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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 | import Mathlib
import OAI.Probability.Ballisticity.Walk.WeakPathEvalTendsto
import OAI.Probability.Ballisticity.Estimates.GaussianPiReal
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
namespace DirectionalTransience
noncomputable def pathPointMean (G : Measure C(unitInterval,ℝ)) (t : unitInterval) : ℝ :=
∫ g, g t ∂G
lemma path_finite_gaussian_marginal (G : ProbabilityMeasure C(unitInterval,ℝ))
(hG : ∀ I : Finset unitInterval,
IsGaussian ((G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g)))
(t : unitInterval) : IsGaussian ((G : Measure C(unitInterval,ℝ)).map (fun g => g t)) := by
let I : Finset unitInterval := {t}
let μ := (G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g)
have : IsGaussian μ := hG I
let L : (I → ℝ) →L[ℝ] ℝ := .proj ⟨t,by simp [I]⟩
have he : μ.map L = (G : Measure C(unitInterval,ℝ)).map (fun g => g t) := by
rw [Measure.map_map L.measurable (continuous_path_restrict I).measurable]
rfl
rw [← he]
infer_instance
lemma path_marginal_eq_gaussian (G : ProbabilityMeasure C(unitInterval,ℝ))
(hG : ∀ I : Finset unitInterval,
IsGaussian ((G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g)))
(t : unitInterval) :
(G : Measure C(unitInterval,ℝ)).map (fun g => g t) =
gaussianReal (pathPointMean G t) (Var[fun g : C(unitInterval,ℝ) => g t; G]).toNNReal := by
have : IsGaussian ((G : Measure C(unitInterval,ℝ)).map (fun g => g t)) := path_finite_gaussian_marginal G hG t
rw [IsGaussian.eq_gaussianReal ((G : Measure C(unitInterval,ℝ)).map (fun g => g t)) inferInstance]
congr 1
· exact integral_map (continuous_eval_const t).measurable.aemeasurable measurable_id.aestronglyMeasurable
· rw [variance_map measurable_id.aemeasurable (continuous_eval_const t).measurable.aemeasurable]; rfl
lemma continuous_pathPointMean (G : ProbabilityMeasure C(unitInterval,ℝ))
(hG : ∀ I : Finset unitInterval,
IsGaussian ((G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g))) :
Continuous (pathPointMean G) :=
continuous_gaussian_path_mean G (pathPointMean G) _ (path_marginal_eq_gaussian G hG)
lemma path_finite_mean (G : ProbabilityMeasure C(unitInterval,ℝ)) (I : Finset unitInterval)
(hG : IsGaussian ((G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g))) :
(∫ x, x ∂(G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g)) =
I.restrict (pathPointMean G) := by
have := hG
ext t
change (ContinuousLinearMap.proj t : (I → ℝ) →L[ℝ] ℝ) (∫ x, x ∂(G : Measure C(unitInterval,ℝ)).map
(fun g : C(unitInterval,ℝ) => I.restrict g)) = pathPointMean G t
rw [← IsGaussian.integral_dual (μ := (G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g)) (.proj t),integral_map
(continuous_path_restrict I).measurable.aemeasurable (by fun_prop)]
rfl
theorem path_iid_difference_centered (G W : ProbabilityMeasure C(unitInterval,ℝ)) (T : ℝ)
(hW : ∀ I : Finset unitInterval,
(W : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g) = gaussianPathFiniteLaw T I)
(hG : ((G : Measure C(unitInterval,ℝ)).prod G).map
(fun p : C(unitInterval,ℝ) × C(unitInterval,ℝ) => p.1-p.2) = W)
(a : ℝ) (ha : a^2 = 1/2) :
∃ m : C(unitInterval,ℝ), (∀ t, m t = pathPointMean G t) ∧
(G : Measure C(unitInterval,ℝ)).map (fun g => g-m) =
(W : Measure C(unitInterval,ℝ)).map (fun g => a • g) := by
have hg := path_of_iid_difference_gaussian G W T hW hG
let m : C(unitInterval,ℝ) := ⟨pathPointMean G,continuous_pathPointMean G hg⟩
refine ⟨m,fun _ => rfl,?_⟩
apply continuous_path_measure_ext
intro I
let V := (G : Measure C(unitInterval,ℝ)).map (fun g : C(unitInterval,ℝ) => I.restrict g)
have hd : (V.prod V).map (fun p : (I → ℝ) × (I → ℝ) => p.1-p.2) = gaussianPathFiniteLaw T I := by
rw [path_restrict_iid_diff,hG,hW]
have hh := iid_difference_centered_eq_half V (gaussianPathFiniteLaw T I)
(gaussianPathFiniteLaw_mean T I) hd a ha
rw [path_finite_mean G I (hg I)] at hh
rw [← hW I] at hh
rw [Measure.map_map (show Measurable (fun x : I → ℝ => x-I.restrict (pathPointMean G)) by fun_prop)
(continuous_path_restrict I).measurable,
Measure.map_map (show Measurable (fun x : I → ℝ => a • x) by fun_prop)
(continuous_path_restrict I).measurable] at hh
rw [Measure.map_map (continuous_path_restrict I).measurable (show Measurable (fun g : C(unitInterval,ℝ) => g-m) by fun_prop),
Measure.map_map (continuous_path_restrict I).measurable (show Measurable (fun g : C(unitInterval,ℝ) => a • g) by fun_prop)]
exact hh
end DirectionalTransience
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal BigOperators Topology Classical
end
end
end OAI
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