Regularity of the backward Monge potential and the Monge–Ampère equation on Wiener space
Publication Date
2023
Advisor
Institution Author
Çağlar, Mine
Demirel, İhsan
Co-Authors
Journal Title
Journal ISSN
Volume Title
Publisher:
Institute of Mathematics. Polish Academy of Sciences
Type
Journal Article
Abstract
In this paper, the Monge–Kantorovich problem is considered in infinite dimensions on an abstract Wiener space (W, H, µ), where H is the 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 some conditions on the density functions, the forward and backward transport maps can be written in terms of Sobolev derivatives of so-called Monge–Brenier maps, or Monge potentials. We show the Sobolev regularity of the backward potential under the assumption that the density of the initial measure is log-concave and prove that the backward potential solves the Monge–Ampère equation.
Description
Subject
Mathematics, applied, Statistics and probability