Technion - Israel Institute of Technology
Department of Mathematics
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
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).
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