Lipschitz transport maps via heat flow

Globally lipschitz transport maps have found many applications in the study of probabilistic functional inequalities such as logarithmic Sobolev and Poincaré inequalities, by transporting an inequality from a nice reference measure to another one. For example, a theorem of Caffarelli states that optimal transport maps from the standard Gaussian measure onto uniformly log-concave measures are $1$-lipschitz. This then recovers the sharp bounds of Bakry and Emery on the logarithmic Sobolev constant of such measures.

In this talk, I will discuss a construction of non-optimal transport maps using the heat flow, due to Kim and Milman, and explain how it allows to get dimension-free lipschitz maps in new settings, including certain Riemannian manifolds. Joint work with D. Mikulincer and Y. Shenfeld.