀ ε : ℝ, 0 < ε → - ∃ j, ∀ x, |f x-D.test j x| ≤ ε * D.test k x := by - obtain ⟨n,hn⟩ := D.exhaustive (tsupport f) f.hasCompactSupport - refine ⟨D.cutoffIndex n,D.cutoff_nonneg n,fun ε hε => ?_⟩ - obtain ⟨j,hj,happrox⟩ := D.dense n f hn ε hε - refine ⟨j,fun x => ?_⟩ - by_cases hx : x ∈ D.compactSet n - · exact (happrox x).trans (le_mul_of_one_le_right hε.le (D.cutoff_one_le n x hx)) - · have hfx : f x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hn h)) - have hjx : D.test j x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hj h)) - rw [hfx,hjx,sub_self,abs_zero] - exact mul_nonneg hε.le (D.cutoff_nonneg n x) - -omit [T2Space X] [LocallyCompactSpace X] in -lemma integral_sub_le_cutoff (μ : Measure X) [IsFiniteMeasureOnCompacts μ] - (f g ψ : C_c(X, ℝ)) (ε : ℝ) - (h : ∀ x, |f x-g x| ≤ ε*ψ x) : - |(∫ x, f x ∂μ)-(∫ x, g x ∂μ)| ≤ ε*(∫ x, ψ x ∂μ) := by - rw [← integral_sub f.integrable g.integrable] - have hh := norm_integral_le_of_norm_le (f := fun x => f x-g x) (ψ.integrable.const_mul ε) - (ae_of_all μ (fun x => by simpa only [Real.norm_eq_abs] using h x)) - simpa only [Real.norm_eq_abs,integral_const_mul] using hh - -omit [T2Space X] [LocallyCompactSpace X] in -/-- Compactly supported integrals are Cauchy once the countable tests -converge. Cutoff convergence supplies the uniform local mass bounds. -/ -theorem LocalTestFamily.integrals_cauchy (D : LocalTestFamily X) - (μ : ℕ → Measure X) [∀ n, IsFiniteMeasureOnCompacts (μ n)] - (htest : ∀ j, ∃ a : ℝ, Tendsto (fun n => ∫ x, D.test j x ∂μ n) atTop (𝓝 a)) - (f : C_c(X, ℝ)) : CauchySeq (fun n => ∫ x, f x ∂μ n) := by - obtain ⟨k,_,hk⟩ := D.weighted_approximation f - obtain ⟨a,ha⟩ := htest k - obtain ⟨B,hB⟩ := (Metric.isBounded_range_of_tendsto _ ha).exists_norm_le - have hB0 : 0 ≤ B := (norm_nonneg _).trans (hB _ (mem_range_self 0)) - have hB' (n : ℕ) : (∫ x, D.test k x ∂μ n) ≤ B := - (le_abs_self _).trans (by simpa only [Real.norm_eq_abs] using hB _ (mem_range_self n)) - apply Metric.cauchySeq_iff.mpr - intro ε hε - let δ : ℝ := ε/(4*(B+1)) - have hδ : 0 < δ := by dsimp [δ]; positivity - have hδB : δ*B < ε/4 := by - dsimp [δ] - rw [div_mul_eq_mul_div,div_lt_div_iff₀ (by positivity) (by norm_num : (0:ℝ)<4)] - nlinarith - obtain ⟨j,hj⟩ := hk δ hδ - obtain ⟨a,ha⟩ := htest j - obtain ⟨N,hN⟩ := Metric.cauchySeq_iff.mp ha.cauchySeq (ε/2) (by positivity) - have herror (n : ℕ) : dist (∫ x, f x ∂μ n) (∫ x, D.test j x ∂μ n) ≤ δ*B := by - rw [Real.dist_eq] - exact (integral_sub_le_cutoff (μ n) f (D.test j) (D.test k) δ hj).trans - (mul_le_mul_of_nonneg_left (hB' n) hδ.le) - refine ⟨N,fun n hn m hm => ?_⟩ - have hmid := hN n hn m hm - have htri := dist_triangle (∫ x, f x ∂μ n) (∫ x, D.test j x ∂μ n) (∫ x, f x ∂μ m) - have htri' := dist_triangle (∫ x, D.test j x ∂μ n) (∫ x, D.test j x ∂μ m) (∫ x, f x ∂μ m) - have hright : dist (∫ x, D.test j x ∂μ m) (∫ x, f x ∂μ m) ≤ δ*B := by - rw [dist_comm] - exact herror m - have hleft := herror n - linarith - -/-- Positive limits of compact continuous tests are represented by one -locally finite regular measure. -/ -theorem radon_limit_of_all_test_limits (μ : ℕ → Measure X) - [∀ n, IsFiniteMeasureOnCompacts (μ n)] - (hlimit : ∀ f : C_c(X, ℝ), ∃ a : ℝ, - Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 a)) : - ∃ ν : Measure X, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ - ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) := by - choose L hL using hlimit - let I (n : ℕ) : C_c(X, ℝ) →ₚ[ℝ] ℝ := - CompactlySupportedContinuousMap.integralPositiveLinearMap (μ n) - have hI (f : C_c(X, ℝ)) : Tendsto (fun n => I n f) atTop (𝓝 (L f)) := hL f - have hadd (f g : C_c(X, ℝ)) : L (f+g) = L f+L g := by - apply tendsto_nhds_unique (hI (f+g)) - simpa only [map_add] using (hI f).add (hI g) - have hsmul (c : ℝ) (f : C_c(X, ℝ)) : L (c • f) = c • L f := by - apply tendsto_nhds_unique (hI (c • f)) - simpa only [map_smul] using tendsto_const_nhds.smul (hI f) - let Λ : C_c(X, ℝ) →ₚ[ℝ] ℝ := - { toFun := L - map_add' := hadd - map_smul' := hsmul - monotone' := fun f g hfg => le_of_tendsto_of_tendsto' (hI f) (hI g) - (fun n => (I n).monotone hfg) } - refine ⟨RealRMK.rieszMeasure Λ,inferInstance,inferInstance,fun f => ?_⟩ - rw [RealRMK.integral_rieszMeasure] - exact hL f - -/-- Reconstruction from a single countable locally determining family. -/ -theorem LocalTestFamily.exists_radon_limit (D : LocalTestFamily X) - (μ : ℕ → Measure X) [∀ n, IsFiniteMeasureOnCompacts (μ n)] - (htest : ∀ j, ∃ a : ℝ, Tendsto (fun n => ∫ x, D.test j x ∂μ n) atTop (𝓝 a)) : - ∃ ν : Measure X, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ - ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) := by - apply radon_limit_of_all_test_limits μ - intro f - exact cauchySeq_tendsto_of_complete (D.integrals_cauchy μ htest f) - -end SLEExactGauge - -end - -end OAI +import Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real +import Mathlib.Topology.MetricSpace.Sequences + +namespace OAI + +/-! Deterministic reconstruction in `mass:construction`. A countable +family locally dense in compactly supported continuous functions, +including nonnegative exhaustion cutoffs, determines one Radon limit. