Sobolev estimates for optimal transport maps on Gaussian spaces

In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and...

Full description

Saved in:
Bibliographic Details
Published inJournal of functional analysis Vol. 266; no. 8; pp. 5045 - 5084
Main Authors Fang, Shizan, Nolot, Vincent
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.04.2014
Subjects
Gaussian measures
Monge–Ampère equations
Optimal transportation
Sobolev estimates
Wiener space
Wiener space
Optimal transportation
Sobolev estimates
Monge–Ampère equations
Gaussian measures
Online AccessGet full text

Cover

More Information
Summary:In this work, we will take the standard Gaussian measure as the reference measure and study the variation of optimal transport maps in Sobolev spaces with respect to it; as a by-product, an inequality which gives a precise link between the variation of entropy, Fisher information between source and target measures, with the Sobolev norm of the optimal transport map will be given. As applications, we will construct strong solutions to Monge–Ampère equations in finite dimension, as well as on the Wiener space, when the target measure satisfies the strong log-concavity condition. A result on the regularity on the optimal transport map on the Wiener space will be obtained.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2014.02.017