Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition

TL;DR

This paper establishes the equivalence between Wasserstein distance contraction properties and the curvature-dimension condition in metric measure spaces, extending known results from Riemannian manifolds and exploring links with functional inequalities.

Contribution

It generalizes the equivalence of contraction inequalities and curvature-dimension conditions from Riemannian manifolds to metric measure spaces, including the converse implication.

Findings

Contraction inequalities are equivalent to curvature-dimension conditions.

Generalization of Wasserstein contraction properties to metric measure spaces.

Connections between contraction inequalities and functional inequalities.

Abstract

The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times. On the other hand, in a compact Riemannian manifold, it implies a same-time Wasserstein contraction property for this semigroup. In this work we generalize the latter result to metric measure spaces and more importantly prove the converse: contraction inequalities are equivalent to curvature-dimension conditions. Links with functional inequalities are also investigated.