Smooth transport map via diffusion process

TL;DR

This paper extends regularity theory for optimal transport to maps generated by heat flow, showing that Langevin transport maps for perturbed Gaussian measures achieve enhanced Hölder regularity, with applications to inequalities and modeling.

Contribution

It introduces a novel regularity result for non-optimal transport maps generated by heat flow, specifically Langevin maps for Gaussian perturbations.

Findings

Langevin maps achieve Hölder regularity $C^{eta + 1}$ for measures with $C^eta$ perturbations

The regularity result includes a logarithmic factor adjustment

Applications to functional inequalities and generative modeling are demonstrated

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 $d\mu(x) = \exp\left(-\frac{|x|^2}{2} + a(x)\right)dx$ on $\mathbb{R}^d$ where $a$ has H\"older regularity $C^\beta$ with $\beta\geq 0$; we show that the Langevin map transporting the $d$-dimensional Gaussian distribution onto $\mu$ achieves H\"older regularity $C^{\beta + 1}$, up to a logarithmic factor. We additionally present applications of this result to functional inequalities and generative modelling.