| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.FiniteStage |
|
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| namespace OAI |
|
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| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal Classical |
| namespace DirectionalTransience |
|
|
| lemma cell_radius_comparison {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))) |
| (C c : ℝ) (hC : 0 < C) (hc : 0 < c) (hc1 : c ≤ 1) : |
| ∃ T : ℝ, 0 < T ∧ ∀ w m i : ℕ, ∀ t : ℝ, T ≤ t → |
| t ≤ c^2/(4*C)*(w:ℝ) → |
| fluctuationRadius (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) t ≤ |
| c*cellRadius ν e f C w m i/2 := by |
| let μ := independentConditionedPairLaw ν (realPosition (step e)) |
| let S := commonIncrementProcess (realPosition (step e)) f 0 |
| let : IsProbabilityMeasure μ := independentConditionedPairLaw_probability ν _ |
| (ne_of_gt (noDrop_positive_of_directionallyTransient ν _ htrans)) |
| have hS : Measurable S := measurable_commonIncrementProcess _ _ _ |
| have hI : Integrable S μ := independent_commonWordIncrement_integrable ν hue _ |
| (signed_direction_unit e) htrans (signedHeight e) (signedHeight_projection e) |
| (signedHeight_step_le e) f |
| have hne : 0 < μ {x | S x ≠ 0} := independent_commonWordIncrement_nonzero ν hue e f hef htrans |
| obtain ⟨R,hR,hr⟩ := fluctuationRadius_scaling_compare μ S hS hI hne |
| refine ⟨max 1 (fluctuationScale μ S R+1),lt_of_lt_of_le (by norm_num) (le_max_left _ _),?_⟩ |
| intro w m i t ht htw |
| have htR : fluctuationScale μ S R < t := by linarith [le_max_right 1 (fluctuationScale μ S R+1)] |
| have hw : (w:ℝ) ≤ cellWidth w m i := by exact_mod_cast cellWidth_min w m i |
| have htw' : t ≤ (c/2)^2*((cellWidth w m i:ℝ)/C) := by |
| calc |
| t ≤ c^2/(4*C)*(w:ℝ) := htw |
| _ ≤ c^2/(4*C)*(cellWidth w m i:ℝ) := mul_le_mul_of_nonneg_left hw (by positivity) |
| _ = (c/2)^2*((cellWidth w m i:ℝ)/C) := by ring |
| have hh := hr ((cellWidth w m i:ℝ)/C) t (c/2) htR htw' (by positivity) (by linarith) |
| simpa only [cellRadius,mul_div_assoc,div_mul_eq_mul_div] using hh |
|
|
| end DirectionalTransience |
|
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| end |
|
|
| section |
|
|
| open scoped Classical |
| namespace DirectionalTransience |
|
|
| noncomputable def stageCellStarts (He H w : ℕ) : Fin (((H-He)/w+1)+1) → ℕ := |
| Fin.lastCases H (fun j => He+j.val*w) |
| noncomputable def stageCellWidths (He H w wplus : ℕ) : Fin (((H-He)/w+1)+1) → ℕ := |
| Fin.lastCases wplus (fun _ => w) |
| noncomputable def stageCellGaps (He H w : ℕ) (Gminus Gplus : ℝ) : Fin (((H-He)/w+1)+1) → ℝ := |
| Fin.lastCases Gplus (fun _ => Gminus) |
|
|
| lemma stage_cells_geometry (He H m W : ℕ) (hw : W ≤ finestCellWidth He m) |
| (hp : W ≤ finestCellWidth H m) (hW : 0<W) (Gminus Gplus : ℝ) : |
| let w := finestCellWidth He m |
| let wp := finestCellWidth H m |
| let v := stageCellStarts He H w |
| let ws := stageCellWidths He H w wp |
| let gs := stageCellGaps He H w Gminus Gplus |
| (∀ j, W ≤ ws j) ∧ (∀ j, 2*cellWidth (ws j) m 0 ≤ v j) ∧ |
| (∀ h : ℕ, He < h → h ≤ H → ∃ j, v j≤h ∧ h≤v j+ws j ∧ Gminus≤gs j) ∧ |
| (∃ j, v j≤H ∧ H≤v j+ws j ∧ Gplus≤gs j) := by |
| dsimp only |
| let w := finestCellWidth He m |
| have hw0 : 0<w := hW.trans_le hw |
| refine ⟨?_,?_,?_,?_⟩ |
| · intro j |
| refine Fin.lastCases ?_ (fun i => ?_) j |
| · simp only [stageCellWidths, Fin.lastCases_last] |
