Optimal transport on the classical Wiener space with different norms

TL;DR

This paper investigates optimal transportation problems on Wiener space with different norms, addressing the Monge problem and convexity properties of relative entropy under various norm settings.

Contribution

It provides new results on the Monge problem for Wiener space with Sobolev and infinite norms, and establishes convexity of relative entropy along geodesics.

Findings

Solved the Monge problem for Wiener space with Sobolev norms.

Proved 1-convexity and C-convexity of relative entropy along geodesics.

Analyzed optimal transport with different norms on Wiener space.

Abstract

In this paper we study two basic facts of optimal transportation on Wiener space W. Our first aim is to answer to the Monge Problem on the Wiener space endowed with the Sobolev type norm (k,gamma) to the power of p (cases p = 1 and p > 1 are considered apart). The second one is to prove 1-convexity (resp. C-convexity) along (constant speed) geodesics of relative entropy in (P2(W);W2), where W is endowed with the infinite norm (resp. with (k,gamma) norm), and W2 is the 2-distance of Wasserstein.