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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 | 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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