| 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 | |