2021/08/27 by Caglar, Mine, Demirel, Ihsan
#49J27 #60H07 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2108.12316
In this paper, Monge-Kantorovich problem is considered in the infinite dimension on an abstract Wiener space (W, H,μ), where H is Cameron-Martin space and μ 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 μ. 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.