| import OAI.Probability.FreeIsing.Brownian.IntegralHorizon |
| import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse |
| import OAI.Probability.FreeIsing.Ising.ObservationIndependence |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology |
|
|
| namespace Problem367.IsingObservationFactor |
| open IsingObservationInnovation |
|
|
| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} |
|
|
| theorem exists_canonical_root_covariance_integrals |
| (hd : 3 ≤ d) (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) : |
| ∃ J : TreeVertex d → ℝ≥0 → Paths d → ℝ, |
| (∀ w, IsStronglyPredictable (filtration d) (J w)) ∧ |
| (∀ w, Martingale (J w) (filtration d) (observationLaw μ)) ∧ |
| (∀ w, ∀ᵐ x ∂observationLaw μ, Continuous (fun t => J w t x)) ∧ |
| (∀ w t, MemLp (J w t) 2 (observationLaw μ)) ∧ |
| (∀ w, J w 0 =ᵐ[observationLaw μ] 0) ∧ |
| (∀ w t, (∫ x, J w t x ∂observationLaw μ) = 0) ∧ |
| (∀ w t, (∫ x, J w t x ^ 2 ∂observationLaw μ) = |
| ∫ s in (0 : ℝ)..(t : ℝ), ∫ x, |
| continuousRootCovariance hb w s.toNNReal x ^ 2 ∂observationLaw μ) ∧ |
| (∀ w (N : ℕ), 0 < N → ∀ t ≤ (N : ℝ≥0), |
| Tendsto (fun n => ∫ x, |
| (BrownianIntegralProcess.stoppedUniformSum |
| (fun s x => innovationMap beta x w s) |
| (continuousRootCovariance hb w) N n t x - J w t x) ^ 2 |
| ∂observationLaw μ) atTop (𝓝 0)) := by |
| have hex (w : TreeVertex d) := |
| BrownianIntegralProcess.exists_global_continuous_uniform_integral |
| (innovationMap_isBrownianReal hμ w).toIsPreBrownianReal |
| (innovationMap_stronglyAdapted beta w) |
| (fun s t hst => BrownianArrayFiltration.coordinate_increment_indep_past |
| (B := fun w s x => innovationMap beta x w s) |
| (F := filtration d) |
| (innovationMap_increment_indep_filtration hμ s t hst) w) |
| (innovationMap_isBrownianReal hμ w).cont |
| (continuousRootCovariance_progressive hb w).stronglyAdapted |
| (continuousRootCovariance_continuous_ae hd hμ hb hcrit w) |
| (show (0 : ℝ) ≤ 1 by norm_num) |
| (Eventually.of_forall (fun x t => continuousRootCovariance_abs_le_one hb w t x)) |
| choose J hJp hJM hJc hJ2 hJ0 hJmean hJenergy hJlim using hex |
| exact ⟨J, hJp, hJM, hJc, hJ2, hJ0, hJmean, hJenergy, hJlim⟩ |
|
|
| end Problem367.IsingObservationFactor |
|
|
| end |
|
|
| end OAI |
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|
