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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 | import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy
import OAI.Probability.FreeIsing.Gibbs.CovarianceL2Limit
import OAI.Probability.FreeIsing.Stochastic.FiniteIsingBrownianEnergy
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal
namespace Problem367
def brownianIsingObservationR {d : ℕ} {Ω : Type*}
(B : TreeVertex d → ℝ≥0 → Ω → ℝ)
(p : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) : TreeVertex d → ℝ :=
fun i => p.1 * spinSign (p.2.1 i) + B i p.1.toNNReal (p.2.2 i)
lemma measurable_brownianIsingObservationR
{d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hmB : ∀ i t, Measurable (B i t))
(hcB : ∀ i omega, Continuous (fun t => B i t omega)) :
Measurable (brownianIsingObservationR B) := by
apply Measurable.of_eval
intro i
apply measurable_uncurry_of_continuous_of_measurable
(u := fun (t : ℝ) (p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) =>
t * spinSign (p.1 i) + B i t.toNNReal (p.2 i))
· intro p
exact (continuous_id.mul continuous_const).add
((hcB i (p.2 i)).comp continuous_real_toNNReal)
· intro t
exact (measurable_const.mul ((measurable_of_finite spinSign).comp
((measurable_pi_apply i).comp measurable_fst))).add
((hmB i t.toNNReal).comp ((measurable_pi_apply i).comp measurable_snd))
lemma finiteGibbsCovariance_eq_finiteIsingBrownianCovariance
{d : ℕ} {Ω : Type*} (beta : ℝ) (v : TreeVertex d) (n : ℕ)
(B : TreeVertex d → ℝ≥0 → Ω → ℝ)
(p : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)))
(u : connectedExhaustion v n) :
finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) =
finiteIsingBrownianCovariance beta (connectedExhaustion v n)
⟨v, mem_connectedExhaustion v n⟩ u B p.1 p.2 := by
classical
simp only [finiteGibbsCovariance, dite_eq_left u.property]
rfl
theorem HasFreeIsingLaw.finiteGibbsCovariance_timeEnergy_le_one
{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 : ∀ 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) (n : ℕ) {T : ℝ} (hT : 0 ≤ T) :
(∫ p, ∑ u ∈ connectedExhaustion v n,
finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2
∂(volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν))) ≤ 1 := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (index : TreeVertex d) : IsProbabilityMeasure (ν index) :=
(hB index).isGaussianProcess.isProbabilityMeasure
let D := connectedExhaustion v n
let ρ := μ.prod (Measure.infinitePi ν)
let τ : Measure ℝ := volume.restrict (Ioc 0 T)
have hX := measurable_brownianIsingObservationR hmB hcB
have hi (u : TreeVertex d) : Integrable
(fun p => finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2)
(τ.prod ρ) := by
apply Integrable.of_bound
(((continuous_finiteGibbsCovariance beta v u n).measurable.comp hX).pow_const 2).aestronglyMeasurable 4
filter_upwards with p
have h := finiteGibbsCovariance_abs_le_two beta v u n (brownianIsingObservationR B p)
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h
norm_num only [sq_abs] at hb
simpa only [Function.comp_def] using hb
have hsum : Integrable (fun p => ∑ u ∈ D,
finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2) (τ.prod ρ) :=
integrable_finsetSum D (fun u _ => hi u)
have heq (p : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) :
