A generalization of Ca¤arellis Contraction Theorem via (reverse) heat-ow

A theorem of L. Ca¤arelli implies the existence of a map T , 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, Ca¤arelli showed that the optimal-transport Brenier map is a contraction in this case.) This theorem has found numerous applications pertaining to correlation inequalities, isoperimetry, spectral-gap estimation, properties of the Gaussian measure, and more. We generalize this result to more general source and target measures, using a condition on the third derivative of the potential. Contrary to the nonconstructive optimal transport map, our map’s inverse T 1 is constructed as a ‡ow along an advection …eld associated to an appropriately modi…ed heat‡ow. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding di¤usion, by using a maximum principle for parabolic PDE. In particular, Ca¤arelli’s original result immediately follows by using the Ornstein–Uhlenbeck process and the Prékopa–Leindler Theorem. We thus avoid using Ca¤arelli’s regularity theory for the Monge-Ampère equation, lending our approach to further generalizations. As applications, we obtain new correlation and isoperimetric inequalities. This is joint work with Young-Heon Kim (UBC).