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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 | import Mathlib
import OAI.Probability.LogConcave.Analysis.CenteringPotential
import OAI.Probability.LogConcave.Dynamics.JointHarmonicFlow
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory
open scoped NNReal
theorem exists_conditional_harmonic_flow {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) :
∃Ξ : Point (d+d) → ℝ → Point (d+d),
(∀y,Continuous (Ξ y) ∧ Ξ y 0=y) ∧
(∀y t,t∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y)
(skewLieField (centeringPotential F x r T) (harmonicSkew d)
(Ξ y t)) (Icc (0:ℝ) 1) t) ∧
Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1) ∧
(∀t∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r T)).map
(fun y => Ξ y t)=gibbs (centeringPotential F x r T)) :=
exists_joint_harmonic_flow
((interpolationPotential_smooth hF x hr hl hT0 hT1).of_le (by norm_cast))
(interpolationPotential_lowerTail hF x hr (by linarith) hT0 hT1)
(interpolationPotential_gradient_lipschitz hF x hr hl hT0 hT1)
end LogConcaveSampling
end
end
end
end OAI
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