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

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

2010-02-02

arxiv.org/abs/1002.0373v3

Mathematics

Analysis of PDEs

33 pages; corrected typos, shortened Section 6 and some of the standard proofs. To appear in Math. Annal

Scientific paper

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.

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