(∑ u ∈ D, finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2) =
∑ u : D, finiteIsingBrownianCovariance beta D
⟨v, mem_connectedExhaustion v n⟩ u B p.1 p.2 ^ 2 := by
rw [← Finset.sum_attach D]
apply Finset.sum_congr rfl
intro u _
rw [finiteGibbsCovariance_eq_finiteIsingBrownianCovariance]
have he := hμ.integral_finiteIsingBrownianEnergy ν hB hmB D
(connectedExhaustion_connected v n) ⟨v, mem_connectedExhaustion v n⟩ hT
dsimp only at he
calc
(∫ p, ∑ u ∈ D,
finiteGibbsCovariance beta v u n (brownianIsingObservationR B p) ^ 2 ∂τ.prod ρ) =
∫ t, ∫ p, ∑ u ∈ D,
finiteGibbsCovariance beta v u n (brownianIsingObservationR B (t, p)) ^ 2 ∂ρ ∂τ :=
integral_prod _ hsum
_ = ∫ t in (0 : ℝ)..T, ∫ p, ∑ u : D,
finiteIsingBrownianCovariance beta D ⟨v, mem_connectedExhaustion v n⟩ u B t p ^ 2 ∂ρ := by
rw [intervalIntegral.integral_of_le hT]
apply integral_congr_ae
filter_upwards with t
apply integral_congr_ae
filter_upwards with p
exact heq (t, p)
_ ≤ 1 := he.2.2.2
theorem HasFreeIsingLaw.finiteGibbsCovariance_tendsto_timeSquareIntegral
{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 : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hcB : ∀ i omega, Continuous (fun t => B i t omega))
(v w : TreeVertex d) {T : ℝ} :
Tendsto (fun n => ∫ p,
finiteGibbsCovariance beta v w n (brownianIsingObservationR B p) ^ 2
∂(volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν))) atTop
(𝓝 (∫ p, infiniteGibbsCovariance beta v w (brownianIsingObservationR B p) ^ 2
∂(volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν)))) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (index : TreeVertex d) : IsProbabilityMeasure (ν index) :=
(hB index).isGaussianProcess.isProbabilityMeasure
let ρ := μ.prod (Measure.infinitePi ν)
let τ : Measure ℝ := volume.restrict (Ioc 0 T)
have hX := measurable_brownianIsingObservationR hmB hcB
have hm (n : ℕ) := ((continuous_finiteGibbsCovariance beta v w n).measurable.comp hX)
have hy := (measurable_infiniteGibbsCovariance beta v w).comp hX
have hlim : ∀ᵐ p ∂τ.prod ρ,
Tendsto (fun n => finiteGibbsCovariance beta v w n (brownianIsingObservationR B p)) atTop
(𝓝 (infiniteGibbsCovariance beta v w (brownianIsingObservationR B p))) := by
apply (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun hm hy)).mpr
filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
have h := hμ.finiteGibbsCovariance_brownian_tendsto_ae ν hB hmB t.toNNReal v w
unfold brownianIsingObservationR
simpa only [Function.comp_def, ρ, Real.coe_toNNReal t ht.1.le] using h
apply tendsto_integral_of_dominated_convergence (fun _ => (4 : ℝ))
(fun n => ((hm n).pow_const 2).aestronglyMeasurable) (integrable_const 4)
· intro n
filter_upwards with p
have h := finiteGibbsCovariance_abs_le_two beta v w n (brownianIsingObservationR B p)
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h
norm_num only [sq_abs] at hb
simpa only [Function.comp_def] using hb
· exact hlim.mono (fun p hp => hp.pow 2)
theorem HasFreeIsingLaw.infiniteGibbsCovariance_timeEnergy
{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 : ∀ 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) {T : ℝ} (hT : 0 ≤ T) :
let P := (volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν))
let C := fun w p => infiniteGibbsCovariance beta v w (brownianIsingObservationR B p)
Summable (fun w => ∫ p, C w p ^ 2 ∂P) ∧
(∑' w, ∫ p, C w p ^ 2 ∂P) ≤ 1 ∧
(∀ᵐ p ∂P, Summable (fun w => C w p ^ 2)) ∧
Integrable (fun p => ∑' w, C w p ^ 2) P ∧
(∫ p, ∑' w, C w p ^ 2 ∂P) ≤ 1 := by
classical
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (index : TreeVertex d) : IsProbabilityMeasure (ν index) :=
(hB index).isGaussianProcess.isProbabilityMeasure
let P := (volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (Measure.infinitePi ν))
let C := fun w p => infiniteGibbsCovariance beta v w (brownianIsingObservationR B p)
let Cn := fun n w p => finiteGibbsCovariance beta v w n (brownianIsingObservationR B p)
have hX := measurable_brownianIsingObservationR hmB hcB
have hm (w : TreeVertex d) : Measurable (C w) :=
(measurable_infiniteGibbsCovariance beta v w).comp hX
have hmn (n : ℕ) (w : TreeVertex d) : Measurable (Cn n w) :=
(continuous_finiteGibbsCovariance beta v w n).measurable.comp hX
have hi (w : TreeVertex d) : Integrable (fun p => C w p ^ 2) P := by
apply Integrable.of_bound ((hm w).pow_const 2).aestronglyMeasurable 4
filter_upwards with p
have h := infiniteGibbsCovariance_abs_le_two beta v w (brownianIsingObservationR B p)
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h
norm_num only [sq_abs] at hb
exact hb
have hin (n : ℕ) (w : TreeVertex d) : Integrable (fun p => Cn n w p ^ 2) P := by
apply Integrable.of_bound ((hmn n w).pow_const 2).aestronglyMeasurable 4
filter_upwards with p
have h := finiteGibbsCovariance_abs_le_two beta v w n (brownianIsingObservationR B p)
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
have hb := (sq_le_sq₀ (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 2)).mpr h
norm_num only [sq_abs] at hb
exact hb
have hevent (S : Finset (TreeVertex d)) :
∀ᶠ n in atTop, S ⊆ connectedExhaustion v n := by
induction S using Finset.induction_on with
| empty => exact Eventually.of_forall fun _ => Finset.empty_subset _
| @insert w S hw ih =>
filter_upwards [eventually_mem_connectedExhaustion v w, ih] with n hn hS
exact Finset.insert_subset hn hS
have hpartial (S : Finset (TreeVertex d)) (n : ℕ)
(hS : S ⊆ connectedExhaustion v n) :
(∑ w ∈ S, ∫ p, Cn n w p ^ 2 ∂P) ≤ 1 := by
rw [← integral_finsetSum S (fun w _ => hin n w)]
apply le_trans (integral_mono_ae
(integrable_finsetSum S (fun w _ => hin n w))
(integrable_finsetSum (connectedExhaustion v n) (fun w _ => hin n w))
(Eventually.of_forall (fun p =>
Finset.sum_le_sum_of_subset_of_nonneg hS (fun w _ _ => sq_nonneg (Cn n w p)))))
exact hμ.finiteGibbsCovariance_timeEnergy_le_one ν hB hmB hcB v n hT
have hbound (S : Finset (TreeVertex d)) :
(∑ w ∈ S, ∫ p, C w p ^ 2 ∂P) ≤ 1 := by
apply le_of_tendsto
(tendsto_finsetSum S (fun w _ =>
hμ.finiteGibbsCovariance_tendsto_timeSquareIntegral ν hB hmB hcB v w))
exact (hevent S).mono (fun n hn => hpartial S n hn)
have hs : Summable (fun w => ∫ p, C w p ^ 2 ∂P) :=
summable_of_sum_le (fun w => integral_nonneg (fun p => sq_nonneg (C w p))) hbound
have hle : (∑' w, ∫ p, C w p ^ 2 ∂P) ≤ 1 := hs.tsum_le_of_sum_le hbound
have hrow := DiscrepancyEnergy.sq_row_integrable_of_summable_integrals hi hs
have hident := DiscrepancyEnergy.hasSum_integral_sq_row
(fun w => (hm w).aestronglyMeasurable) hrow.1 hrow.2
dsimp only
refine ⟨hs, hle, hrow.1, hrow.2, ?_⟩
rw [← hident.tsum_eq]
exact hle
end Problem367
end
end OAI
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