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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Probability.Ballisticity.Estimates.OutwardCapLocality
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter Function
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
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⟩
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⟩
end DirectionalTransience
end
end
end OAI
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