| import Mathlib |
| import OAI.Probability.Ballisticity.Walk.Kernel |
|
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| namespace OAI |
|
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| section |
|
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology |
| namespace TailDecorrelation |
|
|
| lemma sampled_kernel_sections {D E J W : Type*} |
| [MeasurableSpace D] [MeasurableSpace E] [MeasurableSpace J] [MeasurableSpace W] |
| [Countable J] [MeasurableSingletonClass J] |
| (δ : Measure D) (σ : Measure E) (P : Measure W) |
| [IsProbabilityMeasure δ] [IsProbabilityMeasure σ] [IsProbabilityMeasure P] |
| (κ : Kernel D J) (s : D → E → J) |
| (hs : Measurable (Function.uncurry s)) (hmap : ∀ d, σ.map (s d) = κ d) |
| {f : D × W → ℝ} (hf : Measurable f) {Z : J → W → ℝ} |
| (hZ : ∀ a, Measurable (Z a)) |
| (hi : Integrable (fun x : (D × E) × W => f (x.1.1,x.2) * Z (s x.1.1 x.1.2) x.2) |
| ((δ.prod σ).prod P)) : |
| ∀ᵐ x ∂δ.prod P, f x ≠ 0 → Integrable (fun a => Z a x.2) (κ x.1) := by |
| have hz : Measurable (Function.uncurry Z) := measurable_from_prod_countable_right hZ |
| let G : (D × E) × W → ℝ := fun x => f (x.1.1,x.2) * Z (s x.1.1 x.1.2) x.2 |
| have hG : Measurable G := |
| (hf.comp ((measurable_fst.comp measurable_fst).prodMk measurable_snd)).mul |
| (hz.comp ((hs.comp measurable_fst).prodMk measurable_snd)) |
| have hp := swap_seed_preserving δ σ P |
| have hj : Integrable (fun x : (D × W) × E => f x.1 * Z (s x.1.1 x.2) x.1.2) |
| ((δ.prod P).prod σ) := (hp.integrable_comp hG.aestronglyMeasurable).mpr hi |
| filter_upwards [hj.prod_right_ae] with x hx |
| intro hfx |
| have hzx : Integrable (fun u => Z (s x.1 u) x.2) σ := by |
| have hh := hx.const_mul (f x)⁻¹ |
| simpa only [←mul_assoc, inv_mul_cancel₀ hfx, one_mul] using hh |
| rw [←hmap] |
| exact (integrable_map_measure (measurable_of_countable _).aestronglyMeasurable |
| (hs.comp measurable_prodMk_left).aemeasurable).mpr hzx |
|
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| lemma kernel_density_sections {D I A J : Type*} [MeasurableSpace D] [MeasurableSpace A] [Encodable I] |
| [MeasurableSpace J] [Countable J] [MeasurableSingletonClass J] |
| [Nonempty J] [StandardBorelSpace J] |
| (δ : Measure D) (ν : Measure A) [IsProbabilityMeasure δ] [IsProbabilityMeasure ν] |
| {f : D × (I → A) → ℝ} {Z : J → (I → A) → ℝ} |
| {c Q : ℝ} (hc : 0 < c) |
| (hf : Measurable f) (hf0 : ∀ x, 0 ≤ f x) |
| (hfi : Integrable f (δ.prod (Measure.infinitePi (fun _ : I => ν)))) |
| (hH : Integrable (fun x => entropyPlus (f x)) (δ.prod (Measure.infinitePi (fun _ : I => ν)))) |
| (hZ : ∀ a, Measurable (Z a)) (hZ0 : ∀ a ω, 0 ≤ Z a ω) |
