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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import OAI.Probability.FreeIsing.Brownian.CoefficientApproximationIdentification
import OAI.Probability.FreeIsing.Brownian.IntegralTransport
import OAI.Probability.FreeIsing.Ising.CanonicalCovariancePullback
import OAI.Probability.FreeIsing.Ising.CoordinateCompletion
import OAI.Probability.FreeIsing.Ising.ObservationIndependence
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367.IsingInnovation
variable {d : ℕ} {A : Type*} [MeasurableSpace A]
theorem finiteCovariance_tendsto_version_timeMeanSquare
{beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(ν : Measure A) {B : ℝ≥0 → A → ℝ} (hB : IsPreBrownianReal B ν)
(hmB : ∀ t, Measurable (B t)) (hcB : ∀ a, Continuous (fun t => B t a))
(v u : TreeVertex d) (T : ℝ≥0)
{H : ℝ≥0 → State d A → ℝ}
(hver : ∀ t, H t =ᵐ[μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))]
covariance beta B v u t) :
Tendsto (fun n => ∫ t in (0 : ℝ)..(T : ℝ), ∫ q,
(finiteCovariance beta B v u n t.toNNReal q - H t.toNNReal q) ^ 2
∂μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))) atTop (𝓝 0) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let : IsProbabilityMeasure ν := hB.isGaussianProcess.isProbabilityMeasure
let P := μ.prod (Measure.infinitePi (fun _ : TreeVertex d => ν))
let τ := volume.restrict (Set.Ioc (0 : ℝ) (T : ℝ))
have heq (n : ℕ) :
(∫ t in (0 : ℝ)..(T : ℝ), ∫ q,
(finiteCovariance beta B v u n t.toNNReal q - H t.toNNReal q) ^ 2 ∂P) =
∫ p : ℝ × State d A,
(finiteCovariance beta B v u n p.1.toNNReal p.2 -
covariance beta B v u p.1.toNNReal p.2) ^ 2 ∂τ.prod P := by
calc
_ = ∫ t in (0 : ℝ)..(T : ℝ), ∫ q,
(finiteCovariance beta B v u n t.toNNReal q -
covariance beta B v u t.toNNReal q) ^ 2 ∂P := by
apply intervalIntegral.integral_congr
intro t ht
apply integral_congr_ae
filter_upwards [hver t.toNNReal] with q hq
rw [hq]
_ = _ := interval_integral_eq_timeSpace T
((memLp_finiteCovariance_time beta B hmB hcB P τ v u n).sub
(memLp_covariance_time beta B hmB hcB P τ v u)).integrable_sq
exact (finiteCovariance_tendsto_timeMeanSquare hμ ν hB hmB hcB v u T).congr'
(Eventually.of_forall fun n => (heq n).symm)
end Problem367.IsingInnovation
namespace Problem367.IsingObservationFactor
open IsingObservationInnovation
theorem canonical_covariance_coordinate_integral_eq_pullback
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ}
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(u : TreeVertex d) (T : ℝ≥0)
{X : ℕ → IsingInnovation.State d ℝ → ℝ}
{I : IsingInnovation.State d ℝ → ℝ}
(hX : ∀ n, MemLp (X n) 2 (μ.prod (labelMeasure d)))
(hI : MemLp I 2 (μ.prod (labelMeasure d)))
(hXlim : ∀ n, Tendsto (fun k => ∫ q,
(BrownianGrid.uniformSum
(IsingInnovation.innovation beta (fun t r => WienerSampler.sample r t) u)
(IsingInnovation.finiteCovariance beta (fun t r => WienerSampler.sample r t)
(ConnectedHull.root d) u n) T k q - X n q) ^ 2
∂μ.prod (labelMeasure d)) atTop (𝓝 0))
(hXI : Tendsto (fun n => ∫ q, (X n q - I q) ^ 2
∂μ.prod (labelMeasure d)) atTop (𝓝 0))
{J : Paths d → ℝ} (hJ : MemLp J 2 (observationLaw μ))
(hJlim : Tendsto (fun k => ∫ x,
(BrownianGrid.uniformSum (fun t x => innovationMap beta x u t)
(continuousRootCovariance hb u) T k x - J x) ^ 2
∂observationLaw μ) atTop (𝓝 0)) :
