| import Mathlib |
| import OAI.Analysis.HardSphere.Bridges.SupplyIntegral |
|
|
| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Bridge |
| open Wiener GaussianPacking PathCutoff |
| attribute [local irreducible] torusDistance |
|
|
| lemma trialPath_all_continuous (t : ℝ) : |
| Continuous (fun p : (Point×(Point×Point))×(Noise×Noise) ↦ |
| trialPath t p.1.1 p.1.2.1 p.1.2.2 p.2) := by |
| apply ContinuousMap.continuous_of_continuous_uncurry |
| change Continuous (fun p : ((Point×(Point×Point))×(Noise×Noise))×Time ↦ fun j ↦ |
| (mean p.1.1.1 p.1.1.2.2 (leftTime p.2) j+Real.sqrt (2*t)*center (p.1.2.1 j) (leftTime p.2))+ |
| (mean p.1.1.2.1 p.1.1.2.2 (reverseTime (rightTime p.2)) j+ |
| Real.sqrt (2*t)*center (p.1.2.2 j) (reverseTime (rightTime p.2)))-p.1.1.2.2 j) |
| unfold mean |
| have hl := leftTime_continuous |
| have hr := reverseTime.continuous.comp rightTime_continuous |
| have hc := center_continuous |
| fun_prop |
|
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| def exposedTrialKernel (t : ℝ) : Kernel (Point×(Point×Point)) C(Time,Point) := |
| ((Kernel.deterministic id measurable_id).prod |
| (Kernel.const (Point×(Point×Point)) (law.prod law))).map |
| (fun p ↦ trialPath t p.1.1 p.1.2.1 p.1.2.2 p.2) |
|
|
| instance exposedTrialKernel_markov (t : ℝ) : IsMarkovKernel (exposedTrialKernel t) := by |
| unfold exposedTrialKernel |
| exact Kernel.IsMarkovKernel.map _ (trialPath_all_continuous t).measurable |
|
|
| lemma exposedTrialKernel_apply (t : ℝ) (y z u : Point) : |
| exposedTrialKernel t (y,z,u)=trialKernel t y z u := by |
| rw [exposedTrialKernel,Kernel.map_apply _ (trialPath_all_continuous t).measurable, |
| Kernel.prod_apply,Kernel.deterministic_apply,Kernel.const_apply,Measure.dirac_prod, |
| Measure.map_map (trialPath_all_continuous t).measurable |
| (measurable_const.prodMk measurable_id : Measurable (Prod.mk (id (y,z,u)))), |
| trialKernel_apply] |
| rfl |
|
|
| lemma trialCollision_joint_closed {L : Length} (r : ℝ) : |
| IsClosed {p : ConfigurationPath 1 L×C(Time,Point) | TrialCollision r p.1 p.2} := by |
| let E : Set ((ConfigurationPath 1 L×C(Time,Point))×Time) := |
| {p | torusDistance L (projectPoint L (p.1.2 p.2)) (p.1.1 p.2 0) ≤ r} |
| have hE : IsClosed E := by |
| apply isClosed_le _ continuous_const |
| apply (torusDistance_continuous_pair L).comp (f := fun p : (ConfigurationPath 1 L×C(Time,Point))×Time ↦ |
| (projectPoint L (p.1.2 p.2),p.1.1 p.2 0)) |
| apply Continuous.prodMk |
| · change Continuous (fun p : (ConfigurationPath 1 L×C(Time,Point))×Time ↦ |
| fun j : Fin 3 ↦ (p.1.2 p.2 j:AddCircle L.val)) |
| fun_prop |
| · fun_prop |
| convert isClosedMap_fst_of_compactSpace E hE using 1 |
| ext p |
| simp only [TrialCollision,mem_ofPred_eq,mem_image,E] |
| exact ⟨fun ⟨s,hs⟩ ↦ ⟨(p,s),hs,rfl⟩,fun ⟨⟨q,s⟩,hs,hq⟩ ↦ by |
| change q=p at hq |
| subst q |
| exact ⟨s,hs⟩⟩ |
|
|
| lemma successExcept_joint_measurable {L : Length} {N : ℕ} (n : ℕ) |
| (r R C S : ℝ) (w : Time) (v : Point) (i : Fin N) : |
| MeasurableSet {p : (Fin N → ConfigurationPath 1 L)×C(Time,Point) | |
| p.2∈SuccessExcept n r R C S w v p.1 i} := by |
| have hm (j : Fin N) : Measurable (fun p : (Fin N → ConfigurationPath 1 L)×C(Time,Point) ↦ |
| (p.1 j,p.2)) := ((measurable_pi_apply j).comp measurable_fst).prodMk measurable_snd |
| have hh : MeasurableSet {p : (Fin N → ConfigurationPath 1 L)×C(Time,Point) | |
| ∀j,j≠i→¬TrialCollision r (p.1 j) p.2} := by |
| simp only [ofPred_forall] |
| exact MeasurableSet.iInter fun j ↦ MeasurableSet.iInter fun _ ↦ |
| ((trialCollision_joint_closed r).measurableSet.preimage (hm j)).compl |
| exact ((trialCutoff_measurable L n r R C w S v).preimage measurable_snd).inter hh |
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| end HardSphere.Bridge |
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| end |
| end |
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| end |
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| end OAI |
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