| exact hp |
| · simp only [stageCellWidths, Fin.lastCases_castSucc] |
| exact hw |
| · intro j |
| refine Fin.lastCases ?_ (fun i => ?_) j |
| · simp only [stageCellStarts, stageCellWidths, Fin.lastCases_last] |
| exact finestCellWidth_start H m |
| · simp only [stageCellStarts, stageCellWidths, Fin.lastCases_castSucc] |
| exact (finestCellWidth_start He m).trans (Nat.le_add_right _ _) |
| · intro h hHe hH |
| obtain ⟨j,hj₁,hj₂⟩ := integer_cells_cover He H w hw0 h hHe.le hH |
| refine ⟨j.castSucc,?_,?_,?_⟩ |
| · simp only [stageCellStarts, Fin.lastCases_castSucc] |
| exact hj₁ |
| · simp only [stageCellStarts, stageCellWidths, Fin.lastCases_castSucc] |
| exact hj₂ |
| · simp only [stageCellGaps, Fin.lastCases_castSucc] |
| exact le_rfl |
| · refine ⟨Fin.last _,?_,?_,?_⟩ |
| · simp only [stageCellStarts, Fin.lastCases_last] |
| exact le_rfl |
| · simp only [stageCellStarts, stageCellWidths, Fin.lastCases_last] |
| exact Nat.le_add_right _ _ |
| · simp only [stageCellGaps, Fin.lastCases_last] |
| exact le_rfl |
|
|
| lemma stage_cells_gap_bounds {d : ℕ} (ν : MeasureTheory.Measure (Row d)) |
| (e f : Direction d) (C c Gminus Gplus : ℝ) (He H m : ℕ) |
| (hminus : ∀ i, i < m → Gminus ≤ c*cellRadius ν e f C (finestCellWidth He m) m i/2) |
| (hplus : ∀ i, i < m → Gplus ≤ c*cellRadius ν e f C (finestCellWidth H m) m i/2) : |
| ∀ j i, i < m → stageCellGaps He H (finestCellWidth He m) Gminus Gplus j ≤ |
| c*cellRadius ν e f C (stageCellWidths He H (finestCellWidth He m) (finestCellWidth H m) j) m i/2 := by |
| intro j i hi |
| refine Fin.lastCases ?_ (fun l => ?_) j |
| · simpa only [stageCellGaps, stageCellWidths, Fin.lastCases_last] using hplus i hi |
| · simpa only [stageCellGaps, stageCellWidths, Fin.lastCases_castSucc] using hminus i hi |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open Filter |
| open scoped Topology |
| namespace DirectionalTransience |
|
|
| lemma eventually_exp_affine_le (C a r : ℝ) (h : r < a) : |
| ∀ᶠ b : ℝ in atTop, Real.exp (C-a*b) ≤ Real.exp (-r*b)/2 := by |
| filter_upwards [eventually_ge_atTop ((C+Real.log 2)/(a-r))] with b hb |
| have hh := (div_le_iff₀ (sub_pos.mpr h)).mp hb |
| apply (le_div_iff₀ (by norm_num : (0:ℝ)<2)).mpr |
| rw [← Real.exp_log (by norm_num : (0:ℝ)<2), ← Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| nlinarith only [hh] |
|
|
| lemma eventually_mul_exp_le (D a r : ℝ) (h : r < a) : |
| ∀ᶠ b : ℝ in atTop, D*Real.exp (-a*b) ≤ Real.exp (-r*b)/2 := by |
| let E := max D 1 |
| have hE : 0 < E := lt_of_lt_of_le (by norm_num) (le_max_right _ _) |
| filter_upwards [eventually_exp_affine_le (Real.log E) a r h] with b hb |
| calc |
| _ ≤ E*Real.exp (-a*b) := mul_le_mul_of_nonneg_right (le_max_left _ _) (Real.exp_pos _).le |
| _ = Real.exp (Real.log E-a*b) := by rw [sub_eq_add_neg,Real.exp_add,Real.exp_log hE]; congr 2; ring |
| _ ≤ _ := hb |
|
|
| lemma eventually_stage_short_bound (lam C K : ℝ) (k L : ℕ) |
| (hC : 0≤C) (hL : 0 < L) (hk : 2*K+3 < lam*k/2) (hLk : 2*K+3 < (L:ℝ)-k/2) : |
| ∀ᶠ b : ℝ in atTop, |
| (Real.exp (C*k-lam*(k*b/2)/L) + |
| (16*C*k*L^2)*Real.exp ((k*b/2)/L-b))^L ≤ Real.exp (-(2*K+3)*b) := by |
| have hLp : (0:ℝ)<L := by exact_mod_cast hL |
| have hrate₁ : (2*K+3)/(L:ℝ) < lam*k/(2*L) := by |
| rw [div_mul_eq_div_div] |
| exact (div_lt_div_iff_of_pos_right hLp).mpr hk |
| have hrate₂ : (2*K+3)/(L:ℝ) < 1-(k:ℝ)/(2*L) := by |
| have hh := (div_lt_div_iff_of_pos_right hLp).mpr hLk |
