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b317718 | 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 | import OAI.Probability.FreeIsing.Gibbs.PosteriorSymmetry
import OAI.Probability.FreeIsing.Posterior.InteriorLocalization
import OAI.Probability.FreeIsing.Posterior.ProgressiveIsingPosterior
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology
namespace Problem367
theorem HasFreeIsingLaw.continuous_posterior_all_time_field_bound
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(ν : TreeVertex d → Measure Ω)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ v, IsBrownianReal (B v) (ν v))
(hmB : ∀ v t, Measurable (B v t))
{m : TreeVertex d → ℝ≥0 →
((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ}
(hcm : ∀ v, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), Continuous (fun t => m v t p))
(hversion : ∀ v t, m v t =ᵐ[μ.prod (Measure.infinitePi ν)]
(fun p => infiniteGibbsSpinMean beta v
(fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)))) :
∀ᵐ p ∂μ.prod (Measure.infinitePi ν), ∀ v,
(∀ t, m v t p ∈ Set.Ioo (-1 : ℝ) 1 ∧
|Real.artanh (m v t p) - ((t : ℝ) * spinSign (p.1 v) + B v t (p.2 v))| ≤
(d : ℝ) * beta) ∧
Continuous (fun t => Real.artanh (m v t p)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
apply ae_all_iff.mpr
intro v
have hX : ∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
Continuous (fun t : ℝ≥0 => (t : ℝ) * spinSign (p.1 v) + B v t (p.2 v)) := by
have hnoise := (planted_noise_coordinate_measurePreserving μ ν v).quasiMeasurePreserving.ae
(hB v).cont
filter_upwards [hnoise] with p hp
exact (continuous_subtype_val.mul continuous_const).add hp
have hb : ∀ t, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
m v t p ∈ Set.Ioo (-1 : ℝ) 1 ∧
|Real.artanh (m v t p) - ((t : ℝ) * spinSign (p.1 v) + B v t (p.2 v))| ≤
(d : ℝ) * beta := by
intro t
filter_upwards [hversion v t,
hμ.infiniteGibbsSpinMean_brownian_field_bound hbeta ν
(fun i => (hB i).toIsPreBrownianReal) hmB t v] with p hp hb
rw [hp]
exact hb
filter_upwards [hcm v,
PosteriorInteriorLocalization.ae_all_field_bound_of_continuous (hcm v) hX hb]
with p hp hb
exact ⟨hb, PosteriorInteriorLocalization.continuous_artanh_of_interior hp
(fun t => (hb t).1)⟩
theorem HasFreeIsingLaw.continuous_posterior_compact_localization
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta)
(ν : TreeVertex d → Measure Ω)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ v, IsBrownianReal (B v) (ν v))
(hmB : ∀ v t, Measurable (B v t))
{m : TreeVertex d → ℝ≥0 →
((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ}
(hcm : ∀ v, ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), Continuous (fun t => m v t p))
(hversion : ∀ v t, m v t =ᵐ[μ.prod (Measure.infinitePi ν)]
(fun p => infiniteGibbsSpinMean beta v
(fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i)))) :
∀ᵐ p ∂μ.prod (Measure.infinitePi ν), ∀ v, ∀ T : ℝ≥0, ∀ᶠ n in atTop,
∀ t ≤ T, |m v t p| < PosteriorInteriorLocalization.radius n := by
filter_upwards [ae_all_iff.mpr hcm,
hμ.continuous_posterior_all_time_field_bound hbeta ν hB hmB hcm hversion]
with p hc hb
intro v T
have h := PosteriorInteriorLocalization.eventually_uniform_abs_lt_radius
(s := Set.Icc (0 : ℝ≥0) T) isCompact_Icc (hc v).continuousOn
(fun t _ => (hb v).1 t |>.1)
filter_upwards [h] with n hn
exact fun t ht => hn t ⟨bot_le, ht⟩
theorem HasFreeIsingLaw.posterior_version_initial_zero
{d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω]
{μ : Measure (TreeVertex d → IsingSpin)}
(hμ : HasFreeIsingLaw (Real.tanh beta) μ)
(ν : TreeVertex d → Measure Ω)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hB : ∀ v, IsPreBrownianReal (B v) (ν v))
{m : TreeVertex d → ℝ≥0 →
((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ}
(hversion : ∀ v, m v 0 =ᵐ[μ.prod (Measure.infinitePi ν)]
(fun p => infiniteGibbsSpinMean beta v (fun i => B i 0 (p.2 i)))) :
∀ᵐ p ∂μ.prod (Measure.infinitePi ν), ∀ v,
m v 0 p = 0 ∧ Real.artanh (m v 0 p) = 0 := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
filter_upwards [ae_all_iff.mpr hversion,
infiniteGibbsSpinMean_brownian_zero beta μ ν hB] with p hp hz
intro v
have hzero : m v 0 p = 0 := by
rw [hp v]
simpa only [zero_mul, zero_add] using hz v
exact ⟨hzero, by rw [hzero, Real.artanh_zero]⟩
theorem isStronglyProgressive_artanh
{Ω : Type*} [mΩ : MeasurableSpace Ω]
(F : Filtration ℝ≥0 mΩ) {m : ℝ≥0 → Ω → ℝ}
(hm : IsStronglyProgressive F m) :
IsStronglyProgressive F (fun t ω => Real.artanh (m t ω)) := by
have ha : Measurable Real.artanh := by
unfold Real.artanh
fun_prop
intro t
let : MeasurableSpace Ω := F t
exact (ha.comp (hm t).measurable).stronglyMeasurable
end Problem367
end
end OAI
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