File size: 6,231 Bytes
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 | import OAI.Probability.FreeIsing.Brownian.RowComparisonConstruction
import OAI.Probability.FreeIsing.Conditioning.CopyIntegralTransport
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter
open scoped NNReal Topology BigOperators
namespace Problem367.ConditionalIntegralTransport
variable {W Ω ι : Type*} [MeasurableSpace W] [MeasurableSpace Ω]
theorem integral_sq_pair
(ρ : Measure W) [SFinite ρ] (κ : Kernel W Ω) [IsMarkovKernel κ]
(j : Bool) {H : W × Ω → ℝ} (hH : AEStronglyMeasurable H (ρ ⊗ₘ κ)) :
(∫ q, H (pair j q) ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) = ∫ q, H q ^ 2 ∂ρ ⊗ₘ κ := by
simpa using BrownianGrid.integral_sq_sub_comp (pair_measurePreserving ρ κ j)
hH (aestronglyMeasurable_const : AEStronglyMeasurable
(fun _ : W × Ω => (0 : ℝ)) (ρ ⊗ₘ κ))
theorem summable_pair_timeEnergy
(ρ : Measure W) [SFinite ρ] (κ : Kernel W Ω) [IsMarkovKernel κ]
(H : ι → ℝ≥0 → W × Ω → ℝ)
(hmH : ∀ a t, AEStronglyMeasurable (H a t) (ρ ⊗ₘ κ))
(T : ℝ≥0)
(he : Summable fun a => ∫ t in (0 : ℝ)..(T : ℝ),
∫ q, H a t.toNNReal q ^ 2 ∂ρ ⊗ₘ κ) (j : Bool) :
Summable fun a => ∫ t in (0 : ℝ)..(T : ℝ),
∫ q, H a t.toNNReal (pair j q) ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ) := by
simpa only [integral_sq_pair ρ κ j (hmH _ _)] using he
theorem exists_conditionalCopy_row_integrals
(ρ : Measure W) [IsProbabilityMeasure ρ]
(κ : Kernel W Ω) [IsMarkovKernel κ]
(B : ι → ℝ≥0 → W → ℝ)
(F : Filtration ℝ≥0 (inferInstance : MeasurableSpace (W × (Ω × Ω))))
(hB : ∀ a, IsPreBrownianReal (fun (t : ℝ≥0) (q : W × (Ω × Ω)) => B a t q.1)
(ρ ⊗ₘ (κ ×ₖ κ)))
(hAd : ∀ a, StronglyAdapted F (fun (t : ℝ≥0) (q : W × (Ω × Ω)) => B a t q.1))
(hind : ∀ s t, s ≤ t → Indep
(MeasurableSpace.comap
(fun q : W × (Ω × Ω) => fun a => B a t q.1 - B a s q.1) inferInstance)
(F s) (ρ ⊗ₘ (κ ×ₖ κ)))
(hcross : ∀ s t, s ≤ t → Pairwise fun a b =>
IndepFun (fun q : W × (Ω × Ω) => B a t q.1 - B a s q.1)
(fun q : W × (Ω × Ω) => B b t q.1 - B b s q.1) (ρ ⊗ₘ (κ ×ₖ κ)))
(H : ι → ℝ≥0 → W × Ω → ℝ)
(hmH : ∀ a t, AEStronglyMeasurable (H a t) (ρ ⊗ₘ κ))
(hH : ∀ j a, StronglyAdapted F (fun t q => H a t (pair j q)))
(hcH : ∀ a, ∀ᵐ q ∂ρ ⊗ₘ κ, Continuous (fun t => H a t q))
(C : ι → ℝ) (hC : ∀ a, 0 ≤ C a)
(hbH : ∀ a, ∀ᵐ q ∂ρ ⊗ₘ κ, ∀ t, |H a t q| ≤ C a)
(T : ℝ≥0)
(he : Summable fun a => ∫ t in (0 : ℝ)..(T : ℝ),
∫ q, H a t.toNNReal q ^ 2 ∂ρ ⊗ₘ κ) :
∃ (I : Bool → ι → W × (Ω × Ω) → ℝ) (Y : Bool → W × (Ω × Ω) → ℝ),
(∀ j a, MemLp (I j a) 2 (ρ ⊗ₘ (κ ×ₖ κ))) ∧
(∀ j a, Tendsto (fun n => ∫ q,
(BrownianGrid.uniformSum (fun t p => B a t p.1)
(fun t p => H a t (pair j p)) T n q - I j a q) ^ 2
∂ρ ⊗ₘ (κ ×ₖ κ)) atTop (𝓝 0)) ∧
(∀ j, Measurable (Y j)) ∧
