Title: Backward Monge Potential and Monge-Ampere Equation Show Chinese title

Title: 反向Monge势和Monge-Ampere方程

Abstract: In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space $(W, H,\mu)$, where $H$ is Cameron-Martin space and $\mu$ is the Gaussian measure. We study the regularity of optimal transport maps with a quadratic cost function assuming that both initial and target measures have a strictly positive Radon-Nikodym density with respect to $\mu$. Under conditions on the density functions, the forward and backward transport maps can be written in terms of Sobolev derivative of so-called Monge-Brenier maps, or Monge potentials. We show Sobolev regularity of the backward potential under the assumption that the density of the initial measure is log-concave and prove that it solves Monge-Ampere equation. Show Chinese abstract

Abstract: 在本文中，考虑在抽象Wiener空间$(W, H,\mu)$上的无限维Monge-Kantorovich问题，其中$H$是Cameron-Martin空间，$\mu$是高斯测度。 我们研究了具有二次代价函数的最优传输映射的正则性，假设初始测度和目标测度相对于$\mu$都有严格正的Radon-Nikodym密度。 在密度函数的条件下，前向和后向传输映射可以表示为所谓的Monge-Brenier映射或Monge势的Sobolev导数。 我们在初始测度的密度是log-凸的假设下，展示了后向势的Sobolev正则性，并证明它满足Monge-Ampere方程。