Evolution variational inequality and Wasserstein control in variable curvature context

TL;DR

This paper extends the analysis of metric measure spaces with variable Ricci curvature bounds by introducing new curvature-dimension conditions, variational inequalities, and contraction estimates in the Wasserstein space.

Contribution

It introduces a new evolution variational inequality for gradient flows, defines the entropic curvature-dimension condition $CD^e(ppa,N)$, and establishes its stability and equivalence to $CD^*(ppa,N)$ in non-branching spaces, along with a new Wasserstein contraction estimate.

Findings

Introduction of a new evolution variational inequality for gradient flows.

Definition and stability of the entropic curvature-dimension condition $CD^e(ppa,N)$.

Establishment of a differential Wasserstein contraction estimate.

Abstract

In this note we continue the analysis of metric measure space with variable ricci curvature bounds. First, we study $(\kappa,N)$-convex functions on metric spaces where $\kappa$ is a lower semi-continuous function, and gradient flow curves in the sense of a new evolution variational inequality that captures the information that is provided by $\kappa$. Then, in the spirit of previous work by Erbar, Kuwada and Sturm \cite{erbarkuwadasturm} we introduce an entropic curvature-dimension condition $CD^e(\kappa,N)$ for metric measure spaces and lower semi-continuous $\kappa$. This condition is stable with respect to Gromov convergence and we show that is equivalent to the reduced curvature-dimension condition $CD^*(\kappa,N)$ provided the space is non-branching. Finally, we introduce a Riemannian curvature-dimension condition in terms of an evolution variational inequality on the Wasserstein space. A consequence is a new differential Wasserstein contraction estimate.