A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow

TL;DR

This paper extends Caffarelli's contraction theorem to broader classes of measures using heat flow methods, leading to new inequalities in correlation and isoperimetric problems.

Contribution

The authors generalize Caffarelli's contraction theorem by employing heat flow techniques and third derivative conditions, providing two different proofs and new applications.

Findings

Generalized contraction theorem for broader measures

New correlation inequalities derived from the generalization

Enhanced isoperimetric inequalities using the extended framework

Abstract

A theorem of L. Caffarelli implies the existence of a map pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map $T_{opt}$ is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, using two different proofs. The first uses a map $T$, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusion process. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding diffusion semi-group, by using a maximum principle for parabolic PDE. In particular, Caffarelli's original result immediately follows by using the Ornstein-Uhlenbeck process and the Pr\'ekopa--Leindler Theorem. The second uses the map $T_{opt}$ by generalizing Caffarelli's argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.