Smooth transport map via diffusion process

2024/11/15 by Arthur Stéphanovitch, Stéphanovitch, Arthur · 6 citations
Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2411.10235

openalex publication_date 2024/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We extend the classical regularity theory of optimal transport to non-optimal transport maps generated by heat flow for perturbations of Gaussian measures. Considering probability measures of the form dμ(x) = exp(-(|x|²)/(2) + a(x))dx on ℝ^d where a has Hölder regularity C^β with β≥ 0; we show that the Langevin map transporting the d-dimensional Gaussian distribution onto μ achieves Hölder regularity C^{β+ 1}, up to a logarithmic factor. We additionally present applications of this result to functional inequalities and generative modelling.

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