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f61e547 | 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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Probability.FreeIsing.Conditioning.ContinuousCopyIndependence
import OAI.Probability.FreeIsing.Ising.ObservationComparison
import OAI.Probability.FreeIsing.Ising.ObservationIndependence
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367.IsingCopyBrownian
open IsingObservationInnovation IsingObservationFactor
open ContinuousObservationHistory ContinuousNoiseSplit
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
theorem graph_future_indep
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (s : ℝ≥0) :
IndepFun (fun q : Paths d × Paths d =>
(stoppedArray s q.1, stoppedArray s q.2))
(fun q => futureArray s q.1)
(((observationLaw μ).map (innovationMap beta)) ⊗ₘ kernel beta μ) := by
rw [kernel_graphLaw]
unfold IsingConditionalCopyBounds.graphLaw
apply ConstructedConditionalCopies.indepFun_map_of_comp
((measurable_innovationMap beta).prodMk measurable_id)
(((measurable_stoppedArray s).comp measurable_fst).prodMk
((measurable_stoppedArray s).comp measurable_snd))
((measurable_futureArray s).comp measurable_fst)
exact ConstructedContinuousCopies.indepFun_continuous_future
(observationLaw μ) (filtration d) (innovationMap beta) id
(fun v t => (innovationMap_stronglyAdapted beta v t).measurable)
(fun v t => (arrayNaturalFiltration_stronglyAdapted
evaluation measurable_evaluation v t).measurable)
(innovationMap_increment_indep_filtration hμ) s
theorem future_indep_joinedFiltration
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (s : ℝ≥0) :
Indep (MeasurableSpace.comap
(fun q : Paths d × (Paths d × Paths d) => futureArray s q.1) inferInstance)
(ContinuousCopyBrownian.joinedFiltration s) (copies beta μ) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) :=
inferInstance
exact ContinuousCopyBrownian.indep_future_joinedFiltration
((observationLaw μ).map (innovationMap beta)) (kernel beta μ) s
(graph_future_indep hμ s)
theorem increment_indep_joinedFiltration
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) {s t : ℝ≥0} (hst : s ≤ t) :
Indep (MeasurableSpace.comap
(fun q : Paths d × (Paths d × Paths d) => fun v => q.1 v t - q.1 v s)
inferInstance) (ContinuousCopyBrownian.joinedFiltration s) (copies beta μ) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) :=
inferInstance
exact ContinuousCopyBrownian.indep_increment_joinedFiltration
((observationLaw μ).map (innovationMap beta)) (kernel beta μ) hst
(graph_future_indep hμ s)
theorem commonNoise_stronglyAdapted (v : TreeVertex d) :
StronglyAdapted ContinuousCopyBrownian.joinedFiltration
(fun t (q : Paths d × (Paths d × Paths d)) => q.1 v t) := by
intro t
exact ((measurable_pi_apply v).comp
(ContinuousCopyBrownian.measurable_commonNoise_joinedFiltration t)).stronglyMeasurable
omit [IsProbabilityMeasure μ] in
private theorem innovationLaw_coordinate_preBrownian [_hprob : IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (v : TreeVertex d) :
IsPreBrownianReal (fun t (w : Paths d) => w v t)
((observationLaw μ).map (innovationMap beta)) := by
rw [map_innovationMap_observationLaw hμ]
exact ⟨fun I => (WienerSampler.coordinate_isBrownianReal.hasLaw I).comp
(measurePreserving_eval_infinitePi
(fun _ : TreeVertex d => WienerSampler.wienerMeasure) v).hasLaw⟩
theorem commonNoise_isBrownianReal
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (v : TreeVertex d) :
IsBrownianReal (fun t (q : Paths d × (Paths d × Paths d)) => q.1 v t)
(copies beta μ) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) :=
inferInstance
exact ContinuousCopyBrownian.isBrownian_commonNoise v
((observationLaw μ).map (innovationMap beta))
(innovationLaw_coordinate_preBrownian hμ v) (kernel beta μ)
theorem commonNoise_martingale
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (v : TreeVertex d) :
Martingale (fun t (q : Paths d × (Paths d × Paths d)) => q.1 v t)
ContinuousCopyBrownian.joinedFiltration (copies beta μ) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) :=
inferInstance
exact ContinuousCopyBrownian.commonNoise_martingale v
((observationLaw μ).map (innovationMap beta))
(innovationLaw_coordinate_preBrownian hμ v) (kernel beta μ)
(graph_future_indep hμ)
theorem commonNoise_increments_indep
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (s t : ℝ≥0)
{v w : TreeVertex d} (hvw : v ≠ w) :
IndepFun (fun q : Paths d × (Paths d × Paths d) => q.1 v t - q.1 v s)
(fun q => q.1 w t - q.1 w s) (copies beta μ) := by
have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) :=
inferInstance
have hp : MeasurePreserving (Prod.fst : Paths d × (Paths d × Paths d) → Paths d)
(copies beta μ) ((observationLaw μ).map (innovationMap beta)) :=
⟨measurable_fst, Measure.fst_compProd _ _⟩
apply indepFun_comp_measurePreserving hp
((measurable_evaluation v t).sub (measurable_evaluation v s))
((measurable_evaluation w t).sub (measurable_evaluation w s))
rw [map_innovationMap_observationLaw hμ]
exact (iIndepFun_infinitePi (P := fun _ : TreeVertex d => WienerSampler.wienerMeasure)
(fun _ => (continuous_eval_const t).measurable.sub
(continuous_eval_const s).measurable)).indepFun hvw
end Problem367.IsingCopyBrownian
end
end OAI
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