TY - JOUR
T1 - A generalization of Caffarelli's contraction theorem via (reverse) heat flow
AU - Kim, Young Heon
AU - Milman, Emanuel
N1 - Funding Information: Y.-H. Kim and E. Milman were partially supported by the Institute for Advanced Study through NSF Grant DMS-0635607. Y.-H. Kim is also supported by Canadian NSERC discovery Grant 371642-09. E. Milman is also supported by ISF, GIF and the Taub Foundation (Landau Fellow).
PY - 2012/11
Y1 - 2012/11
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84867489721&partnerID=8YFLogxK
U2 - https://doi.org/10.1007/s00208-011-0749-x
DO - https://doi.org/10.1007/s00208-011-0749-x
M3 - مقالة
SN - 0025-5831
VL - 354
SP - 827
EP - 862
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 3
ER -