Transportation inequalities for Markov kernels and their applications.

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 $\mathcal{P}(X)$. We extend results of Luise and Savar\'e on contraction inequalities for the heat semigroup on $\mathcal{P}(X)$ when $X$ is an $RCD(K,\infty)$ metric space, with respect to the Hellinger and Kantorovich--Wasserstein distances, and explore applications to more general Markov kernels satisfying a reverse Poincar\'e inequality. A key idea is a ``dynamic dual'' formulation of these transportation distances. We also modify this formulation to define a new family of divergences on $\mathcal{P}(X)$ which generalize the R\'enyi 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.