| import OAI.Analysis.CircleDomains.Probability.MeanVisitDensity | |
| import OAI.Analysis.CircleDomains.Probability.CodeHitMeasures | |
| namespace OAI | |
| /-! Forgetting the optional tags produces the actual path law. Its full | |
| mean visit function differs from the preterminal function by exactly one | |
| unit at the prescribed terminal point. -/ | |
| open Set MeasureTheory | |
| open scoped ENNReal Classical | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| variable {X Y : Type*} [MetricSpace X] [MetricSpace Y] | |
| [MeasurableSpace X] [MeasurableSpace C(PathTime, X)] | |
| def codePath (d : PathCodeData X Y) (Γ : BasePathCode d) : C(PathTime, X) := Γ.val.1 | |
| theorem measurable_codePath (d : PathCodeData X Y) : Measurable (codePath d) := | |
| measurable_fst.comp measurable_subtype_coe | |
| def codePathLaw (d : PathCodeData X Y) (P : ProbabilityMeasure (BasePathCode d)) : | |
| ProbabilityMeasure C(PathTime, X) := P.map (codePath d) | |
| variable [BorelSpace X] [BorelSpace C(PathTime, X)] [SecondCountableTopology X] | |
| omit [BorelSpace X] [SecondCountableTopology X] in | |
| theorem pathHitProbability_codePathLaw (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) (x : X) : | |
| pathHitProbability (codePathLaw d P) x = codeHitProbability d P x := by | |
| exact Measure.map_apply (measurable_codePath d) (isClosed_paths_hitting x).measurableSet | |
| theorem meanVisitCount_codePathLaw (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) (x : X) : | |
| meanVisitCount (codePathLaw d P) x = | |
| meanCodePreVisits d P x + if d.target = x then 1 else 0 := by | |
| classical | |
| have hv : Measurable (fun γ : C(PathTime, X) => visitCount γ x) := | |
| measurable_visitCount.comp (measurable_id.prodMk measurable_const) | |
| unfold meanVisitCount codePathLaw | |
| rw [ProbabilityMeasure.toMeasure_map, lintegral_map hv (measurable_codePath d)] | |
| have he : ∀ Γ : BasePathCode d, visitCount (codePath d Γ) x = | |
| preterminalVisitCount Γ.val.1 x + if d.target = x then 1 else 0 := by | |
| intro Γ | |
| rw [visitCount_eq_preterminal_add_terminal] | |
| change preterminalVisitCount Γ.val.1 x + terminalVisitCount Γ.val.1 x = _ | |
| rw [terminalVisitCount, Γ.property.2.1.2.1] | |
| simp_rw [he] | |
| have hm : Measurable (fun Γ : BasePathCode d => preterminalVisitCount Γ.val.1 x) := | |
| measurable_preterminalVisitCount.comp | |
| ((measurable_fst.comp measurable_subtype_coe).prodMk measurable_const) | |
| rw [lintegral_add_left hm] | |
| simp only [lintegral_const, measure_univ, mul_one, meanCodePreVisits] | |
| theorem codeHitProbability_le_meanCodeVisits (d : PathCodeData X Y) | |
| (P : ProbabilityMeasure (BasePathCode d)) (x : X) : | |
| codeHitProbability d P x ≤ meanCodePreVisits d P x + if d.target = x then 1 else 0 := by | |
| rw [← meanVisitCount_codePathLaw, ← pathHitProbability_codePathLaw] | |
| exact pathHitProbability_le_meanVisitCount (codePathLaw d P) x | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |