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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 | import OAI.Probability.FreeIsing.Estimates.FiniteIsingCompensatedSquare
import OAI.Probability.FreeIsing.Gibbs.CovarianceCompensator
import OAI.Probability.FreeIsing.Gibbs.SpinMeanL2Limit
import OAI.Probability.FreeIsing.Stochastic.GrowingFiltrationMartingale
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal Topology
namespace Problem367
theorem finiteGibbsCovarianceCompensator_eq_quadraticCompensator
{d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
(beta : ℝ) (v : TreeVertex d) (n : ℕ)
(B : TreeVertex d → ℝ≥0 → Ω → ℝ)
(hcB : ∀ i omega, Continuous (fun t => B i t omega)) (t : ℝ≥0) :
finiteGibbsCovarianceCompensator beta v n B t =
FiniteIsingFiltering.quadraticCompensator beta (connectedExhaustion v n)
⟨v, mem_connectedExhaustion v n⟩ B t := by
classical
funext p
rw [FiniteIsingFiltering.quadraticCompensator_eq_integral_energy _ _ _ _ hcB]
unfold finiteGibbsCovarianceCompensator
rw [intervalIntegral.integral_of_le t.coe_nonneg,
intervalIntegral.integral_of_le t.coe_nonneg]
apply setIntegral_congr_fun measurableSet_Ioc
intro r hr
dsimp only
rw [finiteGibbsCovarianceEnergy_eq_sum, ← Finset.sum_attach]
apply Finset.sum_congr rfl
intro u hu
rw [finiteGibbsCovariance_eq_finiteIsingBrownianCovariance]
simp only [FiniteIsingFiltering.covariance, Real.coe_toNNReal r hr.1.le]
theorem stronglyAdapted_infiniteGibbsSpinMean_brownian
{d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
(beta : ℝ) (v : TreeVertex d)
(B : TreeVertex d → ℝ≥0 → Ω → ℝ)
(hmB : ∀ i t, Measurable (B i t))
(hcB : ∀ i omega, Continuous (fun t => B i t omega)) :
StronglyAdapted
(arrayObservationFiltration B hmB FiniteIsingFiltering.signal
FiniteIsingFiltering.measurable_signal)
(fun t p => infiniteGibbsSpinMean beta v
(brownianIsingObservationR B ((t : ℝ), p))) := by
let F := arrayObservationFiltration B hmB FiniteIsingFiltering.signal
FiniteIsingFiltering.measurable_signal
let X := plantedArrayObservation B FiniteIsingFiltering.signal
have hX (i : TreeVertex d) : IsStronglyProgressive F (X i) := by
apply (arrayNaturalFiltration_stronglyAdapted X
(plantedArrayObservation_measurable hmB FiniteIsingFiltering.measurable_signal) i).isStronglyProgressive_of_continuous
intro p
exact (hcB i (p.2 i)).add (continuous_subtype_val.mul continuous_const)
have h := (isStronglyProgressive_infiniteGibbsSpinMean F beta v X hX).stronglyAdapted
convert h using 1
funext t p
congr 1
funext i
simp only [brownianIsingObservationR, X, plantedArrayObservation,
FiniteIsingFiltering.signal, Real.toNNReal_coe, add_comm]
theorem HasFreeIsingLaw.infiniteGibbsSpinMean_compensatedSquare_martingale
{d : ℕ} (hd : 3 ≤ d) {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta)
(hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1)
(ν : TreeVertex d → Measure Ω) {B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hcB : ∀ i omega, Continuous (fun t => B i t omega))
(v : TreeVertex d) :
Martingale (fun (t : ℝ≥0) p =>
infiniteGibbsSpinMean beta v (brownianIsingObservationR B ((t : ℝ), p)) ^ 2 -
gibbsCovarianceCompensator beta v B t p)
(arrayObservationFiltration B hmB FiniteIsingFiltering.signal
FiniteIsingFiltering.measurable_signal)
(μ.prod (Measure.infinitePi ν)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (i : TreeVertex d) : IsProbabilityMeasure (ν i) :=
(hB i).isGaussianProcess.isProbabilityMeasure
let P := μ.prod (Measure.infinitePi ν)
let F := fun n => FiniteIsingFiltering.observationFiltration B hmB (connectedExhaustion v n)
let G := arrayObservationFiltration B hmB FiniteIsingFiltering.signal
FiniteIsingFiltering.measurable_signal
let X := fun n => FiniteIsingFiltering.mean beta (connectedExhaustion v n)
⟨v, mem_connectedExhaustion v n⟩ B
let Y := fun (t : ℝ≥0) p => infiniteGibbsSpinMean beta v
(brownianIsingObservationR B ((t : ℝ), p))
have hX (n : ℕ) (t : ℝ≥0) : MemLp (X n t) 2 P :=
FiniteIsingFiltering.memLp_mean P beta (connectedExhaustion v n)
⟨v, mem_connectedExhaustion v n⟩ B hmB t 2
have hY : StronglyAdapted G Y :=
stronglyAdapted_infiniteGibbsSpinMean_brownian beta v B hmB hcB
have hY2 (t : ℝ≥0) : MemLp (Y t) 2 P := by
apply MemLp.of_bound (hY.stronglyMeasurable.aestronglyMeasurable) 1
exact Eventually.of_forall fun p => infiniteGibbsSpinMean_abs_le_one beta v _
apply GrowingFiltrationMartingale.compensatedSquare_of_growing_limits F G
(FiniteIsingFiltering.observationFiltration_time_monotone B hmB
(connectedExhaustion_monotone v))
(FiniteIsingFiltering.iSup_observationFiltration_eq_arrayObservationFiltration
B hmB (connectedExhaustion v) (fun w => (eventually_mem_connectedExhaustion v w).exists))
hX
(fun n t => integrable_finiteGibbsCovarianceCompensator beta v n μ ν hmB hcB t)
?_ hY hY2 (stronglyAdapted_gibbsCovarianceCompensator beta v B hmB hcB)
(hμ.integrable_gibbsCovarianceCompensator ν hB hmB hcB v) ?_
(hμ.finiteGibbsCovarianceCompensator_tendsto_L1 hd hb hcrit ν hB hmB hcB v)
· intro n
have h := FiniteIsingFiltering.mean_compensatedSquare_martingale
hμ ν hB hmB hcB (connectedExhaustion v n) (connectedExhaustion_connected v n)
⟨v, mem_connectedExhaustion v n⟩
simpa only [X, finiteGibbsCovarianceCompensator_eq_quadraticCompensator beta v n B hcB] using h
· intro t
apply CompensatedSquareLimit.tendsto_eLpNorm_two_of_meanSquare
(fun n => hX n t) (hY2 t)
exact hμ.finiteGibbsSpinMean_brownian_tendsto_meanSquare ν hB hmB v t
end Problem367
end
end OAI
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