| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.QuenchedMoments |
| import OAI.Probability.Ballisticity.Estimates.SharedLimitOccupation |
| import OAI.Probability.Ballisticity.Estimates.PairWeakUniqueness |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology BoundedContinuousFunction |
| namespace DirectionalTransience |
|
|
| lemma probability_law_tendsto {Ω E : Type*} [MeasurableSpace Ω] |
| [TopologicalSpace E] [MeasurableSpace E] [BorelSpace E] |
| (μ : ℕ → Measure Ω) [∀ i, IsProbabilityMeasure (μ i)] (X : ℕ → Ω → E) |
| (hX : ∀ i, AEMeasurable (X i) (μ i)) (W : ProbabilityMeasure E) |
| (h : TendstoInDistribution X atTop id μ W) : |
| Tendsto (β := ProbabilityMeasure E) (fun i => (⟨(μ i).map (X i), |
| (Measure.isProbabilityMeasure_map_iff (hX i)).mpr inferInstance⟩ : ProbabilityMeasure E)) |
| atTop (𝓝 W) := by |
| have hh := h.tendsto |
| have he : (⟨(W : Measure E).map id, |
| (Measure.isProbabilityMeasure_map_iff h.aemeasurable_limit).mpr inferInstance⟩ : ProbabilityMeasure E)=W := by |
| apply Subtype.ext |
| exact Measure.map_id |
| simpa only [he] using hh |
|
|
| theorem shared_pair_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 ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let n := fun i => fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i) |
| let θ := fun i => recordMedianSlope ν ℓ hp f (r i) |
| ∃ W : ProbabilityMeasure C(unitInterval,ℝ), |
| (∀ I : Finset unitInterval, (W : Measure C(unitInterval,ℝ)).map |
| (fun g : C(unitInterval,ℝ) => I.restrict g)=gaussianPathFiniteLaw (T/(2*commonMeanWidth ν ℓ)) I) ∧ |
| ∀ x : ℕ → Lattice d, |
| ∀ μ : ℕ → ProbabilityMeasure RealPathPair, |
| (∀ i, (μ i : Measure RealPathPair)=(sharedConditionedPairLaw ν ℓ (x i) (x i)).map |
| (sharedLinearPairPath ℓ f (θ i) (r i) (n i) T (x i) (x i))) → |
| Tendsto μ atTop (𝓝 (W.prod W)) := by |
| dsimp only |
| let ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let n := fun i => fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i) |
| let θ := fun i => recordMedianSlope ν ℓ hp f (r i) |
| obtain ⟨W,hW,hlim⟩ := shared_path_marginal_limits ν hue e f hef htrans r hr hT.le |
| refine ⟨W,hW,?_⟩ |
| intro x μ hμ |
| let η := fun i => sharedConditionedPairLaw ν ℓ (x i) (x i) |
| have : ∀ i, IsProbabilityMeasure (η i) := fun i => sharedConditionedPairLaw_probability ν ℓ _ _ |
| (ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans _ _)) |
| let X := fun i (P : Path d × Path d) => recordLinearPath ℓ f (θ i) (r i) (n i) T (fun j => P.1 j-x i) |
| let Y := fun i (P : Path d × Path d) => recordLinearPath ℓ f (θ i) (r i) (n i) T (fun j => P.2 j-x i) |
| have hX (i : ℕ) : Measurable (X i) := (measurable_recordLinearPath _ _ _ _ _ _).comp (by fun_prop) |
| have hY (i : ℕ) : Measurable (Y i) := (measurable_recordLinearPath _ _ _ _ _ _).comp (by fun_prop) |
| obtain ⟨hf,hg⟩ := hlim x x (fun _ => rfl) |
| have hMf : Tendsto (fun i => (μ i).map Prod.fst) atTop (𝓝 W) := by |
| have hh := probability_law_tendsto η X (fun i => (hX i).aemeasurable) W hf |
| apply hh.congr' |
