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9b6e014 | 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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import Mathlib
import OAI.Probability.Ballisticity.Walk.GaussianPathFiniteLawSmul
namespace OAI
section
section
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Pointwise Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Pointwise Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
namespace DirectionalTransience
theorem transverse_firstHit_single_path_limit {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)))
(r : ℕ → ℝ) (hr : IsGaussianSequence (independentConditionedPairLaw ν (realPosition (step e)))
(commonIncrementProcess (realPosition (step e)) f 0) r) {T : ℝ} (hT : 0 ≤ T) :
let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν (realPosition (step e)) htrans)
letI : IsProbabilityMeasure (conditionedLaw ν (realPosition (step e))) :=
conditionedLaw_probability ν _ hp
∃ W : ProbabilityMeasure C(unitInterval,ℝ),
TendstoInDistribution (fun i X => heightPolygon
(fun h => signedCoordinate f (recordIndexPosition (realPosition (step e)) h X)-
(recordMedian ν (realPosition (step e)) hp f h : ℝ)) (r i)
(fluctuationScale (independentConditionedPairLaw ν (realPosition (step e)))
(commonIncrementProcess (realPosition (step e)) f 0) (r i)) T)
atTop id (fun _ => conditionedLaw ν (realPosition (step e))) W ∧
∀ I : Finset unitInterval,
(W : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g) =
gaussianPathFiniteLaw (T/(2*commonMeanWidth ν (realPosition (step e)))) I := by
let ℓ := realPosition (step e)
let μ := conditionedLaw ν ℓ
have hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans)
let : IsProbabilityMeasure μ := conditionedLaw_probability ν ℓ hp
let : IsProbabilityMeasure (independentConditionedPairLaw ν ℓ) :=
independentConditionedPairLaw_probability ν ℓ hp
let F := fun h X => signedCoordinate f (recordIndexPosition ℓ h X)
have hF (h : ℕ) : Measurable (F h) :=
(measurable_of_countable (signedCoordinate f)).comp (measurable_recordIndexPosition ℓ h)
let n := fun i => fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i)
have hi := independent_commonWordIncrement_integrable ν hue ℓ (signed_direction_unit e)
htrans (signedHeight e) (signedHeight_projection e) (signedHeight_step_le e) f
have hne := independent_commonWordIncrement_nonzero ν hue e f hef htrans
have hn : Tendsto n atTop atTop := (fluctuationScale_tendsto _ _
(measurable_commonIncrementProcess ℓ f 0) hi hne).comp hr.1
apply height_median_path_limit μ F hF (fun h => (recordMedian ν ℓ hp f h : ℝ))
(recordMedian_spec ν ℓ hp f) r n (fun _ => fluctuationScale_nonneg _ _ _)
hn (commonMeanWidth ν ℓ) _ hT
intro S hS
obtain ⟨W,hW,hWf⟩ := transverse_firstHit_path_limit ν hue e f hef htrans r hr hS
refine ⟨W,?_,hWf⟩
have h0 : ∀ᵐ P ∂independentConditionedPairLaw ν ℓ, firstHitPairGap ℓ f 0 P = 0 := by
have hx := (conditionedLaw_absolutelyContinuous ν ℓ).ae_le (annealed_initial ν)
filter_upwards [Measure.quasiMeasurePreserving_fst.ae hx,
Measure.quasiMeasurePreserving_snd.ae hx] with P hP1 hP2
simp [firstHitPairGap,recordIndexPosition,recordIndexTime_zero,hP1,hP2]
have he (i : ℕ) : ((heightPathLaw μ F hF (r i) (n i) S).prod
(heightPathLaw μ F hF (r i) (n i) S)).map
(fun paths => paths.1 - paths.2) =
(⟨(independentConditionedPairLaw ν ℓ).map (fun P => scaledPolygon
(heightIncrement (fun h => firstHitPairGap ℓ f h P)) (r i) (n i) S),
inferInstance⟩ : ProbabilityMeasure C(unitInterval,ℝ)) := by
apply Subtype.ext
change ((heightPathLaw μ F hF (r i) (n i) S : Measure C(unitInterval,ℝ)).prod
(heightPathLaw μ F hF (r i) (n i) S : Measure C(unitInterval,ℝ))).map
(fun p => p.1-p.2) = _
rw [heightPathLaw_iid_difference]
apply Measure.map_congr
filter_upwards [h0] with P hP
change heightPolygon (fun h => firstHitPairGap ℓ f h P) (r i) (n i) S = _
simp [heightPolygon,hP]
have hId : (⟨(W : Measure C(unitInterval,ℝ)).map id,
inferInstance⟩ : ProbabilityMeasure C(unitInterval,ℝ)) = W := by
apply Subtype.ext
exact Measure.map_id
have h := hW.tendsto
rw [hId] at h
simpa only [he] using h
end DirectionalTransience
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction
end
end
end OAI
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