File size: 2,877 Bytes
9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 | 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
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