| (he : ∀ a, Integrable (fun ω => Real.exp (c * Z a ω)) (Measure.infinitePi (fun _ : I => ν))) |
| (heQ : ∀ a, (∫ ω, Real.exp (c * Z a ω) ∂Measure.infinitePi (fun _ : I => ν)) ≤ Q) |
| (κ : Kernel D J) [IsMarkovKernel κ] : |
| ∀ᵐ x ∂δ.prod (Measure.infinitePi (fun _ : I => ν)), |
| f x ≠ 0 → Integrable (fun a => Z a x.2) (κ x.1) := by |
| obtain ⟨s,hs,hmap⟩ := Kernel.exists_measurable_map_eq_unitInterval κ |
| let σ : Measure unitInterval := volume |
| let g : (D × unitInterval) × (I → A) → ℝ := fun x => f (x.1.1,x.2) |
| have hg : Measurable g := hf.comp ((measurable_fst.comp measurable_fst).prodMk measurable_snd) |
| obtain ⟨_,hgH,_⟩ := randomized_density δ σ ν hf hfi hH |
| have hi := integrable_density_mul_of_entropy hc hg |
| (measurable_countable_sample hZ hs) (fun x => hf0 _) (fun x => hZ0 _ _) hgH |
| (sampled_exponential_integrable (δ.prod σ) ν hZ he heQ hs).1 |
| exact sampled_kernel_sections δ σ (Measure.infinitePi (fun _ : I => ν)) |
| κ s hs hmap hf hZ hi |
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| end TailDecorrelation |
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| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology |
| namespace DirectionalTransience |
|
|
| lemma cut_neighborhoods_escape {d : ℕ} (e : Direction d) (a : ℝ) |
| (k : Fin d) (L U : ℕ → ℤ) |
| (hL : Tendsto L atTop atBot) (hU : Tendsto U atTop atTop) |
| (N : ℕ) (S : Set (Lattice d)) (hS : S.Finite) : |
| ∀ᶠ r in atTop, ∀ p : Lattice d × Lattice d, |
| p.1 k ≤ L r → U r ≤ p.2 k → Disjoint S (upperPairNeighborhood e a p N) := by |
| have hz : ∀ z ∈ S, ∀ᶠ r in atTop, ∀ p : Lattice d × Lattice d, |
| p.1 k ≤ L r → U r ≤ p.2 k → z ∉ LatticeBall p.1 N ∪ LatticeBall p.2 N := by |
| intro z _ |
| filter_upwards [(tendsto_atBot.mp hL) (z k - (N : ℤ) - 1), |
| (tendsto_atTop.mp hU) (z k + (N : ℤ) + 1)] with r hrx hry |
| intro p hp hq hz |
| rcases hz with hz | hz |
| · have hh := (abs_le.mp (hz k)).2 |
| omega |
| · have hh := (abs_le.mp (hz k)).1 |
| omega |
| filter_upwards [hS.eventually_all.mpr hz] with r hr |
| intro p hp hq |
| exact Set.disjoint_left.mpr fun z hz hz' => hr z hz p hp hq hz'.1 |
|
|
| theorem actual_kernel_tail_decorrelation {d : ℕ} |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] (hue : UniformElliptic ν) |
| (e f : Direction d) (hef : e.1 ≠ f.1) |
| (htrans : DirectionallyTransient ν (realPosition (step e))) : |
| ∃ A : ℝ, 0 < A ∧ |
| (∀ᵐ ω ∂environmentLaw ν, ∀ x : Lattice d × Lattice d, |
| dot (realPosition x.1) (realPosition (step e)) = |
| dot (realPosition x.2) (realPosition (step e)) → |
| ∀ t : ℕ, 0 < outwardKernelMass e f (t+1) x ω ∧ |
