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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 | import Mathlib
import OAI.Probability.LogConcave.Dynamics.ProbabilityFlowIntegral
import OAI.Probability.LogConcave.Sampling.MeasureSmoothLipschitz
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 MeasureTheory ProbabilityTheory
theorem probabilityFlow_pushforward {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) :
(stdGaussian (Point d)).map (probabilityFlow hF x hr hl hT0.le hT1)=
interpolationLaw F x r T := by
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 := probabilityFlow_continuous hF x hr hl hT0.le hT1
let hprob : IsProbabilityMeasure (interpolationLaw F x r T) := by
unfold interpolationLaw
infer_instance
apply measure_eq_of_smooth_lipschitz
intro φ hφ ⟨L,hL⟩ ⟨B,hB⟩
rw [integral_map hc.measurable.aemeasurable hφ.continuous.aestronglyMeasurable]
exact (probabilityFlow_integral hF x hr hl hT0 hT1 (hφ.differentiable (by norm_num)) hL hB).symm
end LogConcaveSampling
end UpperProof
end
end
end
end OAI
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