Smooth transport map via diffusion process

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 $d\hat{I}¼(x)=\exp \left(-\frac{|x{|}^2}{2}+a(x)\right)dx$ on ${\mathbb{R}}^d$ where $a$ has Hölder regularity $C^2$ with ${\hat{I}}^2\ge 0$; we show that the Langevin map transporting the $d$-dimensional Gaussian distribution onto $\hat{I}¼$ achieves Hölder regularity $C^{{\hat{I}}^2+1}$, up to a logarithmic factor. We additionally present applications of this result to functional inequalities and generative modelling.