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 . Extending results of Luise and Savaré on Hellinger–Kantorovich contraction inequalities for the particular case of the heat semigroup on an metric space, we show that more generally, such contraction inequalities are equivalent to reverse Poincaré inequalities. We also adapt the “dynamic dual” formulation of the Hellinger–Kantorovich distance to define a new family of divergences on which generalize the Rényi divergence, and we show that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang Harnack inequalities. We discuss applications including results on the convergence of Markov processes to equilibrium, and on quasi-invariance of heat kernel measures in finite and infinite-dimensional groups.
Fabrice Baudoin has been supported by National Science Foundation grant DMS-1901315. Nathaniel Eldredge has been supported by a grant from the Simons Foundation (#355659, N.E.).
The authors are grateful for helpful discussions with Maria Gordina, Martin Hairer, Ronan Herry, Kazumasa Kuwada, Xue-Mei Li, and Giuseppe Savaré. We also thank the anonymous referee for their careful reading and useful suggestions. This article was completed during a sabbatical visit by author N. Eldredge to the Department of Mathematics at the University of Connecticut; he would like to thank the Department and especially Maria Gordina for their hospitality, especially in view of the difficult circumstances created by the COVID-19 pandemic.
"Transportation inequalities for Markov kernels and their applications." Electron. J. Probab. 26 1 - 30, 2021. https://doi.org/10.1214/21-EJP605