| 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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