On the Lipschitz properties of transportation along heat flows

TL;DR

This paper establishes new Lipschitz bounds for transport maps along heat flows, with applications to eigenvalues, functional inequalities, and distribution domination for specific measure classes.

Contribution

It introduces novel Lipschitz bounds for heat flow transport maps, extending their applicability to semi-log-concave measures and Gaussian mixtures.

Findings

Lipschitz bounds for transport maps are established.

Applications include eigenvalue comparisons and functional inequalities.

Results apply to semi-log-concave measures and Gaussian mixtures.

Abstract

We prove new Lipschitz properties for transport maps along heat flows, constructed by Kim and Milman. For (semi)-log-concave measures and Gaussian mixtures, our bounds have several applications: eigenvalues comparisons, dimensional functional inequalities, and domination of distribution functions.