| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.CurvePolicy |
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| namespace OAI |
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| section |
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| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Classical Topology BigOperators |
| namespace DirectionalTransience |
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| lemma annealed_crossing_translation {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (x : Lattice d) (H : ℝ) : |
| (∫⁻ ω, crossingQuenched ℓ x H ω ∂environmentLaw ν)=annealedLaw ν (Cross ℓ 0 H) := by |
| simp_rw [crossingQuenched_translation ℓ x] |
| rw [← lintegral_map (measurable_crossingQuenched ℓ 0 H) (by fun_prop),environment_translation, |
| annealed_apply ν (measurableSet_cross ℓ 0 H)] |
| rfl |
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| lemma annealed_record_cylinder_cross {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (f : Path d) (n : ℕ) (hn : 0<n) (hr : StrictRecord ℓ f n) |
| (H : ℝ) (hH : 0<H) : |
| annealedLaw ν ((fun X : Path d => fun j => X (n+j)) ⁻¹' Cross ℓ (f n) H ∩ pathCylinder f n)= |
| annealedLaw ν (pathCylinder f n)*annealedLaw ν (Cross ℓ 0 H) := by |
| by_cases hf : f 0=0 |
| · let S : Set (Lattice d) := {y | ∃ j<n, y=f j} |
| let T := Strip ℓ (f n) H |
| have hST : Disjoint S T := by |
| apply Set.disjoint_left.mpr |
| rintro y ⟨j,hj,rfl⟩ hy |
| exact (not_le_of_gt (hr j hj)) hy.1 |
| have hS0 : (0 : Lattice d) ∈ S := ⟨0,hn,hf.symm⟩ |
| have hm : @Measurable _ _ (rowSigma S) _ |
| (fun ω : Environment d => quenchedKernel (ω,0) (pathCylinder f n)) := by |
| apply measurable_of_row_locality _ |
| ((Kernel.measurable_coe _ (measurableSet_pathCylinder f n)).comp |
| (measurable_id.prodMk measurable_const)) S 0 hS0 |
| intro ω η hωη |
| exact pathCylinder_weight_locality S ω η hωη 0 f n (fun j hj => ⟨j,hj,rfl⟩) |
| rw [annealed_apply ν ((measurableSet_cross ℓ (f n) H).preimage (by fun_prop) |>.inter |
| (measurableSet_pathCylinder f n))] |
| simp_rw [quenched_prefix_future _ _ _ hf n (measurableSet_cross ℓ (f n) H)] |
| change (∫⁻ ω, quenchedKernel (ω,0) (pathCylinder f n)*crossingQuenched ℓ (f n) H ω |
| ∂environmentLaw ν)=_ |
| rw [lintegral_mul_eq_lintegral_mul_lintegral_of_independent_measurableSpace |
| (rowSigma_le S) (rowSigma_le T) (environment_indep_rows ν hST) hm |
| (measurable_crossingQuenched_rows ℓ (f n) hH),annealed_crossing_translation, |
| ← annealed_apply ν (measurableSet_pathCylinder f n)] |
| · have hzero : annealedLaw ν (pathCylinder f n)=0 := by |
| rw [annealed_apply ν (measurableSet_pathCylinder f n)] |
| simp_rw [quenched_pathCylinder_zero _ _ _ hf n] |
| simp |
| rw [measure_mono_null Set.inter_subset_right hzero,hzero,zero_mul] |
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| lemma annealed_record_event_cross {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (n : ℕ) (hn : 0<n) (A : Set (Path d)) (hA : PrefixDetermined n A) |
| (hr : ∀ X ∈ A, StrictRecord ℓ X n) (H : ℝ) (hH : 0<H) : |
| annealedLaw ν (A ∩ FutureEvent (fun x => Cross ℓ x H) n)= |
| annealedLaw ν A*annealedLaw ν (Cross ℓ 0 H) := by |
| rw [measure_partition_prefix (annealedLaw ν) (A ∩ _) n, |
| measure_partition_prefix (annealedLaw ν) A n,← ENNReal.tsum_mul_right] |
| apply tsum_congr |
| intro f |
| let g := extendPrefix n f |
| by_cases hg : g ∈ A |
| · have he : (A ∩ FutureEvent (fun x => Cross ℓ x H) n) ∩ pathCylinder g n = |
| (fun X : Path d => fun j => X (n+j)) ⁻¹' Cross ℓ (g n) H ∩ pathCylinder g n := by |
| ext X |
| constructor |
| · rintro ⟨⟨_,hc⟩,hX⟩ |
| exact ⟨by simpa only [FutureEvent,Set.mem_ofPred_eq,Set.mem_preimage,hX n le_rfl] using hc,hX⟩ |
| · rintro ⟨hc,hX⟩ |
| exact ⟨⟨(hA X g hX).mpr hg,by simpa only [FutureEvent,Set.mem_ofPred_eq,Set.mem_preimage,hX n le_rfl] using hc⟩,hX⟩ |
| have hAc : A ∩ pathCylinder g n=pathCylinder g n := by |
| rw [prefixDetermined_inter_cylinder A n hA g,ite_eq_left hg] |
| rw [he,hAc] |
| exact annealed_record_cylinder_cross ν ℓ g n hn (hr g hg) H hH |
| · have hzero : A ∩ pathCylinder g n=∅ := by |
| rw [prefixDetermined_inter_cylinder A n hA g,ite_eq_right hg] |
| have he : (A ∩ FutureEvent (fun x => Cross ℓ x H) n) ∩ pathCylinder g n=∅ := by |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| rintro X ⟨⟨hXA,_⟩,hX⟩ |
| exact (Set.eq_empty_iff_forall_notMem.mp hzero) X ⟨hXA,hX⟩ |
| rw [he,hzero,measure_empty,zero_mul] |
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| end DirectionalTransience |
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| end |
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| end OAI |
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