Technion - Israel Institute of Technology Department of Mathematics ============================================ PDE AND APPLIED MATHEMATICS SEMINAR ============================================ DATE: Tuesday, May 24, 2011 SPEAKER: Emanuel Milman, Technion TITLE: A generalization of Caffarelli's Contraction Theorem via (reverse) heat-flow PLACE: Room 814, Amado Mathematics Building, Technion TIME: 14:30 Abstract: A theorem of L. Caffarelli 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, Caffarelli 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 non-constructive optimal-transport map, our map T 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 Prékopa-Leindler Theorem. We thus avoid using Caffarelli'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). For further info: Gershon Wolansky <gershonw@math.technion.ac.il> For past and future Applied Math/PDE seminars: <http://www.math.technion.ac.il/pde/seminar.html> --------------------------------------------------------- Technion Math. Net (TECHMATH) Editor: Michael Cwikel <techm@math.technion.ac.il> Announcement from: Gershon Wolansky <gershonw@math.technion.ac.il>