openAI-math-dataset / lean /OAI /Probability /Ballisticity /Walk /TransverseFirstHitSinglePathLimit.lean
| 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 | |