Eulerian calculus for the displacement convexity in the Wasserstein distance

TL;DR

This paper presents a novel Eulerian approach to prove the displacement convexity of certain functionals on Riemannian manifolds with Ricci curvature bounds, avoiding reliance on optimal transport map regularity.

Contribution

It introduces a new proof method based on Eulerian calculus and metric gradient flow characterization, differing from traditional approaches.

Findings

Provides a new proof of displacement convexity without optimal transport map regularity

Extends convexity results to Riemannian manifolds with Ricci curvature bounds

Utilizes Eulerian calculus and metric gradient flow framework

Abstract

In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto-Westdickenberg and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.