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import OAI.Probability.FreeIsing.Energy.NormalizedResponseEnergy
import OAI.Probability.FreeIsing.Energy.NormalizedResponseTime
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal ENNReal Topology
namespace Problem367.NormalizedInfiniteTimeEnergy
open GaussianTreeResponse
variable {d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
def observation (B : TreeVertex d → ℝ≥0 → Ω → ℝ) (t : ℝ)
(p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) : TreeVertex d → ℝ :=
fun i => t * spinSign (p.1 i) + B i t.toNNReal (p.2 i)
def finiteRow (beta : ℝ) (D : Finset (TreeVertex d)) (v : TreeVertex d)
(B : TreeVertex d → ℝ≥0 → Ω → ℝ) (w : TreeVertex d)
(q : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) : ℝ :=
observedRow beta D v q.1 w (q.2.1, fun i => B i q.1.toNNReal (q.2.2 i))
def infiniteRow (beta : ℝ) (v : TreeVertex d)
(B : TreeVertex d → ℝ≥0 → Ω → ℝ) (w : TreeVertex d)
(q : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) : ℝ :=
infiniteGibbsResponse beta v w
(fun i => q.1 * spinSign (q.2.1 i) + B i q.1.toNNReal (q.2.2 i))
lemma measurable_observation
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2)) :
Measurable (fun q : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) =>
observation B q.1 q.2) := by
apply Measurable.of_eval
intro i
exact (measurable_fst.mul ((measurable_of_finite spinSign).comp
((measurable_pi_apply i).comp (measurable_fst.comp measurable_snd)))).add
((hjoint i).comp ((measurable_real_toNNReal.comp measurable_fst).prodMk
((measurable_pi_apply i).comp (measurable_snd.comp measurable_snd))))
lemma measurable_finiteRow (beta : ℝ) (D : Finset (TreeVertex d))
(v : TreeVertex d) {B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(w : TreeVertex d) : Measurable (finiteRow beta D v B w) := by
classical
by_cases hw : w ∈ D
· unfold finiteRow
simp only [observedRow, ite_eq_left hw]
exact (measurable_response beta D v w).comp (measurable_observation hjoint)
· unfold finiteRow
simp only [observedRow, ite_eq_right hw]
exact measurable_const
lemma measurable_infiniteRow (beta : ℝ) (v : TreeVertex d)
{B : TreeVertex d → ℝ≥0 → Ω → ℝ}
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(w : TreeVertex d) : Measurable (infiniteRow beta v B w) :=
(measurable_infiniteGibbsResponse beta v w).comp (measurable_observation hjoint)
omit [MeasurableSpace Ω] in
lemma finiteRow_eq_normalizedCovariance (beta : ℝ) (D : Finset (TreeVertex d))
(v w : D) (B : TreeVertex d → ℝ≥0 → Ω → ℝ) (t : ℝ)
(p : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) :
finiteRow beta D v B w (t, p) =
finiteIsingBrownianNormalizedCovariance beta D v w B t p := by
classical
simp only [finiteRow, observedRow, ite_eq_left w.property]
unfold response TreePathResponse.response TreePathResponse.covariance
LeafGibbs.covariance TreePathResponse.mean
finiteIsingBrownianNormalizedCovariance FinitePosterior.normalizedFieldCovariance
FinitePosterior.fieldCovariance
simp only [extendFiniteSpin_apply D _ v, extendFiniteSpin_apply D _ w,
GibbsLeaf.expectation_eq_fieldMean]
rfl
omit [MeasurableSpace Ω] in
lemma finiteRow_eq_zero (beta : ℝ) (D : Finset (TreeVertex d)) (v w : TreeVertex d)
(B : TreeVertex d → ℝ≥0 → Ω → ℝ) (hw : w ∉ D)
(q : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) :
finiteRow beta D v B w q = 0 := by
simp [finiteRow, observedRow, hw]
omit [MeasurableSpace Ω] in
lemma tsum_finiteRow_sq (beta : ℝ) (D : Finset (TreeVertex d)) (v : D)
(B : TreeVertex d → ℝ≥0 → Ω → ℝ)
(q : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω))) :
(∑' w, ENNReal.ofReal (finiteRow beta D v B w q ^ 2)) =
ENNReal.ofReal (∑ w : D,
finiteIsingBrownianNormalizedCovariance beta D v w B q.1 q.2 ^ 2) := by
classical
rcases q with ⟨t, p⟩
rw [tsum_eq_sum (s := D)]
· rw [← Finset.sum_coe_sort D]
simp_rw [finiteRow_eq_normalizedCovariance]
exact (ENNReal.ofReal_sum_of_nonneg (fun _ _ => sq_nonneg _)).symm
· intro w hw
simp [finiteRow_eq_zero beta D v w B hw]
theorem finiteRow_tendsto_ae
{beta : ℝ} {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 : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (w : TreeVertex d) (T : ℝ) :
∀ᵐ q ∂(volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν)),
Tendsto (fun n => finiteRow beta (D n) v B w q) atTop
(𝓝 (infiniteRow beta v B w q)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
apply (Measure.ae_prod_iff_ae_ae (measurableSet_tendsto_fun
(fun n => measurable_finiteRow beta (D n) v hjoint w)
(measurable_infiniteRow beta v hjoint w))).mpr
filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
have h := NormalizedResponseTime.observedRow_tendsto_ae_brownian
hμ hbeta ν hB hmB D hD hconn hex v hv w t.toNNReal
simpa only [finiteRow, infiniteRow, Real.coe_toNNReal t ht.1.le] using h
theorem lintegral_infiniteRow_le_of_finite
{beta : ℝ} {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 : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(D : ℕ → Finset (TreeVertex d)) (hD : Monotone D)
(hconn : ∀ n, ConnectedFinset (D n)) (hex : ∀ u, ∃ n, u ∈ D n)
(v : TreeVertex d) (hv : ∀ n, v ∈ D n) (T : ℝ) {K : ℝ≥0∞}
(henergy : ∀ n,
(∫⁻ q : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)),
∑ u : D n, ENNReal.ofReal
(finiteIsingBrownianNormalizedCovariance beta (D n) ⟨v, hv n⟩ u B q.1 q.2 ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν))) ≤ K) :
(∫⁻ q, ∑' w, ENNReal.ofReal (infiniteRow beta v B w q ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν))) ≤ K := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
let P := (volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν))
rw [lintegral_tsum (fun w =>
((measurable_infiniteRow beta v hjoint w).pow_const 2).ennreal_ofReal.aemeasurable)]
apply tsum_lintegral_sq_le_of_ae_tendsto P
(fun n w => (measurable_finiteRow beta (D n) v hjoint w).aestronglyMeasurable)
(fun w => finiteRow_tendsto_ae hμ hbeta ν hB hmB hjoint D hD hconn hex v hv w T)
intro n
rw [← lintegral_tsum (fun w =>
((measurable_finiteRow beta (D n) v hjoint w).pow_const 2).ennreal_ofReal.aemeasurable)]
calc
_ = ∫⁻ q : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)),
∑ u : D n, ENNReal.ofReal
(finiteIsingBrownianNormalizedCovariance beta (D n) ⟨v, hv n⟩ u B q.1 q.2 ^ 2)
∂P := by
apply lintegral_congr
intro q
rw [tsum_finiteRow_sq beta (D n) ⟨v, hv n⟩ B q]
exact ENNReal.ofReal_sum_of_nonneg (fun _ _ => sq_nonneg _)
_ ≤ K := henergy n
theorem lintegral_infiniteRow_le
{beta : ℝ} {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 : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
(∫⁻ q, ∑' w, ENNReal.ofReal (infiniteRow beta v B w q ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν))) ≤
ENNReal.ofReal (2 * ((d : ℝ) * beta + T + Real.sqrt T)) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
apply lintegral_infiniteRow_le_of_finite hμ hbeta ν hB hmB hjoint
(connectedExhaustion v) (connectedExhaustion_monotone v)
(connectedExhaustion_connected v)
(fun u => (eventually_mem_connectedExhaustion v u).exists)
v (mem_connectedExhaustion v) T
intro n
let D := connectedExhaustion v n
let r : D := ⟨v, mem_connectedExhaustion v n⟩
let R : ℝ × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → ℝ≥0∞ :=
fun q => ENNReal.ofReal (∑ u : D,
finiteIsingBrownianNormalizedCovariance beta D r u B q.1 q.2 ^ 2)
have hR : Measurable R := by
have heq : R = fun q => ∑' w, ENNReal.ofReal (finiteRow beta D v B w q ^ 2) := by
funext q
exact (tsum_finiteRow_sq beta D r B q).symm
rw [heq]
exact Measurable.tsum (fun w =>
((measurable_finiteRow beta D v hjoint w).pow_const 2).ennreal_ofReal)
calc
_ = ∫⁻ q, R q
∂(volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν)) := by
apply lintegral_congr
intro q
exact (ENNReal.ofReal_sum_of_nonneg (fun _ _ => sq_nonneg _)).symm
_ = ∫⁻ t in Ioc 0 T, ∫⁻ p, R (t, p) ∂μ.prod (Measure.infinitePi ν) :=
lintegral_prod R hR.aemeasurable
_ ≤ _ := hμ.lintegral_finiteIsingBrownianNormalizedEnergy_le hbeta ν hB hmB
D (connectedExhaustion_connected v n) r hT
theorem row_energy_integrable
{beta : ℝ} {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 : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
(∀ᵐ q ∂(volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν)),
Summable (fun w => infiniteRow beta v B w q ^ 2)) ∧
Integrable (fun q => ∑' w, infiniteRow beta v B w q ^ 2)
((volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν))) ∧
(∫ q, ∑' w, infiniteRow beta v B w q ^ 2
∂(volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν))) ≤
2 * ((d : ℝ) * beta + T + Real.sqrt T) := by
let P := (volume.restrict (Ioc 0 T)).prod (μ.prod (Measure.infinitePi ν))
have henergy := lintegral_infiniteRow_le hμ hbeta ν hB hmB hjoint v hT
rw [lintegral_tsum (fun w =>
((measurable_infiniteRow beta v hjoint w).pow_const 2).ennreal_ofReal.aemeasurable)] at henergy
have hm (w : TreeVertex d) : AEStronglyMeasurable (infiniteRow beta v B w) P :=
(measurable_infiniteRow beta v hjoint w).aestronglyMeasurable
exact ⟨ae_summable_sq_of_tsum_lintegral_sq_ne_top P hm
(ne_top_of_le_ne_top ENNReal.ofReal_ne_top henergy),
integral_tsum_sq_le_of_tsum_lintegral_sq_le P hm (by positivity) henergy⟩
theorem ae_path_row_energy_integrable
{beta : ℝ} {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 : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
∀ᵐ p ∂μ.prod (Measure.infinitePi ν),
IntervalIntegrable (fun t => ∑' w, infiniteRow beta v B w (t, p) ^ 2)
volume 0 T := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
filter_upwards [(row_energy_integrable hμ hbeta ν hB hmB hjoint v hT).2.1.prod_left_ae]
with p hp
exact (intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mpr hp
theorem integral_mean_row_energy_le
{beta : ℝ} {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 : ∀ i, IsPreBrownianReal (B i) (ν i))
(hmB : ∀ i t, Measurable (B i t))
(hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2))
(v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
IntervalIntegrable (fun t => ∫ p, ∑' w, infiniteRow beta v B w (t, p) ^ 2
∂μ.prod (Measure.infinitePi ν)) volume 0 T ∧
(∫ t in 0..T, ∫ p, ∑' w, infiniteRow beta v B w (t, p) ^ 2
∂μ.prod (Measure.infinitePi ν)) ≤ 2 * ((d : ℝ) * beta + T + Real.sqrt T) := by
let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure
have (vertex : TreeVertex d) : IsProbabilityMeasure (ν vertex) :=
(hB vertex).isGaussianProcess.isProbabilityMeasure
have h := row_energy_integrable hμ hbeta ν hB hmB hjoint v hT
refine ⟨(intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mpr
h.2.1.integral_prod_left, ?_⟩
rw [intervalIntegral.integral_of_le hT, ← integral_prod _ h.2.1]
exact h.2.2
end Problem367.NormalizedInfiniteTimeEnergy
end
end OAI
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