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dd14c26 | 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 | import Mathlib
import OAI.Probability.LogConcave.Dynamics.TriangularEquiv
import OAI.Probability.LogConcave.Sampling.ProbabilityVelocity
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology
section UpperProof
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open Function MeasureTheory Set Filter
open scoped Topology NNReal
namespace LogConcaveSampling
open Set Function MeasureTheory Filter ProbabilityTheory
open scoped Topology
theorem probabilityFlow_integral {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0≤r)
(hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1)
{φ : Point d → ℝ} (hφ : Differentiable ℝ φ) {L : ℝ≥0} (hφL : LipschitzWith L φ)
{B : ℝ} (hφB : ∀ z,‖φ z‖≤B) :
(∫ y,φ y ∂interpolationLaw F x r T) =
∫ z,φ (probabilityFlow hF x hr hl hT0.le hT1 z) ∂stdGaussian (Point d) := by
let μ := gibbs (primitivePotential F x r)
let ν := stdGaussian (Point d)
let := probability_gibbs_of_partition
(partition_pos_of_continuous (hF.continuous_potential x r)).ne'
(partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr (by linarith)))
have hc := clampedProbabilityVelocity_continuous hF x hr hl hT0.le hT1
have hL := fun t (_ : t∈Icc 0 T) => clampedProbabilityVelocity_lipschitz hF x hr hl hT0.le hT1 t
obtain ⟨g,hg,hgb⟩ := interpolationPath_defect_bound hF x hr hl hT0.le hT1
have hh := GlobalODE.transport_expectation_of_flux (μ:=μ.prod ν) hT0 hc hL
(clampedProbabilityDerivative_continuous hF x hr hl hT0.le hT1)
(clampedProbabilityVelocity_hasFDerivAt hF x hr hl hT0.le hT1)
(X:=interpolationPath) (X':=interpolationPathDerivative)
interpolationPath_measurable interpolationPath_continuous
(fun t ht p => interpolationPath_hasDerivAt (by
have hh := (probability_time ht.1.le (ht.2.trans hT1)).1; linarith) p)
interpolationPathDerivative_measurable hg (fun t ht => hgb t ⟨ht.1.le,ht.2.le⟩)
(fun t ht S hS ⟨C,hC⟩ => by
simp_rw [clampedProbabilityVelocity_eq F x r T hT0.le ⟨ht.1.le,ht.2.le⟩]
exact interpolationPath_flux hF x hr hl ht.1.le (ht.2.trans hT1) hS hC)
hφ hφL hφB
have hfc := probabilityFlow_continuous hF x hr hl hT0.le hT1
have he := integral_map (μ:=μ.prod ν) measurable_snd.aemeasurable
(hφ.continuous.comp hfc).aestronglyMeasurable
rw [Measure.map_snd_prod,measure_univ,one_smul] at he
change (∫ y,φ y ∂(μ.prod ν).map (interpolationPath T))=_
rw [integral_map (interpolationPath_measurable T).aemeasurable hφ.continuous.aestronglyMeasurable]
change (∫ p,φ (interpolationPath T p) ∂μ.prod ν)=_
rw [hh]
trans ∫ p,φ (probabilityFlow hF x hr hl hT0.le hT1 p.2) ∂μ.prod ν
· apply integral_congr_ae
filter_upwards [] with p
rw [show interpolationPath 0 p=p.2 by simp [interpolationPath]]
rfl
· exact he.symm
end LogConcaveSampling
end UpperProof
end
end
end
end OAI
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