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + +/-- A countable determining family with explicit support control. +Density is in the uniform norm within each compact exhaustion set. -/ +structure LocalTestFamily (X : Type*) [TopologicalSpace X] where + compactSet : ℕ → Set X + isCompact : ∀ n, IsCompact (compactSet n) + exhaustive : ∀ K : Set X, IsCompact K → ∃ n, K ⊆ compactSet n + test : ℕ → C_c(X, ℝ) + cutoffIndex : ℕ → ℕ + cutoff_nonneg : ∀ n x, 0 ≤ test (cutoffIndex n) x + cutoff_one_le : ∀ n x, x ∈ compactSet n → 1 ≤ test (cutoffIndex n) x + dense : ∀ n (f : C_c(X, ℝ)), tsupport f ⊆ compactSet n → + ∀ ε : ℝ, 0 < ε → ∃ j, tsupport (test j) ⊆ compactSet n ∧ + ∀ x, |f x-test j x| ≤ ε + +omit [T2Space X] [LocallyCompactSpace X] [MeasurableSpace X] [BorelSpace X] in +lemma LocalTestFamily.weighted_approximation (D : LocalTestFamily X) (f : C_c(X, ℝ)) : + ∃ k, (∀ x, 0 ≤ D.test k x) ∧ ∀ ε : ℝ, 0 < ε → + ∃ j, ∀ x, |f x-D.test j x| ≤ ε * D.test k x := by + obtain ⟨n,hn⟩ := D.exhaustive (tsupport f) f.hasCompactSupport + refine ⟨D.cutoffIndex n,D.cutoff_nonneg n,fun ε hε => ?_⟩ + obtain ⟨j,hj,happrox⟩ := D.dense n f hn ε hε + refine ⟨j,fun x => ?_⟩ + by_cases hx : x ∈ D.compactSet n + · exact (happrox x).trans (le_mul_of_one_le_right hε.le (D.cutoff_one_le n x hx)) + · have hfx : f x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hn h)) + have hjx : D.test j x=0 := image_eq_zero_of_notMem_tsupport (fun h => hx (hj h)) + rw [hfx,hjx,sub_self,abs_zero] + exact mul_nonneg hε.le (D.cutoff_nonneg n x) + +omit [T2Space X] [LocallyCompactSpace X] in +lemma integral_sub_le_cutoff (μ : Measure X) [IsFiniteMeasureOnCompacts μ] + (f g ψ : C_c(X, ℝ)) (ε : ℝ) + (h : ∀ x, |f x-g x| ≤ ε*ψ x) : + |(∫ x, f x ∂μ)-(∫ x, g x ∂μ)| ≤ ε*(∫ x, ψ x ∂μ) := by + rw [← integral_sub f.integrable g.integrable] + have hh := norm_integral_le_of_norm_le (f := fun x => f x-g x) (ψ.integrable.const_mul ε) + (ae_of_all μ (fun x => by simpa only [Real.norm_eq_abs] using h x)) + simpa only [Real.norm_eq_abs,integral_const_mul] using hh + +omit [T2Space X] [LocallyCompactSpace X] in +/-- Compactly supported integrals are Cauchy once the countable tests +converge. Cutoff convergence supplies the uniform local mass bounds. -/ +theorem LocalTestFamily.integrals_cauchy (D : LocalTestFamily X) + (μ : ℕ → Measure X) [∀ n, IsFiniteMeasureOnCompacts (μ n)] + (htest : ∀ j, ∃ a : ℝ, Tendsto (fun n => ∫ x, D.test j x ∂μ n) atTop (𝓝 a)) + (f : C_c(X, ℝ)) : CauchySeq (fun n => ∫ x, f x ∂μ n) := by + obtain ⟨k,_,hk⟩ := D.weighted_approximation f + obtain ⟨a,ha⟩ := htest k + obtain ⟨B,hB⟩ := (Metric.isBounded_range_of_tendsto _ ha).exists_norm_le + have hB0 : 0 ≤ B := (norm_nonneg _).trans (hB _ (mem_range_self 0)) + have hB' (n : ℕ) : (∫ x, D.test k x ∂μ n) ≤ B := + (le_abs_self _).trans (by simpa only [Real.norm_eq_abs] using hB _ (mem_range_self n)) + apply Metric.cauchySeq_iff.mpr + intro ε hε + let δ : ℝ := ε/(4*(B+1)) + have hδ : 0 < δ := by dsimp [δ]; positivity + have hδB : δ*B < ε/4 := by + dsimp [δ] + rw [div_mul_eq_mul_div,div_lt_div_iff₀ (by positivity) (by norm_num : (0:ℝ)<4)] + nlinarith + obtain ⟨j,hj⟩ := hk δ hδ + obtain ⟨a,ha⟩ := htest j + obtain ⟨N,hN⟩ := Metric.cauchySeq_iff.mp ha.cauchySeq (ε/2) (by positivity) + have herror (n : ℕ) : dist (∫ x, f x ∂μ n) (∫ x, D.test j x ∂μ n) ≤ δ*B := by + rw [Real.dist_eq] + exact (integral_sub_le_cutoff (μ n) f (D.test j) (D.test k) δ hj).trans + (mul_le_mul_of_nonneg_left (hB' n) hδ.le) + refine ⟨N,fun n hn m hm => ?_⟩ + have hmid := hN n hn m hm + have htri := dist_triangle (∫ x, f x ∂μ n) (∫ x, D.test j x ∂μ n) (∫ x, f x ∂μ m) + have htri' := dist_triangle (∫ x, D.test j x ∂μ n) (∫ x, D.test j x ∂μ m) (∫ x, f x ∂μ m) + have hright : dist (∫ x, D.test j x ∂μ m) (∫ x, f x ∂μ m) ≤ δ*B := by + rw [dist_comm] + exact herror m + have hleft := herror n + linarith + +/-- Positive limits of compact continuous tests are represented by one +locally finite regular measure. -/ +theorem radon_limit_of_all_test_limits (μ : ℕ → Measure X) + [∀ n, IsFiniteMeasureOnCompacts (μ n)] + (hlimit : ∀ f : C_c(X, ℝ), ∃ a : ℝ, + Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 a)) : + ∃ ν : Measure X, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ + ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) := by + choose L hL using hlimit + let I (n : ℕ) : C_c(X, ℝ) →ₚ[ℝ] ℝ := + CompactlySupportedContinuousMap.integralPositiveLinearMap (μ n) + have hI (f : C_c(X, ℝ)) : Tendsto (fun n => I n f) atTop (𝓝 (L f)) := hL f + have hadd (f g : C_c(X, ℝ)) : L (f+g) = L f+L g := by + apply tendsto_nhds_unique (hI (f+g)) + simpa only [map_add] using (hI f).add (hI g) + have hsmul (c : ℝ) (f : C_c(X, ℝ)) : L (c • f) = c • L f := by + apply tendsto_nhds_unique (hI (c • f)) + simpa only [map_smul] using tendsto_const_nhds.smul (hI f) + let Λ : C_c(X, ℝ) →ₚ[ℝ] ℝ := + { toFun := L + map_add' := hadd + map_smul' := hsmul + monotone' := fun f g hfg => le_of_tendsto_of_tendsto' (hI f) (hI g) + (fun n => (I n).monotone hfg) } + refine ⟨RealRMK.rieszMeasure Λ,inferInstance,inferInstance,fun f => ?_⟩ + rw [RealRMK.integral_rieszMeasure] + exact hL f + +/-- Reconstruction from a single countable locally determining family. -/ +theorem LocalTestFamily.exists_radon_limit (D : LocalTestFamily X) + (μ : ℕ → Measure X) [∀ n, IsFiniteMeasureOnCompacts (μ n)] + (htest : ∀ j, ∃ a : ℝ, Tendsto (fun n => ∫ x, D.test j x ∂μ n) atTop (𝓝 a)) : + ∃ ν : Measure X, ν.Regular ∧ IsFiniteMeasureOnCompacts ν ∧ + ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν)) := by + apply radon_limit_of_all_test_limits μ + intro f + exact cauchySeq_tendsto_of_complete (D.integrals_cauchy μ htest f) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonTestLimitProperties.lean b/lean/OAI/Probability/SLE/Mass/RadonTestLimitProperties.lean index ed4d511ae2774e243d50421f3a439ed928667ca5..43a1f1b81ebf2e967972c521e81565e8ab8ee94b 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonTestLimitProperties.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonTestLimitProperties.lean @@ -1,101 +1,101 @@ -import OAI.Probability.SLE.Mass.RadonTestFamily - -namespace OAI - -/-! Order and closed-support properties of the reconstructed Radon limits. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported ENNReal - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [MeasurableSpace X] [BorelSpace X] - -lemma compact_measure_le_test_integral (μ : Measure X) [μ.Regular] - (f : C_c(X, ℝ)) (hf : ∀ x, 0 ≤ f x) {K : Set X} (hK : IsCompact K) - (hfone : ∀ x ∈ K, f x=1) : μ K ≤ ENNReal.ofReal (∫ x, f x ∂μ) := by - simpa only [RealRMK.rieszMeasure_integralPositiveLinearMap, - CompactlySupportedContinuousMap.integralPositiveLinearMap_apply] using - RealRMK.rieszMeasure_le_of_eq_one - (CompactlySupportedContinuousMap.integralPositiveLinearMap μ) hf hK hfone - -lemma test_integral_le_measure (μ : Measure X) [μ.Regular] - (f : C_c(X, ℝ)) (hf : ∀ x, 0 ≤ f x ∧ f x ≤ 1) {U : Set X} - (hs : tsupport f ⊆ U) : ENNReal.ofReal (∫ x, f x ∂μ) ≤ μ U := by - simpa only [RealRMK.rieszMeasure_integralPositiveLinearMap, - CompactlySupportedContinuousMap.integralPositiveLinearMap_apply] using - RealRMK.le_rieszMeasure_tsupport_subset - (CompactlySupportedContinuousMap.integralPositiveLinearMap μ) hf hs - -/-- The positive compact-test order determines the order of Radon measures. -/ -theorem radon_measure_le_of_nonneg_tests (μ ν : Measure X) [μ.Regular] [ν.Regular] - (h : ∀ f : C_c(X, ℝ), (∀ x, 0 ≤ f x) → (∫ x, f x ∂μ) ≤ ∫ x, f x ∂ν) : μ ≤ ν := by - have hc (K : Set X) (hK : IsCompact K) : μ K ≤ ν K := by - apply ENNReal.le_of_forall_pos_le_add - intro ε hε _ - obtain ⟨U,hKU,hU,hνU⟩ := K.exists_isOpen_le_add ν - (ne_of_gt (ENNReal.coe_lt_coe.mpr hε)) - obtain ⟨f,hf1,hfc,hfs,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen - hK hU hKU - let g : C_c(X, ℝ) := ⟨f,hfc⟩ - have hg0 : ∀ x, 0 ≤ g x := fun x => (hf01 x).1 - calc - μ K ≤ ENNReal.ofReal (∫ x, g x ∂μ) := compact_measure_le_test_integral μ g hg0 hK hf1 - _ ≤ ENNReal.ofReal (∫ x, g x ∂ν) := ENNReal.ofReal_le_ofReal (h g hg0) - _ ≤ ν U := test_integral_le_measure ν g hf01 hfs - _ ≤ ν K+ε := hνU - have ho (U : Set X) (hU : IsOpen U) : μ U ≤ ν U := by - rw [Measure.Regular.innerRegular.measure_eq_iSup hU] - exact iSup_le fun K => iSup_le fun hKU => iSup_le fun hK => (hc K hK).trans (measure_mono hKU) - apply Measure.le_iff'.mpr - intro A - rw [A.measure_eq_iInf_isOpen ν] - exact le_iInf fun U => le_iInf fun hAU => le_iInf fun hU => (measure_mono hAU).trans (ho U hU) - -/-- Order of positive approximants passes to every compact-test limit. -/ -theorem radon_limits_mono (μ ν : ℕ → Measure X) - [∀ n, IsFiniteMeasureOnCompacts (μ n)] [∀ n, IsFiniteMeasureOnCompacts (ν n)] - (μ₀ ν₀ : Measure X) [μ₀.Regular] [ν₀.Regular] - (hμ : ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂μ₀))) - (hν : ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂ν n) atTop (𝓝 (∫ x, f x ∂ν₀))) - (hle : ∀ n, μ n ≤ ν n) : μ₀ ≤ ν₀ := by - apply radon_measure_le_of_nonneg_tests - intro f hf - exact le_of_tendsto_of_tendsto' (hμ f) (hν f) (fun n => - integral_mono_measure (hle n) (ae_of_all _ hf) f.integrable) - -/-- Vanishing compact tests inside an open set force the Radon mass of -that open set to vanish. -/ -theorem radon_open_eq_zero_of_tests (μ : Measure X) [μ.Regular] (U : Set X) (hU : IsOpen U) - (hzero : ∀ f : C_c(X, ℝ), tsupport f ⊆ U → (∫ x, f x ∂μ)=0) : μ U=0 := by - apply le_antisymm ?_ bot_le - rw [Measure.Regular.innerRegular.measure_eq_iSup hU] - refine iSup_le fun K => iSup_le fun hKU => iSup_le fun hK => ?_ - obtain ⟨f,hf1,hfc,hfs,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen - hK hU hKU - let g : C_c(X, ℝ) := ⟨f,hfc⟩ - have hg0 : ∀ x, 0 ≤ g x := fun x => (hf01 x).1 - have hb := compact_measure_le_test_integral μ g hg0 hK hf1 - change μ K ≤ 0 - simpa only [hzero g hfs,ENNReal.ofReal_zero] using hb - -/-- The compact-test limit is carried by a closed set if each compact -test separated from it eventually vanishes. -/ -theorem radon_limit_carried_by_closed (μ : ℕ → Measure X) (ν : Measure X) [ν.Regular] - (hlim : ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν))) - (S : Set X) (hS : IsClosed S) - (hvanish : ∀ f : C_c(X, ℝ), tsupport f ⊆ Sᶜ → - ∀ᶠ n in atTop, (∫ x, f x ∂μ n)=0) : ν Sᶜ=0 := by - apply radon_open_eq_zero_of_tests ν Sᶜ hS.isOpen_compl - intro f hf - have heq : (fun _ : ℕ => (0:ℝ)) =ᶠ[atTop] (fun n => ∫ x, f x ∂μ n) := - (hvanish f hf).mono (fun _ h => h.symm) - exact tendsto_nhds_unique (hlim f) (tendsto_const_nhds.congr' heq) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonTestFamily + +namespace OAI + +/-! Order and closed-support properties of the reconstructed Radon limits. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported ENNReal + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [MeasurableSpace X] [BorelSpace X] + +lemma compact_measure_le_test_integral (μ : Measure X) [μ.Regular] + (f : C_c(X, ℝ)) (hf : ∀ x, 0 ≤ f x) {K : Set X} (hK : IsCompact K) + (hfone : ∀ x ∈ K, f x=1) : μ K ≤ ENNReal.ofReal (∫ x, f x ∂μ) := by + simpa only [RealRMK.rieszMeasure_integralPositiveLinearMap, + CompactlySupportedContinuousMap.integralPositiveLinearMap_apply] using + RealRMK.rieszMeasure_le_of_eq_one + (CompactlySupportedContinuousMap.integralPositiveLinearMap μ) hf hK hfone + +lemma test_integral_le_measure (μ : Measure X) [μ.Regular] + (f : C_c(X, ℝ)) (hf : ∀ x, 0 ≤ f x ∧ f x ≤ 1) {U : Set X} + (hs : tsupport f ⊆ U) : ENNReal.ofReal (∫ x, f x ∂μ) ≤ μ U := by + simpa only [RealRMK.rieszMeasure_integralPositiveLinearMap, + CompactlySupportedContinuousMap.integralPositiveLinearMap_apply] using + RealRMK.le_rieszMeasure_tsupport_subset + (CompactlySupportedContinuousMap.integralPositiveLinearMap