Smooth transport map via diffusion process

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