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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 | import OAI.Probability.FreeIsing.Brownian.IntegralTransport
import OAI.Probability.FreeIsing.Conditioning.Copies
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter TopologicalSpace
open scoped Topology
namespace Problem367.ConditionalIntegralTransport
variable {W Ω : Type*}
def pair (j : Bool) (q : W × (Ω × Ω)) : W × Ω :=
(q.1, if j then q.2.1 else q.2.2)
theorem pair_measurePreserving [MeasurableSpace W] [MeasurableSpace Ω]
(ρ : Measure W) [SFinite ρ] (κ : Kernel W Ω) [IsMarkovKernel κ] (j : Bool) :
MeasurePreserving (pair (W := W) (Ω := Ω) j) (ρ ⊗ₘ (κ ×ₖ κ)) (ρ ⊗ₘ κ) := by
cases j
· exact measurePreserving_conditionalCopy_snd ρ κ
· exact measurePreserving_conditionalCopy_fst ρ κ
theorem integral_transport
{V T : Type*} [MeasurableSpace W] [MeasurableSpace Ω] [Countable V]
[TopologicalSpace T] [SeparableSpace T] [Nonempty T]
(ρ : Measure W) [SFinite ρ] (κ : Kernel W Ω) [IsMarkovKernel κ]
{X : V → T → ℕ → W × Ω → ℝ} {I : V → T → W × Ω → ℝ}
{J : Bool → V → T → W × (Ω × Ω) → ℝ}
(hX : ∀ v t n, MemLp (X v t n) 2 (ρ ⊗ₘ κ))
(hI : ∀ v t, MemLp (I v t) 2 (ρ ⊗ₘ κ))
(hJ : ∀ j v t, MemLp (J j v t) 2 (ρ ⊗ₘ (κ ×ₖ κ)))
(hcI : ∀ v, ∀ᵐ q ∂ρ ⊗ₘ κ, Continuous (fun t => I v t q))
(hcJ : ∀ j v, ∀ᵐ q ∂ρ ⊗ₘ (κ ×ₖ κ), Continuous (fun t => J j v t q))
(hlimI : ∀ v t, Tendsto (fun n => ∫ q, (X v t n q - I v t q) ^ 2 ∂ρ ⊗ₘ κ)
atTop (𝓝 0))
(hlimJ : ∀ j v t, Tendsto (fun n => ∫ q,
(X v t n (pair j q) - J j v t q) ^ 2 ∂ρ ⊗ₘ (κ ×ₖ κ)) atTop (𝓝 0)) :
∀ᵐ q ∂ρ ⊗ₘ (κ ×ₖ κ), ∀ j v t, I v t (pair j q) = J j v t q := by
have hj : ∀ j, (fun q v t => I v t (pair j q)) =ᵐ[ρ ⊗ₘ (κ ×ₖ κ)]
(fun q v t => J j v t q) := fun j =>
BrownianGrid.ae_eq_array_of_meanSquare_limits_comp (pair_measurePreserving ρ κ j)
hX hI (hJ j) hcI (hcJ j) hlimI (hlimJ j)
filter_upwards [ae_all_iff.mpr hj] with q hq
intro j v t
exact congrFun (congrFun (hq j) v) t
end Problem367.ConditionalIntegralTransport
end
end OAI
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