μ) hf hs + +/-- The positive compact-test order determines the order of Radon measures. -/ +theorem radon_measure_le_of_nonneg_tests (μ ν : Measure X) [μ.Regular] [ν.Regular] + (h : ∀ f : C_c(X, ℝ), (∀ x, 0 ≤ f x) → (∫ x, f x ∂μ) ≤ ∫ x, f x ∂ν) : μ ≤ ν := by + have hc (K : Set X) (hK : IsCompact K) : μ K ≤ ν K := by + apply ENNReal.le_of_forall_pos_le_add + intro ε hε _ + obtain ⟨U,hKU,hU,hνU⟩ := K.exists_isOpen_le_add ν + (ne_of_gt (ENNReal.coe_lt_coe.mpr hε)) + obtain ⟨f,hf1,hfc,hfs,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen + hK hU hKU + let g : C_c(X, ℝ) := ⟨f,hfc⟩ + have hg0 : ∀ x, 0 ≤ g x := fun x => (hf01 x).1 + calc + μ K ≤ ENNReal.ofReal (∫ x, g x ∂μ) := compact_measure_le_test_integral μ g hg0 hK hf1 + _ ≤ ENNReal.ofReal (∫ x, g x ∂ν) := ENNReal.ofReal_le_ofReal (h g hg0) + _ ≤ ν U := test_integral_le_measure ν g hf01 hfs + _ ≤ ν K+ε := hνU + have ho (U : Set X) (hU : IsOpen U) : μ U ≤ ν U := by + rw [Measure.Regular.innerRegular.measure_eq_iSup hU] + exact iSup_le fun K => iSup_le fun hKU => iSup_le fun hK => (hc K hK).trans (measure_mono hKU) + apply Measure.le_iff'.mpr + intro A + rw [A.measure_eq_iInf_isOpen ν] + exact le_iInf fun U => le_iInf fun hAU => le_iInf fun hU => (measure_mono hAU).trans (ho U hU) + +/-- Order of positive approximants passes to every compact-test limit. -/ +theorem radon_limits_mono (μ ν : ℕ → Measure X) + [∀ n, IsFiniteMeasureOnCompacts (μ n)] [∀ n, IsFiniteMeasureOnCompacts (ν n)] + (μ₀ ν₀ : Measure X) [μ₀.Regular] [ν₀.Regular] + (hμ : ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂μ₀))) + (hν : ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂ν n) atTop (𝓝 (∫ x, f x ∂ν₀))) + (hle : ∀ n, μ n ≤ ν n) : μ₀ ≤ ν₀ := by + apply radon_measure_le_of_nonneg_tests + intro f hf + exact le_of_tendsto_of_tendsto' (hμ f) (hν f) (fun n => + integral_mono_measure (hle n) (ae_of_all _ hf) f.integrable) + +/-- Vanishing compact tests inside an open set force the Radon mass of +that open set to vanish. -/ +theorem radon_open_eq_zero_of_tests (μ : Measure X) [μ.Regular] (U : Set X) (hU : IsOpen U) + (hzero : ∀ f : C_c(X, ℝ), tsupport f ⊆ U → (∫ x, f x ∂μ)=0) : μ U=0 := by + apply le_antisymm ?_ bot_le + rw [Measure.Regular.innerRegular.measure_eq_iSup hU] + refine iSup_le fun K => iSup_le fun hKU => iSup_le fun hK => ?_ + obtain ⟨f,hf1,hfc,hfs,hf01⟩ := exists_continuousMap_one_of_isCompact_subset_isOpen + hK hU hKU + let g : C_c(X, ℝ) := ⟨f,hfc⟩ + have hg0 : ∀ x, 0 ≤ g x := fun x => (hf01 x).1 + have hb := compact_measure_le_test_integral μ g hg0 hK hf1 + change μ K ≤ 0 + simpa only [hzero g hfs,ENNReal.ofReal_zero] using hb + +/-- The compact-test limit is carried by a closed set if each compact +test separated from it eventually vanishes. -/ +theorem radon_limit_carried_by_closed (μ : ℕ → Measure X) (ν : Measure X) [ν.Regular] + (hlim : ∀ f : C_c(X, ℝ), Tendsto (fun n => ∫ x, f x ∂μ n) atTop (𝓝 (∫ x, f x ∂ν))) + (S : Set X) (hS : IsClosed S) + (hvanish : ∀ f : C_c(X, ℝ), tsupport f ⊆ Sᶜ → + ∀ᶠ n in atTop, (∫ x, f x ∂μ n)=0) : ν Sᶜ=0 := by + apply radon_open_eq_zero_of_tests ν Sᶜ hS.isOpen_compl + intro f hf + have heq : (fun _ : ℕ => (0:ℝ)) =ᶠ[atTop] (fun n => ∫ x, f x ∂μ n) := + (hvanish f hf).mono (fun _ h => h.symm) + exact tendsto_nhds_unique (hlim f) (tendsto_const_nhds.congr' heq) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RadonTestMeasurability.lean b/lean/OAI/Probability/SLE/Mass/RadonTestMeasurability.lean index f093013be00cbcdf355ad78c1a22796d68eeed0b..855956fb156d6b0fc43be01dce141a1e55c09d63 100644 --- a/lean/OAI/Probability/SLE/Mass/RadonTestMeasurability.lean +++ b/lean/OAI/Probability/SLE/Mass/RadonTestMeasurability.lean @@ -1,138 +1,138 @@ -import OAI.Probability.SLE.Mass.RadonTestLimitProperties -import Mathlib.MeasureTheory.Measure.GiryMonad -import Mathlib.MeasureTheory.Constructions.Polish.Basic - -namespace OAI - -/-! A Radon measure determined by measurable compact continuous tests is -a measurable measure-valued function. Local finite restrictions are used, -so no finite total-mass hypothesis is imposed. -/ - -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported ENNReal - -namespace SLEExactGauge - -variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] - [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] - -/-- A fixed countable list of compact cutoffs recovers the mass of a fixed -open set simultaneously for all Radon measures. -/ -theorem exists_open_cutoff_sequence (U : Set X) (hU : IsOpen U) : - ∃ f : ℕ → C_c(X, ℝ), (∀ n x, 0 ≤ f n x ∧ f n x ≤ 1) ∧ - (∀ n, tsupport (f n) ⊆ U) ∧ ∀ (μ : Measure X), μ.Regular → - μ U = ⨆ n, ENNReal.ofReal (∫ x, f n x ∂μ) := by - let : LocallyCompactSpace U := hU.locallyCompactSpace - let K := CompactExhaustion.choice U - let A (n : ℕ) : Set X := Subtype.val '' K n - have hAc (n : ℕ) : IsCompact (A n) := (K.isCompact n).image continuous_subtype_val - have hAU (n : ℕ) : A n ⊆ U := by rintro _ ⟨x,_,rfl⟩; exact x.2 - have hcut (n : ℕ) := exists_continuousMap_one_of_isCompact_subset_isOpen (hAc n) hU (hAU n) - choose f hf1 hfc hfs hf01 using hcut - let g (n : ℕ) : C_c(X, ℝ) := ⟨f n,hfc n⟩ - have hAm : Monotone A := fun _ _ h => image_mono (K.subset h) - have hUnion : (⋃ n, A n)=U := by - ext x - constructor - · intro hx - obtain ⟨n,hn⟩ := mem_iUnion.mp hx - exact hAU n hn - · intro hx - obtain ⟨n,hn⟩ := K.exists_mem (⟨x,hx⟩ : U) - exact mem_iUnion.mpr ⟨n,⟨⟨x,hx⟩,hn,rfl⟩⟩ - refine ⟨g,hf01,hfs,fun μ hμ => ?