Transportation inequalities for Markov kernels and their applications

Abstract

We study the relationship between functional inequalities for a Markov kernel on a metric space X and inequalities of transportation distances on the space of probability measures P(X). We extend results of Luise and Savaré on contraction inequalities for the heat semigroup on P(X) when X is an RCD(K, ∞) metric space, with respect to the Hellinger and Kantorovich-Wasserstein distances, and explore applications to more general Markov kernels satisfying a reverse Poincaré inequality. A key idea is a "dynamic dual" formulation of these transportation distances. We also modify this formulation to define a new family of distancelike functions on P(X) which generalize the Rényi divergence, and relate them to reverse logarithmic Sobolev inequalities. Applications include results on the convergence of Markov processes to equilibrium, and on quasi-invariance of heat kernel measures in finite and infinite-dimensional groups.