Caffarelli收缩定理的（逆）热流推广
A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow
摘要：
L. Caffarelli暗示了存在一个映射，将源高斯测度向前推到比源测度更对数凹的目标测度，这收缩了欧几里得距离（事实上，Caffarelli证明了最优传输Brenier映射$T_{opt}$在这种情况下是一个收缩）。我们推广这一结果更一般的源和目标的措施，使用条件的三阶导数的潜力，使用两种不同的证明。第一个使用的地图$T$，其逆被构造为一个流沿平流场相关联的一个适当的热扩散过程。收缩属性，然后减少到显示，对数的保留沿相应的扩散半群，通过使用一个最大值原理抛物型偏微分方程。特别是，卡法雷利的原始结果立即遵循使用Ornstein-Uhlenbeck过程和Prékopa-Leindler定理。第二个使用的地图$T_{opt}$通过推广Caffarelli的论点，采用另外进一步的结果Caffarelli。作为应用，我们得到了新的相关不等式和等周不等式。
英文摘要：
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 $T_{opt}$ 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, using 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 $T_{opt}$ by generalizing Caffarelli's argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.
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