_⟩ - let : μ.Regular := hμ - apply le_antisymm - · rw [← hUnion,hAm.measure_iUnion] - exact iSup_mono fun n => compact_measure_le_test_integral μ (g n) - (fun x => (hf01 n x).1) (hAc n) (hf1 n) - · exact iSup_le fun n => test_integral_le_measure μ (g n) (hf01 n) (hfs n) - -/-- Measurability of all compact continuous test integrals implies -measurability of every open-set evaluation. -/ -theorem measurable_open_evaluation_of_tests - {Ω : Type*} [MeasurableSpace Ω] (ν : Ω → Measure X) - (hν : ∀ ω, (ν ω).Regular) - (htest : ∀ f : C_c(X, ℝ), Measurable (fun ω => ∫ x, f x ∂ν ω)) - (U : Set X) (hU : IsOpen U) : Measurable (fun ω => ν ω U) := by - obtain ⟨f,_,_,hf⟩ := exists_open_cutoff_sequence U hU - have he : (fun ω => ν ω U)=(fun ω => ⨆ n, ENNReal.ofReal (∫ x, f n x ∂ν ω)) := - funext fun ω => hf (ν ω) (hν ω) - rw [he] - exact Measurable.iSup fun n => (htest (f n)).ennreal_ofReal - -omit [T2Space X] in -/-- Locally finite Radon kernels are measurable once all their open-set -evaluations are measurable. -/ -theorem measurable_radon_of_open_evaluations - {Ω : Type*} [MeasurableSpace Ω] (ν : Ω → Measure X) - (hν : ∀ ω, (ν ω).Regular) - (hopen : ∀ U : Set X, IsOpen U → Measurable (fun ω => ν ω U)) : Measurable ν := by - let K := CompactExhaustion.choice X - let U (n : ℕ) := interior (K n) - have hU (n : ℕ) : IsOpen (U n) := isOpen_interior - have hfinite (n : ℕ) (ω : Ω) : ν ω (U n) < ⊤ := by - let : (ν ω).Regular := hν ω - exact (measure_mono interior_subset).trans_lt (K.isCompact n).measure_lt_top - have hlocal (n : ℕ) : ∀ A : Set X, MeasurableSet A → - Measurable (fun ω => ν ω (A ∩ U n)) := by - apply MeasurableSet.induction_on_open - · intro A hA - exact hopen _ (hA.inter (hU n)) - · intro A hA hm - have he (ω : Ω) : ν ω (Aᶜ ∩ U n)=ν ω (U n)-ν ω (A ∩ U n) := by - have hset : Aᶜ ∩ U n=U n \ (A ∩ U n) := by ext x; simp only [Set.mem_inter_iff,Set.mem_compl_iff,Set.mem_sdiff]; tauto - rw [hset] - exact measure_sdiff inter_subset_right (hA.inter (hU n).measurableSet).nullMeasurableSet - ((measure_mono inter_subset_right).trans_lt (hfinite n ω)).ne - simp_rw [he] - exact (hopen _ (hU n)).sub hm - · intro A hd hA hm - have hd' : Pairwise (fun i j => Disjoint (A i ∩ U n) (A j ∩ U n)) := - fun i j hij => (hd hij).mono inter_subset_left inter_subset_left - have he (ω : Ω) : ν ω ((⋃ i, A i) ∩ U n)=∑' i, ν ω (A i ∩ U n) := by - rw [iUnion_inter] - exact measure_iUnion hd' (fun i => (hA i).inter (hU n).measurableSet) - simp_rw [he] - exact Measurable.tsum hm - have hUnion : (⋃ n, U n)=univ := by - apply eq_univ_of_forall - intro x - obtain ⟨n,hn⟩ := K.exists_mem x - exact mem_iUnion.mpr ⟨n+1,K.subset_interior_succ n hn⟩ - have hUm : Monotone U := fun _ _ h => interior_mono (K.subset h) - apply Measure.measurable_of_measurable_coe - intro A hA - have he (ω : Ω) : ν ω A=⨆ n, ν ω (A ∩ U n) := by - have hmono : Monotone (fun n => A ∩ U n) := monotone_const.inter hUm - rw [← hmono.measure_iUnion,← inter_iUnion,hUnion,inter_univ] - simp_rw [he] - exact Measurable.iSup fun n => hlocal n A hA - -/-- A reconstruction with measurable compact-test integrals is a genuine -measurable measure-valued map, not only a collection of scalar tests. -/ -theorem measurable_radon_of_tests - {Ω : Type*} [MeasurableSpace Ω] (ν : Ω → Measure X) - (hν : ∀ ω, (ν ω).Regular) - (htest : ∀ f : C_c(X, ℝ), Measurable (fun ω => ∫ x, f x ∂ν ω)) : Measurable ν := - measurable_radon_of_open_evaluations ν hν (measurable_open_evaluation_of_tests ν hν htest) - -/-- Pointwise convergence of measurable compact-test integrals provides -measurability of the reconstructed Radon kernel. -/ -theorem measurable_radon_of_test_limits - {Ω : Type*} [MeasurableSpace Ω] (μ : ℕ → Ω → Measure X) (ν : Ω → Measure X) - (hν : ∀ ω, (ν ω).Regular) - (hmeas : ∀ n (f : C_c(X, ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω)) - (hlim : ∀ ω (f : C_c(X, ℝ)), Tendsto (fun n => ∫ x, f x ∂μ n ω) - atTop (𝓝 (∫ x, f x ∂ν ω))) : Measurable ν := by - apply measurable_radon_of_tests ν hν - intro f - exact measurable_of_tendsto_metrizable (fun n => hmeas n f) - (tendsto_pi_nhds.mpr (fun ω => hlim ω f)) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonTestLimitProperties +import Mathlib.MeasureTheory.Measure.GiryMonad +import Mathlib.MeasureTheory.Constructions.Polish.Basic + +namespace OAI + +/-! A Radon measure determined by measurable compact continuous tests is +a measurable measure-valued function. Local finite restrictions are used, +so no finite total-mass hypothesis is imposed. -/ + +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported ENNReal + +namespace SLEExactGauge + +variable {X : Type*} [TopologicalSpace X] [T2Space X] [LocallyCompactSpace X] + [SecondCountableTopology X] [MeasurableSpace X] [BorelSpace X] + +/-- A fixed countable list of compact cutoffs recovers the mass of a fixed +open set simultaneously for all Radon measures. -/ +theorem exists_open_cutoff_sequence (U : Set X) (hU : IsOpen U) : + ∃ f : ℕ → C_c(X, ℝ), (∀ n x, 0 ≤ f n x ∧ f n x ≤ 1) ∧ + (∀ n, tsupport (f n) ⊆ U) ∧ ∀ (μ : Measure X), μ.Regular → + μ U = ⨆ n, ENNReal.ofReal (∫ x, f n x ∂μ) := by + let : LocallyCompactSpace U := hU.locallyCompactSpace + let K := CompactExhaustion.choice U + let A (n : ℕ) : Set X := Subtype.val '' K n + have hAc (n : ℕ) : IsCompact (A n) := (K.isCompact n).image continuous_subtype_val + have hAU (n : ℕ) : A n ⊆ U := by rintro _ ⟨x,_,rfl⟩; exact x.2 + have hcut (n : ℕ) := exists_continuousMap_one_of_isCompact_subset_isOpen (hAc n) hU (hAU n) + choose f hf1 hfc hfs hf01 using hcut + let g (n : ℕ) : C_c(X, ℝ) := ⟨f n,hfc n⟩ + have hAm : Monotone A := fun _ _ h => image_mono (K.subset h) + have hUnion : (⋃ n, A n)=U := by + ext x + constructor + · intro hx + obtain ⟨n,hn⟩ := mem_iUnion.mp hx + exact hAU n hn + · intro hx + obtain ⟨n,hn⟩ := K.exists_mem (⟨x,hx⟩ : U) + exact mem_iUnion.mpr ⟨n,⟨⟨x,hx⟩,hn,rfl⟩⟩ + refine ⟨g,hf01,hfs,fun μ hμ => ?_⟩ + let : μ.Regular := hμ + apply le_antisymm + · rw [← hUnion,hAm.measure_iUnion] + exact iSup_mono fun n => compact_measure_le_test_integral μ (g n) + (fun x => (hf01 n x).1) (hAc n) (hf1 n) + · exact iSup_le fun n => test_integral_le_measure μ (g n) (hf01 n) (hfs n) + +/-- Measurability of all compact continuous test integrals implies +measurability of every open-set evaluation. -/ +theorem measurable_open_evaluation_of_tests + {Ω : Type*} [MeasurableSpace Ω] (ν : Ω → Measure X) + (hν : ∀ ω, (ν ω).Regular) + (htest : ∀ f : C_c(X, ℝ), Measurable (fun ω => ∫ x, f x ∂ν ω)) + (U : Set X) (hU : IsOpen U) : Measurable (fun ω => ν ω U) := by + obtain ⟨f,_,_,hf⟩ := exists_open_cutoff_sequence U hU + have he : (fun ω => ν ω U)=(fun ω => ⨆ n, ENNReal.ofReal (∫ x, f n x ∂ν ω)) := + funext fun ω => hf (ν ω) (hν ω) + rw [he] + exact Measurable.iSup fun n => (htest (f n)).ennreal_ofReal + +omit [T2Space X] in +/-- Locally finite Radon kernels are measurable once all their open-set +evaluations are measurable. -/ +theorem measurable_radon_of_open_evaluations + {Ω : Type*} [MeasurableSpace Ω] (ν : Ω → Measure X) + (hν : ∀ ω, (ν ω).Regular) + (hopen : ∀ U : Set X, IsOpen U → Measurable (fun ω => ν ω U)) : Measurable ν := by + let K := CompactExhaustion.choice X + let U (n : ℕ) := interior (K n) + have hU (n : ℕ) : IsOpen (U n) := isOpen_interior + have hfinite (n : ℕ) (ω : Ω) : ν ω (U n) < ⊤ := by + let : (ν ω).Regular := hν ω + exact (measure_mono interior_subset).trans_lt (K.isCompact n).measure_lt_top + have hlocal (n : ℕ) : ∀ A : Set X, MeasurableSet A → + Measurable (fun ω => ν ω (A ∩ U n)) := by + apply MeasurableSet.induction_on_open + · intro A hA + exact hopen _ (hA.inter (hU n)) + · intro A hA hm + have he (ω : Ω) : ν ω (Aᶜ ∩ U n)=ν ω (U n)-ν ω (A ∩ U n) := by + have hset : Aᶜ ∩ U n=U n \ (A ∩ U n) := by ext x; simp only [Set.mem_inter_iff,Set.mem_compl_iff,Set.mem_sdiff]; tauto + rw [hset] + exact measure_sdiff inter_subset_right (hA.inter (hU n).measurableSet).nullMeasurableSet + ((measure_mono inter_subset_right).trans_lt (hfinite n ω)).ne + simp_rw [he] + exact (hopen _ (hU n)).sub hm + · intro A hd hA hm + have hd' : Pairwise (fun i j => Disjoint (A i ∩ U n) (A j ∩ U n)) := + fun i j hij => (hd hij).mono inter_subset_left inter_subset_left + have he (ω : Ω) : ν ω ((⋃ i, A i) ∩ U n)=∑' i, ν ω (A i ∩ U n) := by + rw [iUnion_inter] + exact measure_iUnion hd' (fun i => (hA i).inter (hU n).measurableSet) + simp_rw [he] + exact Measurable.tsum hm + have hUnion : (⋃ n, U n)=univ := by + apply eq_univ_of_forall + intro x + obtain ⟨n,hn⟩ := K.exists_mem x + exact mem_iUnion.mpr ⟨n+1,K.subset_interior_succ n hn⟩ + have hUm : Monotone U := fun _ _ h => interior_mono (K.subset h) + apply Measure.measurable_of_measurable_coe + intro A hA + have he (ω : Ω) : ν ω A=⨆ n, ν ω (A ∩ U n) := by + have hmono : Monotone (fun n => A ∩ U n) := monotone_const.inter hUm + rw [← hmono.measure_iUnion,← inter_iUnion,hUnion,inter_univ] + simp_rw [he] + exact Measurable.iSup fun n => hlocal n A hA + +/-- A reconstruction with measurable compact-test integrals is a genuine +measurable measure-valued map, not only a collection of scalar tests. -/ +theorem measurable_radon_of_tests + {Ω : Type*} [MeasurableSpace Ω] (ν : Ω → Measure X) + (hν : ∀ ω, (ν ω).Regular) + (htest : ∀ f : C_c(X, ℝ), Measurable (fun ω => ∫ x, f x ∂ν ω)) : Measurable ν := + measurable_radon_of_open_evaluations ν hν (measurable_open_evaluation_of_tests ν hν htest) + +/-- Pointwise convergence of measurable compact-test integrals provides +measurability of the reconstructed Radon kernel. -/ +theorem measurable_radon_of_test_limits + {Ω : Type*} [MeasurableSpace Ω] (μ : ℕ → Ω → Measure X) (ν : Ω → Measure X) + (hν : ∀ ω, (ν ω).Regular) + (hmeas : ∀ n (f : C_c(X, ℝ)), Measurable (fun ω => ∫ x, f x ∂μ n ω)) + (hlim : ∀ ω (f : C_c(X, ℝ)), Tendsto (fun n => ∫ x, f x ∂μ n ω) + atTop (𝓝 (∫ x, f x ∂ν ω))) : Measurable ν := by + apply measurable_radon_of_tests ν hν + intro f + exact measurable_of_tendsto_metrizable (fun n => hmeas n f) + (tendsto_pi_nhds.mpr (fun ω => hlim ω f)) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RandomDilationMass.lean b/lean/OAI/Probability/SLE/Mass/RandomDilationMass.lean index 6ced430bd1cfb6f6ce7606046b5d470494f5d08c..2ec5bf18568c7e0b65aa5e17c3fd713b4e15a8a3 100644 --- a/lean/OAI/Probability/SLE/Mass/RandomDilationMass.lean +++ b/lean/OAI/Probability/SLE/Mass/RandomDilationMass.lean @@ -1,36 +1,36 @@ -import OAI.Probability.SLE.Mass.DilationMass -import OAI.Probability.SLE.Mass.WeightedImageRandomMass -import OAI.Probability.SLE.Stochastic.ProgressiveParameterTime - -namespace OAI - -/-! Measurability of the actual mass normalization when its scale and -evaluation time are selected from the stopped past. -/ - -noncomputable section -open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory -open scoped Topology NNReal ENNReal - -namespace SLEExactGauge.Multipoint - -lemma dilationMass_measurable_family {Ω : Type*} [MeasurableSpace Ω] - (κ : ℝ) (c : Ω → ℝ≥0) (hc : Measurable c) (hpos : ∀ ω, 0 dilationMass κ (c ω) (hpos ω) (μ ω)) := by - let J : Ω × upperHalfPlane → upperHalfPlane := fun p => halfPlaneDilation (c p.1) (hpos p.1) p.2 - have hJ : Measurable J := by - apply Measurable.subtype_mk - exact (Complex.measurable_ofReal.comp - (NNReal.continuous_coe.measurable.comp (hc.comp measurable_fst))).mul - (measurable_subtype_coe.comp measurable_snd) - let w : Ω × upperHalfPlane → ℝ≥0∞ := fun p => ENNReal.ofReal ((c p.1:ℝ)^dimension κ) - have hw : Measurable