| -Real.log (outwardKernelMass e f (t+1) x ω) ≤ |
| 2*A*Real.log ((t:ℝ)+2)+outwardCanonicalExcess e f A x ω) ∧ |
| let J := {p : Lattice d × Lattice d |
| let Z : J → Environment d → ℝ := fun p => outwardCanonicalExcess e f A p.1 |
| BddAbove (Set.range (fun p => ∫ ω, Z p ω ∂environmentLaw ν)) ∧ |
| ∀ (D : Type) [MeasurableSpace D] (δ : Measure D) [IsProbabilityMeasure δ] |
| (g : D × Environment d → ℝ), |
| Measurable g → (∀ x, 0 ≤ g x) → Integrable g (δ.prod (environmentLaw ν)) → |
| (∫ x, g x ∂δ.prod (environmentLaw ν)) = 1 → |
| Integrable (fun x => TailDecorrelation.entropyPlus (g x)) (δ.prod (environmentLaw ν)) → |
| ∀ (κ : ℕ → Kernel D J) [∀ r, IsMarkovKernel (κ r)] (L U : ℕ → D → ℤ), |
| (∀ᵐ z ∂δ, Tendsto (fun r => L r z) atTop atBot ∧ Tendsto (fun r => U r z) atTop atTop) → |
| (∀ r, ∀ᵐ z ∂δ, ∀ᵐ p ∂κ r z, p.1.1 f.1 ≤ L r z ∧ U r z ≤ p.1.2 f.1) → |
| (∀ r, ∀ᵐ z ∂δ.prod (environmentLaw ν), g z ≠ 0 → |
| Integrable (fun p => Z p z.2) (κ r z.1)) ∧ |
| (∀ r, Integrable (fun z => g z * (∫ p, Z p z.2 ∂κ r z.1)) (δ.prod (environmentLaw ν))) ∧ |
| (∀ ε : ℝ, 0 < ε → ∀ᶠ r in atTop, |
| (∫ z, g z * (∫ p, Z p z.2 ∂κ r z.1) ∂δ.prod (environmentLaw ν)) ≤ |
| (⨆ p, ∫ ω, Z p ω ∂environmentLaw ν) + ε) ∧ |
| Filter.limsup (fun r => ∫ z, g z * (∫ p, Z p z.2 ∂κ r z.1) |
| ∂δ.prod (environmentLaw ν)) atTop ≤ ⨆ p, ∫ ω, Z p ω ∂environmentLaw ν := by |
| classical |
| obtain ⟨A,c,B,hA,hc,hB,hbound,hloc⟩ := actual_outward_order ν hue e f hef htrans |
| let J := {p : Lattice d × Lattice d |
| let Z : J → Environment d → ℝ := fun p => outwardCanonicalExcess e f A p.1 |
| have hZm : ∀ p : J, Measurable (Z p) := fun p => |
| ((hbound 0 p.1 p.2).2.1).mono (rowSigma_le _) le_rfl |
| have hZ0 : ∀ p : J, ∀ ω, 0 ≤ Z p ω := fun p => (hbound 0 p.1 p.2).1 |
| have he : ∀ p : J, Integrable (fun ω => Real.exp (c * Z p ω)) (environmentLaw ν) := |
| fun p => (hbound 0 p.1 p.2).2.2.1 |
| have heB : ∀ p : J, (∫ ω, Real.exp (c * Z p ω) ∂environmentLaw ν) ≤ B := |
| fun p => (hbound 0 p.1 p.2).2.2.2.1 |
| have hZi : ∀ p : J, Integrable (Z p) (environmentLaw ν) := fun p => |
| (integrable_and_mean_le_of_exp (environmentLaw ν) hc (hZm p) (hZ0 p) (he p) (heB p)).1 |
| have hZB : ∀ p : J, (∫ ω, Z p ω ∂environmentLaw ν) ≤ B / c := fun p => |
| (integrable_and_mean_le_of_exp (environmentLaw ν) hc (hZm p) (hZ0 p) (he p) (heB p)).2 |
| have hba : BddAbove (Set.range (fun p : J => ∫ ω, Z p ω ∂environmentLaw ν)) := |
| ⟨B/c, by rintro _ ⟨p,rfl⟩; exact hZB p⟩ |
| let p0 : J := ⟨(0,0), by simp [PairAtHeight, dot, realPosition]⟩ |
| let : Nonempty J := ⟨p0⟩ |
| have hC : 0 ≤ ⨆ p : J, ∫ ω, Z p ω ∂environmentLaw ν := |
