Transportation inequalities for Markov kernels and their applications

We study the relationship between functional inequalities for a Markov kernel on metric space X and of transportation distances probability measures P(X). Extending results Luise Savaré Hellinger–Kantorovich contraction particular case heat semigroup an RCD(K,∞) space, we show that more generally, such are equivalent to reverse Poincaré inequalities. also adapt “dynamic dual” formulation distance define new family divergences P(X) which generalize Rényi divergence, these logarithmic Sobolev Wang Harnack discuss applications including convergence processes equilibrium, quasi-invariance in finite infinite-dimensional groups.