w := ENNReal.measurable_ofReal.comp - ((NNReal.continuous_coe.measurable.comp (hc.comp measurable_fst)).pow measurable_const) - exact weightedImageMass_measurable μ hμ hfin J hJ w hw - -end SLEExactGauge.Multipoint - -end - -end OAI +import OAI.Probability.SLE.Mass.DilationMass +import OAI.Probability.SLE.Mass.WeightedImageRandomMass +import OAI.Probability.SLE.Stochastic.ProgressiveParameterTime + +namespace OAI + +/-! Measurability of the actual mass normalization when its scale and +evaluation time are selected from the stopped past. -/ + +noncomputable section +open Set Filter _root_.MeasureTheory _root_.OAI.MeasureTheory +open scoped Topology NNReal ENNReal + +namespace SLEExactGauge.Multipoint + +lemma dilationMass_measurable_family {Ω : Type*} [MeasurableSpace Ω] + (κ : ℝ) (c : Ω → ℝ≥0) (hc : Measurable c) (hpos : ∀ ω, 0 dilationMass κ (c ω) (hpos ω) (μ ω)) := by + let J : Ω × upperHalfPlane → upperHalfPlane := fun p => halfPlaneDilation (c p.1) (hpos p.1) p.2 + have hJ : Measurable J := by + apply Measurable.subtype_mk + exact (Complex.measurable_ofReal.comp + (NNReal.continuous_coe.measurable.comp (hc.comp measurable_fst))).mul + (measurable_subtype_coe.comp measurable_snd) + let w : Ω × upperHalfPlane → ℝ≥0∞ := fun p => ENNReal.ofReal ((c p.1:ℝ)^dimension κ) + have hw : Measurable w := ENNReal.measurable_ofReal.comp + ((NNReal.continuous_coe.measurable.comp (hc.comp measurable_fst)).pow measurable_const) + exact weightedImageMass_measurable μ hμ hfin J hJ w hw + +end SLEExactGauge.Multipoint + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RandomMeasureDifference.lean b/lean/OAI/Probability/SLE/Mass/RandomMeasureDifference.lean index 4a15675b7e97bcfad9ec73f1c7e9362da3f2122f..28ff434b2a5e2bad88dd64a1326035b4bfc22c57 100644 --- a/lean/OAI/Probability/SLE/Mass/RandomMeasureDifference.lean +++ b/lean/OAI/Probability/SLE/Mass/RandomMeasureDifference.lean @@ -1,78 +1,78 @@ -import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract -import OAI.Probability.SLE.Mass.RationalTraceMassAdapted - -namespace OAI - -/-! Positive differences of locally finite random Radon measures admit -measurable versions. No finite total mass is required. -/ - -noncomputable section -open Set Filter MeasureTheory Classical -open scoped ENNReal Topology CompactlySupported - -namespace SLEExactGauge - -local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - -lemma halfPlane_sub_finite_on_compacts (μ ν : Measure upperHalfPlane) - [IsFiniteMeasureOnCompacts μ] : IsFiniteMeasureOnCompacts (μ-ν) := - ⟨fun K hK => lt_of_le_of_lt (Measure.sub_le K) hK.measure_lt_top⟩ - -lemma integral_halfPlane_sub (μ ν : Measure upperHalfPlane) [μ.Regular] [ν.Regular] - (hle : ν ≤ μ) (f : C_c(upperHalfPlane,ℝ)) : - (∫ z, f z ∂(μ-ν)) = (∫ z, f z ∂μ) - ∫ z, f z ∂ν := by - let := halfPlane_sub_finite_on_compacts μ ν - have hi := integral_add_measure (μ := μ-ν) (ν := ν) f.integrable f.integrable - rw [sigmaFinite_sub_add_cancel_of_le hle] at hi - linarith - -theorem measurable_halfPlane_sub_of_le - {Ω : Type*} [MeasurableSpace Ω] (μ ν : Ω → Measure upperHalfPlane) - (hμ : Measurable μ) (hν : Measurable ν) - (hμr : ∀ ω, (μ ω).Regular) (hνr : ∀ ω, (ν ω).Regular) - (hle : ∀ ω, ν ω ≤ μ ω) : Measurable (fun ω => μ ω-ν ω) := by - let (ω : Ω) : (μ ω).Regular := hμr ω - let (ω : Ω) : (ν ω).Regular := hνr ω - let (ω : Ω) : IsFiniteMeasureOnCompacts (μ ω-ν ω) := - halfPlane_sub_finite_on_compacts (μ ω) (ν ω) - apply measurable_radon_of_tests _ (fun _ => inferInstance) - intro f - simp_rw [integral_halfPlane_sub _ _ (hle _) f] - exact (measurable_halfPlane_compact_test_integral μ hμ hμr f).sub - (measurable_halfPlane_compact_test_integral ν hν hνr f) - -theorem exists_measurable_halfPlane_difference - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ ν : Ω → Measure upperHalfPlane) - (hμ : Measurable μ) (hν : Measurable ν) - (hμr : ∀ ω, (μ ω).Regular) (hνr : ∀ ω, (ν ω).Regular) - (hle : ∀ᵐ ω ∂P, ν ω ≤ μ ω) : - ∃ η : Ω → Measure upperHalfPlane, Measurable η ∧ - (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ - (∀ᵐ ω ∂P, η ω + ν ω = μ ω) := by - obtain ⟨E,hE,hEm,hEsub⟩ := hle.exists_measurable_mem - let μ' (ω : Ω) := if ω ∈ E then μ ω else 0 - let ν' (ω : Ω) := if ω ∈ E then ν ω else 0 - have hμ' : Measurable μ' := hμ.ite hEm measurable_const - have hν' : Measurable ν' := hν.ite hEm measurable_const - have hμr' (ω : Ω) : (μ' ω).Regular := by - dsimp only [μ']; split_ifs; exacts [hμr ω,inferInstance] - have hνr' (ω : Ω) : (ν' ω).Regular := by - dsimp only [ν']; split_ifs; exacts [hνr ω,inferInstance] - have hle' (ω : Ω) : ν' ω ≤ μ' ω := by - dsimp only [μ',ν']; split_ifs with he - · exact hEsub ω he - · exact le_rfl - refine ⟨fun ω => μ' ω-ν' ω,measurable_halfPlane_sub_of_le μ' ν' hμ' hν' hμr' hνr' hle',?_,?_⟩ - · intro ω - let := hμr' ω - let := halfPlane_sub_finite_on_compacts (μ' ω) (ν' ω) - exact ⟨inferInstance,inferInstance⟩ - · filter_upwards [hE] with ω hω - let := hνr ω - simpa only [μ',ν',ite_eq_left hω] using sigmaFinite_sub_add_cancel_of_le (hEsub ω hω) - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.LocallyFiniteMeasureSubtract +import OAI.Probability.SLE.Mass.RationalTraceMassAdapted + +namespace OAI + +/-! Positive differences of locally finite random Radon measures admit +measurable versions. No finite total mass is required. -/ + +noncomputable section +open Set Filter MeasureTheory Classical +open scoped ENNReal Topology CompactlySupported + +namespace SLEExactGauge + +local instance : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + +lemma halfPlane_sub_finite_on_compacts (μ ν : Measure upperHalfPlane) + [IsFiniteMeasureOnCompacts