(∀ j, AEStronglyMeasurable[F T] (Y j) (ρ ⊗ₘ (κ ×ₖ κ))) ∧
(∀ j, MemLp (Y j) 2 (ρ ⊗ₘ (κ ×ₖ κ))) ∧
(∀ j, (∫ q, Y j q ∂ρ ⊗ₘ (κ ×ₖ κ)) = 0) ∧
(∀ j, (∫ q, Y j q ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) =
∑' a, ∫ t in (0 : ℝ)..(T : ℝ), ∫ q, H a t.toNNReal q ^ 2 ∂ρ ⊗ₘ κ) ∧
(∀ j, Tendsto (fun s : Finset ι => ∫ q,
((∑ a ∈ s, I j a q) - Y j q) ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) atTop (𝓝 0)) ∧
Summable (fun a => ∫ t in (0 : ℝ)..(T : ℝ),
∫ q, (H a t.toNNReal (pair true q) - H a t.toNNReal (pair false q)) ^ 2
∂ρ ⊗ₘ (κ ×ₖ κ)) ∧
(∫ q, (Y true q - Y false q) ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) =
(∑' a, ∫ t in (0 : ℝ)..(T : ℝ),
∫ q, (H a t.toNNReal (pair true q) - H a t.toNNReal (pair false q)) ^ 2
∂ρ ⊗ₘ (κ ×ₖ κ)) ∧
(∫ q, (Y true q - Y false q) ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) ≤
4 * (∑' a, ∫ t in (0 : ℝ)..(T : ℝ), ∫ q, H a t.toNNReal q ^ 2 ∂ρ ⊗ₘ κ) := by
have hc (j : Bool) (a : ι) : ∀ᵐ q ∂ρ ⊗ₘ (κ ×ₖ κ),
Continuous (fun t => H a t (pair j q)) :=
(pair_measurePreserving ρ κ j).quasiMeasurePreserving.ae (hcH a)
have hb (j : Bool) (a : ι) : ∀ᵐ q ∂ρ ⊗ₘ (κ ×ₖ κ),
∀ t, |H a t (pair j q)| ≤ C a :=
(pair_measurePreserving ρ κ j).quasiMeasurePreserving.ae (hbH a)
have he' (j : Bool) := summable_pair_timeEnergy ρ κ H hmH T he j
obtain ⟨I, J, Y, Z, hI, hJ, haI, haJ, hmY, hmZ, htY, htZ,
hY, hZ, hzY, hzZ, heY, heZ, hsY, hsZ, heD, hdiff, hbound⟩ :=
BrownianArrayFiltration.exists_paired_infinite_integrals hB hAd hind
(fun a t q => H a t (pair true q)) (fun a t q => H a t (pair false q))
(hH true) (hH false) (hc true) (hc false) C C hC hC (hb true) (hb false)
T (he' true) (he' false) hcross
have heY' : (∫ q, Y q ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) =
∑' a, ∫ t in (0 : ℝ)..(T : ℝ), ∫ q, H a t.toNNReal q ^ 2 ∂ρ ⊗ₘ κ := by
simpa only [integral_sq_pair ρ κ true (hmH _ _)] using heY
have heZ' : (∫ q, Z q ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) =
∑' a, ∫ t in (0 : ℝ)..(T : ℝ), ∫ q, H a t.toNNReal q ^ 2 ∂ρ ⊗ₘ κ := by
simpa only [integral_sq_pair ρ κ false (hmH _ _)] using heZ
refine ⟨(fun j => if j then I else J), (fun j => if j then Y else Z),
?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, heD, hdiff, ?_⟩
· intro j; cases j <;> assumption
· intro j; cases j <;> assumption
· intro j; cases j <;> assumption
· intro j; cases j <;> assumption
· intro j; cases j <;> assumption
· intro j; cases j <;> assumption
· intro j; cases j <;> assumption
· intro j; cases j <;> assumption
· change (∫ q, (Y q - Z q) ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) ≤ _
simp only [integral_sq_pair ρ κ true (hmH _ _),
integral_sq_pair ρ κ false (hmH _ _)] at hbound
linarith
end Problem367.ConditionalIntegralTransport
end
end OAI
|