Title: Lipschitz properties of transport maps under a log-Lipschitz condition

Abstract: Consider the problem of realizing a target probability measure as a push forward, by a transport map, of a given source measure. Typically one thinks about the target measure as being ‘complicated’ while the source is simpler and often more structured. In such a setting, for applications, it is desirable to find Lipschitz transport maps which afford the transfer of analytic properties from the source to the target. The talk will focus on Lipschitz regularity when the target measure satisfies a log-Lipschitz condition.

I will present a construction of a transport map, constructed infinitesimally along the Langevin flow, and explain how to analyze its Lipschitz constant. The analysis of this map leads to several new results which apply both to Euclidean spaces and manifolds, and which, at the moment, seem to be out of reach of the classically studied optimal transport theory.

Joint work with Max Fathi and Yair Shenfeld.