| (integral_nonneg fun ω => hZ0 p0 ω).trans (le_ciSup hba p0) |
| refine ⟨A,hA,outward_order_countable_event ν e f A (fun a x hx => (hbound a x hx).2.2.2.2),hba,?_⟩ |
| intro D mD δ hδ g hgm hg0 hgi hg1 hgH κ hκ L U hcuts htail |
| have hcap : ∀ K η : ℝ, 0 < K → 0 < η → ∃ N : ℕ, |
| ∃ F : J → Environment d → ℝ, |
| (∀ p, @Measurable _ _ (TailDecorrelation.rows (upperPairNeighborhood e 0 p.1 N)) _ (F p)) ∧ |
| (∀ p ω, 0 ≤ F p ω ∧ F p ω ≤ K) ∧ |
| (∀ p, (∫ ω, |min K (Z p ω) - F p ω| ∂environmentLaw ν) ≤ η) := by |
| intro K η hK hη |
| obtain ⟨N,hN⟩ := outward_cap_locality_L1 ν e f A hloc hK hη |
| have hh : ∀ p : J, ∃ F : Environment d → ℝ, |
| @Measurable _ _ (rowSigma (upperPairNeighborhood e 0 p.1 N)) _ F ∧ |
| (∀ ω, 0 ≤ F ω ∧ F ω ≤ K) ∧ |
| (∫ ω, |min (Z p ω) K - F ω| ∂environmentLaw ν) < η := |
| fun p => hN 0 p.1 p.2 |
| choose F hFm hFb hFe using hh |
| refine ⟨N,F,?_,hFb,?_⟩ |
| · intro p |
| rw [analytic_rows_eq] |
| exact hFm p |
| · intro p |
| simpa only [min_comm] using (hFe p).le |
| have hesc : ∀ N (S : Set (Lattice d)), S.Finite → ∀ᵐ z ∂δ, |
| ∀ᶠ r in atTop, ∀ᵐ p ∂κ r z, Disjoint S (upperPairNeighborhood e 0 p.1 N) := by |
| intro N S hS |
| filter_upwards [hcuts, ae_all_iff.mpr htail] with z hz ht |
| filter_upwards [cut_neighborhoods_escape e 0 f.1 (fun r => L r z) (fun r => U r z) |
| hz.1 hz.2 N S hS] with r hr |
| filter_upwards [ht r] with p hp |
| exact hr p.1 hp.1 hp.2 |
| have he' : ∀ p : J, Integrable (fun ω => Real.exp ((2*(c/2)) * Z p ω)) (environmentLaw ν) := by |
| simpa only [mul_div_cancel₀ _ (by norm_num : (2:ℝ) ≠ 0)] using he |
| have heB' : ∀ p : J, (∫ ω, Real.exp ((2*(c/2)) * Z p ω) ∂environmentLaw ν) ≤ B := by |
| simpa only [show 2*(c/2) = c by ring] using heB |
| refine ⟨fun r => TailDecorrelation.kernel_density_sections δ ν hc hgm hg0 hgi hgH |
| hZm hZ0 he heB (κ r), ?_⟩ |
| refine ⟨fun r => TailDecorrelation.kernel_density_integrable δ ν hc hgm hg0 hgi hgH |
| hZm hZ0 he heB (κ r), ?_⟩ |
| have hE := TailDecorrelation.kernel_entropy_decorrelation δ ν (by linarith : 0 < c/2) hC |
| hgm hg0 hgi hg1 hgH hZm hZ0 he' heB' hZi (fun p => le_ciSup hba p) |
| (fun N p => upperPairNeighborhood e 0 p.1 N) κ hcap hesc |
| exact ⟨hE, TailDecorrelation.limsup_le_of_nonneg_eventual |
| (fun r => integral_nonneg fun z => mul_nonneg (hg0 z) (integral_nonneg fun p => hZ0 _ _)) hE⟩ |
|
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| end DirectionalTransience |
|
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| end |
|
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| end OAI |
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|