| have heq : ((L:ℝ)-(k:ℝ)/2)/L = 1-(k:ℝ)/(2*L) := by field_simp |
| rwa [heq] at hh |
| filter_upwards [eventually_exp_affine_le (C*k) (lam*k/(2*L)) ((2*K+3)/L) hrate₁, |
| eventually_mul_exp_le (16*C*k*L^2) (1-(k:ℝ)/(2*L)) ((2*K+3)/L) hrate₂] with b h₁ h₂ |
| have hsum : Real.exp (C*k-lam*(k*b/2)/L)+ |
| (16*C*k*L^2)*Real.exp ((k*b/2)/L-b) ≤ Real.exp (-((2*K+3)/L)*b) := by |
| have h₁' : Real.exp (C*k-lam*(k*b/2)/L) ≤ Real.exp (-((2*K+3)/L)*b)/2 := by convert h₁ using 1; congr 1; ring |
| have h₂' : (16*C*k*L^2)*Real.exp ((k*b/2)/L-b) ≤ Real.exp (-((2*K+3)/L)*b)/2 := by convert h₂ using 1; congr 2; ring |
| linarith only [h₁',h₂'] |
| calc |
| _ ≤ (Real.exp (-((2*K+3)/L)*b))^L := pow_le_pow_left₀ (by positivity) hsum L |
| _ = _ := by rw [← Real.exp_nat_mul]; congr 1; field_simp |
|
|
| lemma cell_count_exp_bound (b j D : ℝ) (k m : ℕ) (n : ℝ) |
| (hk : 2 ≤ k) (hm : j*b-1 ≤ (m:ℝ)) (hn : n ≤ Real.exp (D*b)) (hn0 : 0≤n) : |
| n*(2:ℝ)^(m-1)*Real.exp (-2*k*m) ≤ |
| Real.exp ((2*k-Real.log 2)-((2*k-Real.log 2)*j-D)*b) := by |
| have hlog : Real.log 2 ≤ 1 := by convert Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2) using 1; norm_num |
| have hk' : (2:ℝ)≤k := by exact_mod_cast hk |
| have hpos : 0 ≤ 2*k-Real.log 2 := by linarith |
| have hp : (2:ℝ)^(m-1) ≤ 2^m := pow_le_pow_right₀ (by norm_num) (Nat.sub_le m 1) |
| calc |
| _ ≤ Real.exp (D*b)*(2:ℝ)^m*Real.exp (-2*k*m) := by |
| apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le |
| exact (mul_le_mul_of_nonneg_left hp hn0).trans |
| (mul_le_mul_of_nonneg_right hn (by positivity)) |
| _ = Real.exp (D*b+Real.log 2*m-2*k*m) := by |
| rw [sub_eq_add_neg,Real.exp_add,Real.exp_add,show Real.log 2*(m:ℝ)=(m:ℝ)*Real.log 2 by ring, |
| Real.exp_nat_mul,Real.exp_log (by norm_num : (0:ℝ)<2)] |
| congr 2; ring |
| _ ≤ _ := by |
| apply Real.exp_le_exp.mpr |
| have hh := mul_le_mul_of_nonneg_left hm hpos |
| nlinarith only [hh] |
|
|
| lemma eventually_stage_probability_sum (lam lamq C Cq j D K : ℝ) (k L : ℕ) |
| (hk : 2 ≤ k) (hL : 0<L) (hCr : 0≤C) |
| (hshort : 2*K+3 < lam*k/2) (hLk : 2*K+3 < (L:ℝ)-k/2) |
| (hmass : 2*K+3 < lamq*k/4) (hcell : 2*K+3 < (2*k-Real.log 2)*j-D) : |
| ∀ᶠ b : ℝ in atTop, ∀ m : ℕ, j*b-1 ≤ m → ∀ n : ℝ, 0≤n → n≤Real.exp (D*b) → |
| (Real.exp (C*k-lam*(k*b/2)/L)+(16*C*k*L^2)*Real.exp ((k*b/2)/L-b))^L + |
| Real.exp (Cq*k)*(Real.exp (-k*b/4))^lamq + n*2^(m-1)*Real.exp (-2*k*m) ≤ Real.exp (-2*K*b) := by |
| filter_upwards [eventually_stage_short_bound lam C K k L hCr hL hshort hLk, |
| eventually_exp_affine_le (Cq*k) (lamq*k/4) (2*K+3) hmass, |
| eventually_exp_affine_le (2*k-Real.log 2) ((2*k-Real.log 2)*j-D) (2*K+3) hcell, |
| eventually_ge_atTop (1:ℝ)] with b hs hq hc hb m hm n hn0 hn |
| have hq' : Real.exp (Cq*k)*(Real.exp (-k*b/4))^lamq ≤ Real.exp (-(2*K+3)*b)/2 := by |
| rw [Real.rpow_def_of_pos (Real.exp_pos _),Real.log_exp,← Real.exp_add] |
| convert hq using 1; congr 1; ring |
| have hc' := (cell_count_exp_bound b j D k m n hk hm hn hn0).trans hc |
| have htwo : 2*Real.exp (-(2*K+3)*b) ≤ Real.exp (-2*K*b) := by |
| have hlog : Real.log 2 ≤ 1 := by convert Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2) using 1; norm_num |
| rw [← Real.exp_log (by norm_num : (0:ℝ)<2),← Real.exp_add] |
| exact Real.exp_le_exp.mpr (by nlinarith only [hb,hlog]) |
| linarith only [hs,hq',hc',htwo] |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