I =ᵐ[μ.prod (labelMeasure d)] J ∘ uniformObservation := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
let B : ℝ≥0 → ℝ → ℝ := fun t r => WienerSampler.sample r t
have hmB (t : ℝ≥0) : Measurable (B t) :=
(continuous_eval_const t).measurable.comp WienerSampler.measurable_sample
have hcB (r : ℝ) : Continuous (fun t => B t r) :=
(WienerSampler.sample r).continuous
have hB : IsPreBrownianReal B uniform01 :=
WienerSampler.sample_isBrownianReal.toIsPreBrownianReal
have hφ : MeasurePreserving (uniformObservation (d := d))
(μ.prod (labelMeasure d)) (observationLaw μ) :=
⟨measurable_uniformObservation, rfl⟩
let H : ℝ≥0 → IsingInnovation.State d ℝ → ℝ :=
fun t q => continuousRootCovariance hb u t (uniformObservation q)
have hH := continuousRootCovariance_uniformObservation_stronglyAdapted hmB hb u
have hcH := continuousRootCovariance_uniformObservation_continuous_ae hd hμ hb hcrit u
have hbH : ∀ᵐ q ∂μ.prod (labelMeasure d), ∀ t, |H t q| ≤ (1 : ℝ) :=
Eventually.of_forall fun q t =>
continuousRootCovariance_uniformObservation_abs_le_one hb u t q
have hver := continuousRootCovariance_uniformObservation_eq_covariance_ae hd hμ hb hcrit u
have hC2 (t : ℝ≥0) : MemLp (continuousRootCovariance hb u t) 2
(observationLaw μ) := by
apply MemLp.of_bound
(((continuousRootCovariance_progressive hb u).stronglyAdapted t).mono
((filtration d).le t)).aestronglyMeasurable 1
exact Eventually.of_forall fun x => by
simpa only [Real.norm_eq_abs] using continuousRootCovariance_abs_le_one hb u t x
have hi (s t : ℝ≥0) (hst : s ≤ t) :=
BrownianArrayFiltration.coordinate_increment_indep_past
(B := fun v t x => innovationMap beta x v t) (F := filtration d)
(s := s) (t := t)
(innovationMap_increment_indep_filtration hμ s t hst) u
have hS (k : ℕ) := BrownianFiltrationGrid.uniformSum_memLp
(innovationMap_isBrownianReal hμ u).toIsPreBrownianReal
(innovationMap_stronglyAdapted beta u) hi
(continuousRootCovariance hb u)
(continuousRootCovariance_progressive hb u).stronglyAdapted hC2 T k
have hJpull := BrownianGrid.meanSquare_limit_comp hφ
(fun k => (hS k).aestronglyMeasurable) hJ.aestronglyMeasurable hJlim
have hcoeff := IsingInnovation.finiteCovariance_tendsto_version_timeMeanSquare
hμ uniform01 hB hmB hcB (ConnectedHull.root d) u T hver
apply BrownianGrid.ae_eq_of_mean_square_limits hX hI
(hJ.comp_measurePreserving hφ) hXI
exact BrownianFiltrationGrid.tendsto_meanSquare_of_coefficient_interval
(IsingInnovation.isBrownianReal_innovation hμ uniform01 hB hmB hcB u).toIsPreBrownianReal
(IsingInnovation.innovation_martingale hμ uniform01 hB hmB hcB u).stronglyAdapted
(fun s t hst => BrownianArrayFiltration.coordinate_increment_indep_past
(IsingInnovation.innovation_increment_indep_filtration hμ uniform01 hB hmB hcB s t hst) u)
(fun n => IsingInnovation.finiteCovariance beta B (ConnectedHull.root d) u n) H
(fun n => IsingInnovation.finiteCovariance_stronglyAdapted beta B hmB
(ConnectedHull.root d) u n) hH
(fun n => Eventually.of_forall
(IsingInnovation.finiteCovariance_continuous beta B hcB (ConnectedHull.root d) u n)) hcH
(fun _ => 2) 1 (fun _ => by norm_num) (by norm_num)
(fun n => Eventually.of_forall fun q t =>
IsingInnovation.finiteCovariance_abs_le_two beta B (ConnectedHull.root d) u n t q) hbH
T hX (hJ.comp_measurePreserving hφ) hXlim hJpull hcoeff
end Problem367.IsingObservationFactor
end
end OAI
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