Sobolev estimates for optimal transport maps on Gaussian spaces
Shizan Fang, Vincent Nolot
TL;DR
This paper investigates Sobolev regularity of optimal transport maps with Gaussian measures, establishing dimension-free inequalities and constructing solutions to Monge-Ampère equations in finite and infinite-dimensional spaces.
Contribution
It introduces new Sobolev estimates for Gaussian optimal transport maps and applies these to solve Monge-Ampère equations in finite and infinite dimensions.
Findings
Dimension-free Sobolev inequalities for Gaussian optimal transport maps
Construction of solutions to Monge-Ampère equations in finite dimensions
Extension of solutions to the Wiener space
Abstract
We will study variations in Sobolev spaces of optimal transport maps with the standard Gaussian measure as the reference measure. Some dimension free inequalities will be obtained. As application, we construct solutions to Monge-Ampere equations in finite dimension, as well as on the Wiener space.
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Taxonomy
TopicsGeometry and complex manifolds · Geometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations