| 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 |
| 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] |
|
|
| 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 |
|
|
