Exponential contraction in Wasserstein distance on static and evolving manifolds

TL;DR

This paper establishes exponential contraction rates in Wasserstein distance for heat semigroups on static and evolving Riemannian manifolds under weak curvature conditions, extending previous results and providing explicit estimates.

Contribution

It provides explicit exponential contraction estimates for heat semigroups on static and evolving manifolds without requiring non-negative Ricci curvature.

Findings

Exponential contraction in Wasserstein distance is proven under weak curvature conditions.

Results extend to manifolds evolving under geometric flows.

Gradient estimates with exponential contraction are obtained for time-inhomogeneous semigroups.

Abstract

In this article, exponential contraction in Wasserstein distance for heat semigroups of diffusion processes on Riemannian manifolds is established under curvature conditions where Ricci curvature is not necessarily required to be non-negative. Compared to the results of Wang (2016), we focus on explicit estimates for the exponential contraction rate. Moreover, we show that our results extend to manifolds evolving under a geometric flow. As application, for the time-inhomogeneous semigroups, we obtain a gradient estimate with an exponential contraction rate under weak curvature conditions, as well as uniqueness of the corresponding evolution system of measures.