| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.InteriorPairMeasureExt |
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| namespace OAI |
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| section |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology BoundedContinuousFunction |
| namespace DirectionalTransience |
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| lemma probability_prod_initial_zero (μ ν : Measure C(unitInterval,ℝ)) |
| (hμ : ∀ᵐ P ∂μ,P 0=0) (hν : ∀ᵐ P ∂ν,P 0=0) [SFinite ν] : |
| ∀ᵐ P : RealPathPair ∂μ.prod ν,P.1 0=0 ∧ P.2 0=0 := by |
| apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr |
| filter_upwards [hμ] with P hP |
| filter_upwards [hν] with Q hQ |
| exact ⟨hP,hQ⟩ |
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| lemma normalJointPast_product_identification (μ : Measure RealPathPair) [IsProbabilityMeasure μ] |
| (c : ℝ≥0) (h : NormalJointPast μ c) {A K : ℝ} (hA : 0 < A) (hK : 0 ≤ K) |
| (hcross : SeparatedCrossBound μ A K) |
| (hz : (μ.prod volume) {z : RealPathPair × unitInterval | z.1.1 z.2-z.1.2 z.2=0}=0) |
| (hzero : ∀ᵐ P ∂μ,P.1 0=0 ∧ P.2 0=0) : |
| μ=(μ.map Prod.fst).prod (μ.map Prod.snd) := by |
| let : IsProbabilityMeasure (μ.map Prod.fst) := inferInstance |
| let : IsProbabilityMeasure (μ.map Prod.snd) := inferInstance |
| have hf : NormalSinglePast (μ.map Prod.fst) c := normalJointPast_single μ c h false |
| have hg : NormalSinglePast (μ.map Prod.snd) c := normalJointPast_single μ c h true |
| have h0f : ∀ᵐ P ∂μ.map Prod.fst,P 0=0 := by |
| rw [ae_map_iff measurable_fst.aemeasurable ((isClosed_eq (continuous_eval_const 0) continuous_const).measurableSet)] |
| exact hzero.mono fun P hP => hP.1 |
| have h0g : ∀ᵐ P ∂μ.map Prod.snd,P 0=0 := by |
| rw [ae_map_iff measurable_snd.aemeasurable ((isClosed_eq (continuous_eval_const 0) continuous_const).measurableSet)] |
| exact hzero.mono fun P hP => hP.2 |
| have hfull := fullNormalJointPast_of_cross μ c h hA hK hcross hz |
| have hprod := normalSinglePast_prod (μ.map Prod.fst) (μ.map Prod.snd) c hf hg |
| apply interior_pair_measure_ext |
| intro I hI |
| exact fullNormalJointPast_finite_uniqueness μ _ c hfull hprod hzero |
| (probability_prod_initial_zero _ _ h0f h0g) I hI |
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| lemma wiener_path_initial_zero (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) : |
| ∀ᵐ P ∂(W : Measure C(unitInterval,ℝ)),P 0=0 := by |
| have he := wiener_path_eval W T hW 0 |
| have he0 : ((0:unitInterval):ℝ)=0 := rfl |
| simp only [he0,mul_zero,Real.toNNReal_zero,gaussianReal_zero_var] at he |
| have hh : ∀ᵐ x ∂(W : Measure C(unitInterval,ℝ)).map (fun P => P 0),x=0 := by |
| rw [he] |
| simp |
| exact (ae_map_iff (continuous_eval_const 0).measurable.aemeasurable (by measurability)).mp hh |
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| end DirectionalTransience |
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| end |
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| end |
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| end OAI |
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