μ] : IsFiniteMeasureOnCompacts (μ-ν) := + ⟨fun K hK => lt_of_le_of_lt (Measure.sub_le K) hK.measure_lt_top⟩ + +lemma integral_halfPlane_sub (μ ν : Measure upperHalfPlane) [μ.Regular] [ν.Regular] + (hle : ν ≤ μ) (f : C_c(upperHalfPlane,ℝ)) : + (∫ z, f z ∂(μ-ν)) = (∫ z, f z ∂μ) - ∫ z, f z ∂ν := by + let := halfPlane_sub_finite_on_compacts μ ν + have hi := integral_add_measure (μ := μ-ν) (ν := ν) f.integrable f.integrable + rw [sigmaFinite_sub_add_cancel_of_le hle] at hi + linarith + +theorem measurable_halfPlane_sub_of_le + {Ω : Type*} [MeasurableSpace Ω] (μ ν : Ω → Measure upperHalfPlane) + (hμ : Measurable μ) (hν : Measurable ν) + (hμr : ∀ ω, (μ ω).Regular) (hνr : ∀ ω, (ν ω).Regular) + (hle : ∀ ω, ν ω ≤ μ ω) : Measurable (fun ω => μ ω-ν ω) := by + let (ω : Ω) : (μ ω).Regular := hμr ω + let (ω : Ω) : (ν ω).Regular := hνr ω + let (ω : Ω) : IsFiniteMeasureOnCompacts (μ ω-ν ω) := + halfPlane_sub_finite_on_compacts (μ ω) (ν ω) + apply measurable_radon_of_tests _ (fun _ => inferInstance) + intro f + simp_rw [integral_halfPlane_sub _ _ (hle _) f] + exact (measurable_halfPlane_compact_test_integral μ hμ hμr f).sub + (measurable_halfPlane_compact_test_integral ν hν hνr f) + +theorem exists_measurable_halfPlane_difference + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) (μ ν : Ω → Measure upperHalfPlane) + (hμ : Measurable μ) (hν : Measurable ν) + (hμr : ∀ ω, (μ ω).Regular) (hνr : ∀ ω, (ν ω).Regular) + (hle : ∀ᵐ ω ∂P, ν ω ≤ μ ω) : + ∃ η : Ω → Measure upperHalfPlane, Measurable η ∧ + (∀ ω, (η ω).Regular ∧ IsFiniteMeasureOnCompacts (η ω)) ∧ + (∀ᵐ ω ∂P, η ω + ν ω = μ ω) := by + obtain ⟨E,hE,hEm,hEsub⟩ := hle.exists_measurable_mem + let μ' (ω : Ω) := if ω ∈ E then μ ω else 0 + let ν' (ω : Ω) := if ω ∈ E then ν ω else 0 + have hμ' : Measurable μ' := hμ.ite hEm measurable_const + have hν' : Measurable ν' := hν.ite hEm measurable_const + have hμr' (ω : Ω) : (μ' ω).Regular := by + dsimp only [μ']; split_ifs; exacts [hμr ω,inferInstance] + have hνr' (ω : Ω) : (ν' ω).Regular := by + dsimp only [ν']; split_ifs; exacts [hνr ω,inferInstance] + have hle' (ω : Ω) : ν' ω ≤ μ' ω := by + dsimp only [μ',ν']; split_ifs with he + · exact hEsub ω he + · exact le_rfl + refine ⟨fun ω => μ' ω-ν' ω,measurable_halfPlane_sub_of_le μ' ν' hμ' hν' hμr' hνr' hle',?_,?_⟩ + · intro ω + let := hμr' ω + let := halfPlane_sub_finite_on_compacts (μ' ω) (ν' ω) + exact ⟨inferInstance,inferInstance⟩ + · filter_upwards [hE] with ω hω + let := hνr ω + simpa only [μ',ν',ite_eq_left hω] using sigmaFinite_sub_add_cancel_of_le (hEsub ω hω) + +end SLEExactGauge + +end + +end OAI diff --git a/lean/OAI/Probability/SLE/Mass/RandomRadonEquality.lean b/lean/OAI/Probability/SLE/Mass/RandomRadonEquality.lean index b67ebe48d2a056329ce6b51ab59e6ae847ae53f5..7c05e16d5addbd26425f48de13cee894fe46e25d 100644 --- a/lean/OAI/Probability/SLE/Mass/RandomRadonEquality.lean +++ b/lean/OAI/Probability/SLE/Mass/RandomRadonEquality.lean @@ -1,35 +1,35 @@ -import OAI.Probability.SLE.Mass.RadonProcessAEUniqueness -import OAI.Probability.SLE.LoewnerCharts -import OAI.Probability.SLE.Mass.RadonTestFamily - -namespace OAI - -/-! Equality of nonnegative tests yields equality of random Radon measures -on one common event, using the existing countable determining family. -/ -noncomputable section -open Set Filter MeasureTheory -open scoped Topology CompactlySupported NNReal ENNReal - -namespace SLEExactGauge - -lemma random_halfPlane_measure_eq_of_lintegrals - {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) - (N M : Ω → Measure upperHalfPlane) - (hN : ∀ ω, (N ω).Regular) (hM : ∀ ω, (M ω).Regular) - (htest : ∀ f : C_c(upperHalfPlane,ℝ), (∀ z, 0 ≤ f z) → - ∀ᵐ ω ∂P, (∫⁻ z, ENNReal.ofReal (f z) ∂N ω)=∫⁻ z, ENNReal.ofReal (f z) ∂M ω) : - ∀ᵐ ω ∂P, N ω=M ω := by - let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace - have h := random_radon_process_eq_of_positive_tests P (localTestFamily upperHalfPlane) - (fun _ => N) (fun _ => M) (fun _ => hN) (fun _ => hM) (fun f hf => ?_) - · exact h.mono (fun _ hω => hω 0) - · filter_upwards [htest f hf] with ω hω - intro _ - rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable, - integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable,hω] - -end SLEExactGauge - -end - -end OAI +import OAI.Probability.SLE.Mass.RadonProcessAEUniqueness +import OAI.Probability.SLE.LoewnerCharts +import OAI.Probability.SLE.Mass.RadonTestFamily + +namespace OAI + +/-! Equality of nonnegative tests yields equality of random Radon measures +on one common event, using the existing countable determining family. -/ +noncomputable section +open Set Filter MeasureTheory +open scoped Topology CompactlySupported NNReal ENNReal + +namespace SLEExactGauge + +lemma random_halfPlane_measure_eq_of_lintegrals + {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) + (N M : Ω → Measure upperHalfPlane) + (hN : ∀ ω, (N ω).Regular) (hM : ∀ ω, (M ω).Regular) + (htest : ∀ f : C_c(upperHalfPlane,ℝ), (∀ z, 0 ≤ f z) → + ∀ᵐ ω ∂P, (∫⁻ z, ENNReal.ofReal (f z) ∂N ω)=∫⁻ z, ENNReal.ofReal (f z) ∂M ω) : + ∀ᵐ ω ∂P, N ω=M ω := by + let : LocallyCompactSpace upperHalfPlane := isOpen_upperHalfPlane.locallyCompactSpace + have h := random_radon_process_eq_of_positive_tests P (localTestFamily upperHalfPlane) + (fun _ => N) (fun _ => M) (fun _ => hN) (fun _ => hM) (fun f hf => ?_) + · exact h.mono (fun _ hω => hω 0) + · filter_upwards [htest f hf] with ω hω + intro _ + rw [integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable, + integral_eq_lintegral_of_nonneg_ae (ae_of_all _ hf) f.continuous.measurable.aestronglyMeasurable,hω] + +end SLEExactGauge + +end + +end OAI