| filter_upwards [] with i |
| symm |
| apply Subtype.ext |
| change (μ i : Measure RealPathPair).map Prod.fst=_ |
| rw [hμ i,Measure.map_map measurable_fst (measurable_sharedLinearPairPath _ _ _ _ _ _ _ _)] |
| rfl |
| have hMg : Tendsto (fun i => (μ i).map Prod.snd) atTop (𝓝 W) := by |
| have hh := probability_law_tendsto η Y (fun i => (hY i).aemeasurable) W hg |
| apply hh.congr' |
| filter_upwards [] with i |
| symm |
| apply Subtype.ext |
| change (μ i : Measure RealPathPair).map Prod.snd=_ |
| rw [hμ i,Measure.map_map measurable_snd (measurable_sharedLinearPairPath _ _ _ _ _ _ _ _)] |
| rfl |
| apply weak_limit_of_tight_identification μ (W.prod W) (pair_weak_tight μ W W hMf hMg) |
| intro j hj V hV |
| have hVf : (V : Measure RealPathPair).map Prod.fst=W := by |
| have he := tendsto_nhds_unique |
| ((ProbabilityMeasure.continuous_map continuous_fst).tendsto V |>.comp hV) (hMf.comp hj) |
| exact congrArg (fun M : ProbabilityMeasure C(unitInterval,ℝ) => (M : Measure C(unitInterval,ℝ))) he |
| have hVg : (V : Measure RealPathPair).map Prod.snd=W := by |
| have he := tendsto_nhds_unique |
| ((ProbabilityMeasure.continuous_map continuous_snd).tendsto V |>.comp hV) (hMg.comp hj) |
| exact congrArg (fun M : ProbabilityMeasure C(unitInterval,ℝ) => (M : Measure C(unitInterval,ℝ))) he |
| apply Subtype.ext |
| exact shared_limit_product ν hue e f hef htrans (r ∘ j) (hr.comp hj) T hT (x ∘ j) |
| (μ ∘ j) V hV W hW hVf hVg (fun i => hμ (j i)) |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology BoundedContinuousFunction |
| namespace DirectionalTransience |
|
|
| lemma medianFirstHitPath_sub_linear {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (hp : annealedLaw ν (NoDrop ℓ 0) ≠ 0) (f : Direction d) |
| (x : Lattice d) (θ r n T : ℝ) (X : Path d) : |
| medianFirstHitPath ν ℓ hp f x r n T X- |
| recordLinearPath ℓ f θ r n T (fun j => X j-x) = |
| heightPolygon (fun h => (h:ℝ)*θ-(recordMedian ν ℓ hp f h:ℝ)) r n T := by |
| exact heightPolygon_change_center _ _ _ _ _ _ |
|
|
| lemma linear_add_median_error {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (hp : annealedLaw ν (NoDrop ℓ 0) ≠ 0) (f : Direction d) |
| (x : Lattice d) (θ r n T : ℝ) (X : Path d) : |
| recordLinearPath ℓ f θ r n T (fun j => X j-x)+ |
| heightPolygon (fun h => (h:ℝ)*θ-(recordMedian ν ℓ hp f h:ℝ)) r n T = |
| medianFirstHitPath ν ℓ hp f x r n T X := by |
| rw [← medianFirstHitPath_sub_linear ν ℓ hp f x θ r n T X] |
| abel |
|
|
| lemma median_center_error_tendsto {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 ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let n := fun i => fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i) |
| let θ := fun i => recordMedianSlope ν ℓ hp f (r i) |
| Tendsto (fun i => heightPolygon (fun h => (h:ℝ)*θ i-(recordMedian ν ℓ hp f h:ℝ)) |
| (r i) (n i) T) atTop (𝓝 0) := by |
| dsimp only |
| apply heightPolygon_tendsto_zero_of_grid _ _ _ |
| (recordFluctuationScale_tendsto ν hue e f hef htrans r hr.1) T hT |
| intro ε hε |
| filter_upwards [recordMedian_linearization ν hue e f hef htrans r hr (T+1) hε] with i hi |
| intro j hj |
| simpa only [abs_div,abs_sub_comm] using hi j hj |
|
|
| theorem shared_median_pair_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 ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let n := fun i => fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i) |
| ∃ W : ProbabilityMeasure C(unitInterval,ℝ), |
| (∀ I : Finset unitInterval, (W : Measure C(unitInterval,ℝ)).map |
| (fun g : C(unitInterval,ℝ) => I.restrict g)=gaussianPathFiniteLaw (T/(2*commonMeanWidth ν ℓ)) I) ∧ |
| ∀ x : ℕ → Lattice d, |
| ∀ μ : ℕ → ProbabilityMeasure RealPathPair, |
| (∀ i, (μ i : Measure RealPathPair)=(sharedConditionedPairLaw ν ℓ (x i) (x i)).map |
| (fun P : Path d × Path d => (medianFirstHitPath ν ℓ hp f (x i) (r i) (n i) T P.1, |
| medianFirstHitPath ν ℓ hp f (x i) (r i) (n i) T P.2))) → |
| Tendsto μ atTop (𝓝 (W.prod W)) := by |
| dsimp only |
| let ℓ := realPosition (step e) |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) |
| let n := fun i => fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) (r i) |
| let θ := fun i => recordMedianSlope ν ℓ hp f (r i) |
| obtain ⟨W,hW,hlim⟩ := shared_pair_path_limit ν hue e f hef htrans r hr T hT |
| refine ⟨W,hW,?_⟩ |
| intro x μ hμ |
| let η := fun i => sharedConditionedPairLaw ν ℓ (x i) (x i) |
| have : ∀ i, IsProbabilityMeasure (η i) := fun i => sharedConditionedPairLaw_probability ν ℓ _ _ |
| (ne_of_gt (sharedNoDropMass_pos ν hue ℓ (signed_direction_unit e) htrans _ _)) |
| let L := fun i => sharedLinearPairPath ℓ f (θ i) (r i) (n i) T (x i) (x i) |
| have hL (i : ℕ) : Measurable (L i) := measurable_sharedLinearPairPath _ _ _ _ _ _ _ _ |
| let V : ℕ → ProbabilityMeasure RealPathPair := fun i => ⟨(η i).map (L i), |
| (Measure.isProbabilityMeasure_map_iff (hL i).aemeasurable).mpr inferInstance⟩ |
| have hV : Tendsto V atTop (𝓝 (W.prod W)) := hlim x V (fun i => rfl) |
| let a := fun i => heightPolygon (fun h => (h:ℝ)*θ i-(recordMedian ν ℓ hp f h:ℝ)) (r i) (n i) T |
| have ha : Tendsto a atTop (𝓝 0) := median_center_error_tendsto ν hue e f hef htrans r hr hT.le |
| let addPair : RealPathPair × C(unitInterval,ℝ) → RealPathPair := fun z => (z.1.1+z.2,z.1.2+z.2) |
| have hadd : Continuous addPair := by fun_prop |
| have hh := weak_map_varying_parameter V (W.prod W) hV a 0 ha addPair hadd |
| have he0 : (W.prod W).map (fun pair => addPair (pair,0))=W.prod W := by |
| apply Subtype.ext |
| change (↑(W.prod W) : Measure RealPathPair).map (fun P => addPair (P,0))=_ |
| simp only [addPair,add_zero,Prod.eta] |
| exact Measure.map_id |
| rw [he0] at hh |
| apply hh.congr' |
| filter_upwards [] with i |
| apply Subtype.ext |
| change ((η i).map (L i)).map (fun P => addPair (P,a i))=(μ i : Measure RealPathPair) |
| rw [Measure.map_map (by fun_prop) (hL i),hμ i] |
| congr 1 |
| funext P |
| apply Prod.ext |
| · exact linear_add_median_error ν ℓ hp f (x i) (θ i) (r i) (n i) T P.1 |
| · exact linear_add_median_error ν ℓ hp f (x i) (θ i) (r i) (n i) T P.2 |
| |
| end DirectionalTransience |
| |
| end |
| |
| end |
| |
| end OAI |
| |
