| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.OutwardCapLocality |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
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| lemma outward_canonical_bound_of_log_bound {d : ℕ} |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (e f : Direction d) {A D B : ℝ} (hA : 0 < A) (hD : 0 < D) (hB : 0 < B) |
| (hdata : ∀ (a : ℝ) (x : Lattice d × Lattice d), x ∈ PairAtHeight (realPosition (step e)) a → |
| ∃ W : Environment d → ℝ, Measurable W ∧ (∀ ω, 0 ≤ W ω) ∧ |
| Integrable (fun ω => Real.exp (W ω)) (environmentLaw ν) ∧ |
| (∫ ω, Real.exp (W ω) ∂environmentLaw ν) ≤ B ∧ |
| (∀ᵐ ω ∂environmentLaw ν, ∀ t : ℕ, |
| 0 < outwardKernelMass e f (t+1) x ω ∧ |
| -Real.log (outwardKernelMass e f (t+1) x ω) ≤ |
| A*Real.log ((t:ℝ)+2)+D*(W ω+1))) : |
| ∃ c B' : ℝ, 0 < c ∧ 0 < B' ∧ |
| ∀ (a : ℝ) (x : Lattice d × Lattice d), x ∈ PairAtHeight (realPosition (step e)) a → |
| let Z := outwardCanonicalExcess e f A x |
| (∀ ω, 0 ≤ Z ω) ∧ |
| @Measurable _ _ (rowSigma {y | a ≤ dot (realPosition y) (realPosition (step e))}) _ Z ∧ |
| Integrable (fun ω => Real.exp (c*Z ω)) (environmentLaw ν) ∧ |
| (∫ ω, Real.exp (c*Z ω) ∂environmentLaw ν) ≤ B' ∧ |
| (∀ᵐ ω ∂environmentLaw ν, ∀ 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)+Z ω) := by |
| refine ⟨D⁻¹,Real.exp 1*B,inv_pos.mpr hD,mul_pos (Real.exp_pos _) hB,?_⟩ |
| intro a x hx |
| obtain ⟨W,hW,hW0,hWi,hWB,hbound⟩ := hdata a x hx |
| let Z := outwardCanonicalExcess e f A x |
| have hZm : Measurable Z := measurable_outwardCanonicalExcess e f A x |
| have hZbound : ∀ᵐ ω ∂environmentLaw ν, Z ω ≤ D*(W ω+1) ∧ |
| ∀ n, -Real.log (outwardKernelMass e f (n+1) x ω) ≤ 2*A*Real.log ((n:ℝ)+2)+Z ω := by |
| filter_upwards [hbound] with ω hω |
| exact canonicalLogExcess_bounds hA.le hD.le (hW0 ω) _ ω (fun n => (hω n).2) |
| have hdom : Integrable (fun ω => Real.exp 1*Real.exp (W ω)) (environmentLaw ν) := hWi.const_mul _ |
| have hexp : ∀ᵐ ω ∂environmentLaw ν, Real.exp (D⁻¹*Z ω) ≤ Real.exp 1*Real.exp (W ω) := by |
| filter_upwards [hZbound] with ω hω |
| rw [←Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| have hh := mul_le_mul_of_nonneg_left hω.1 (inv_nonneg.mpr hD.le) |
| rw [←mul_assoc,inv_mul_cancel₀ hD.ne',one_mul] at hh |
| linarith |
| have hZi : Integrable (fun ω => Real.exp (D⁻¹*Z ω)) (environmentLaw ν) := |
| hdom.mono_nonneg (hZm.const_mul _ |>.exp |>.aestronglyMeasurable) |
| (Eventually.of_forall (fun ω => (Real.exp_pos _).le)) hexp |
| refine ⟨fun ω => canonicalLogExcess_nonneg _ _ ω,outwardCanonicalExcess_upper_rows e f A a x hx,hZi,?_,?_⟩ |
| · calc |
| _ ≤ ∫ ω, Real.exp 1*Real.exp (W ω) ∂environmentLaw ν := integral_mono_ae hZi hdom hexp |
| _ = Real.exp 1*(∫ ω, Real.exp (W ω) ∂environmentLaw ν) := integral_const_mul _ _ |
| _ ≤ Real.exp 1*B := mul_le_mul_of_nonneg_left hWB (Real.exp_pos _).le |
| · filter_upwards [hbound,hZbound] with ω hω hZω |
| exact fun n => ⟨(hω n).1,hZω.2 n⟩ |
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| theorem actual_outward_order {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 c B : ℝ, 0 < A ∧ 0 < c ∧ 0 < B ∧ |
| (∀ (a : ℝ) (x : Lattice d × Lattice d), x ∈ PairAtHeight (realPosition (step e)) a → |
| let Z := outwardCanonicalExcess e f A x |
| (∀ ω, 0 ≤ Z ω) ∧ |
| @Measurable _ _ (rowSigma {y | a ≤ dot (realPosition y) (realPosition (step e))}) _ Z ∧ |
| Integrable (fun ω => Real.exp (c*Z ω)) (environmentLaw ν) ∧ |
| (∫ ω, Real.exp (c*Z ω) ∂environmentLaw ν) ≤ B ∧ |
| (∀ᵐ ω ∂environmentLaw ν, ∀ 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)+Z ω)) ∧ |
| (∀ C ε δ : ℝ, 0 < C → 0 < ε → 0 < δ → ∃ R : ℕ, |
| ∀ (a : ℝ) (x : Lattice d × Lattice d), x ∈ PairAtHeight (realPosition (step e)) a → |
| ∃ F : Environment d → ℝ, |
| @Measurable _ _ (rowSigma (upperPairNeighborhood e a x R)) _ F ∧ |
| (∀ ω, 0 ≤ F ω ∧ F ω ≤ C) ∧ |
| (environmentLaw ν).real {ω | ε < |min (outwardCanonicalExcess e f A x ω) C-F ω|} < δ) := by |
| obtain ⟨A,D,B,hA,hD,hB,hdata⟩ := actual_outward_log_bound ν hue e f hef htrans |
| obtain ⟨c,B',hc,hB',hbound⟩ := outward_canonical_bound_of_log_bound ν e f hA hD hB hdata |
| exact ⟨A,c,B',hA,hc,hB',hbound, |
| outward_canonical_cap_locality_of_log_bound ν hue e f hA hD hB hdata⟩ |
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| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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