Self-improvement of the Bakry-\'Emery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces

TL;DR

This paper establishes that heat flow in RCD(K,∞) metric measure spaces exhibits Wasserstein contraction properties, linking Ricci curvature bounds with Bakry-Émery conditions through advanced non-smooth analysis techniques.

Contribution

It extends Bakry's argument to non-smooth spaces, providing new estimates and demonstrating Wasserstein contraction in RCD(K,∞) spaces.

Findings

Proves Wasserstein contraction for heat flow in RCD(K,∞) spaces.

Provides optimal W_infinity-coupling estimates.

Establishes equivalence between Ricci bounds and Bakry-Émery condition in non-smooth settings.

Abstract

We prove that the linear heat flow in a RCD(K,\infty) metric measure space (X,d,m) satisfies a contraction property with respect to every L^p-Kantorovich-Rubinstein-Wasserstein distance. In particular, we obtain a precise estimate for the optimal W_\infty-coupling between two fundamental solutions in terms of the distance of the initial points. The result is a consequence of the equivalence between the RCD(K,\infty) lower Ricci bound and the corresponding Bakry-\'Emery condition for the canonical Cheeger-Dirichlet form in (X,d,m). The crucial tool is the extension to the non-smooth metric measure setting of the Bakry's argument, that allows to improve the commutation estimates between the Markov semigroup and the Carr\'e du Champ associated to the Dirichlet form. This extension is based on a new a priori estimate and a capacitary argument for regular and tight Dirichlet forms that are of independent interest.