A generalization of Caffarelli’s contraction theorem via (reverse) heat flow

By Young-Heon Kim, Emanuel Milman. Mathematische Annalen, 2010

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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 Topt 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, by providing 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 Prékopa–Leindler Theorem. The second uses the map Topt by generalizing Caffarelli’s argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/65cf532074f5a5f34b381774782d27b276661063

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In Applied Mathematics

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  4. Jon A. Wellner
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