On Sobolev regularity of mass transport and transportation inequalities

TL;DR

This paper establishes Sobolev regularity estimates for optimal transport maps between probability measures, linking second derivatives to Fisher information and extending classical contraction and transportation inequalities.

Contribution

It provides new Sobolev a priori estimates for optimal transport maps under convexity assumptions, generalizes Caffarelli's contraction theorem, and connects these results with transportation inequalities.

Findings

Control of second derivatives by Fisher information

Extension of Caffarelli's contraction theorem to L^p norms

Dimension-free Fisher information inequalities

Abstract

We study Sobolev a priori estimates for the optimal transportation $T = \nabla \Phi$ between probability measures $\mu=e^{-V} \ dx$ and $\nu=e^{-W} \ dx$ on $\R^d$. Assuming uniform convexity of the potential $W$ we show that $\int \| D^2 \Phi\|^2_{HS} \ d\mu$, where $\|\cdot\|_{HS}$ is the Hilbert-Schmidt norm, is controlled by the Fisher information of $\mu$. In addition, we prove similar estimate for the $L^p(\mu)$-norms of $\|D^2 \Phi\|$ and obtain some $L^p$-generalizations of the well-known Caffarelli contraction theorem. We establish a connection of our results with the Talagrand transportation inequality. We also prove a corresponding dimension-free version for the relative Fisher information with respect to a Gaussian measure.