Contraction and regularizing properties of heat flows in metric measure spaces

Transportation inequalities for Markov kernels and their applications

“…In Section 4, we extend the results of [23] in a different direction. By introducing a logarithmic term in the dynamic dual definition of the Hellinger-Kantorovich distance, we obtain a new family of "entropic" distance-like functions T a,b which generalize the Rényi divergence.…”

“…The goal of this paper is to build upon recent results of G. Luise and G. Savaré [23] on contraction properties of the flow of a heat semigroup in spaces of measures. There, the authors study a "dynamic dual" formulation of various distances between probability measures on a metric measure space, including the Kantorovich-Wasserstein and Hellinger distances as well as a family of Hellinger-Kantorovich distances HK α introduced in [22].…”

“…The first goal of the present paper is to observe that the techniques of [23] are not limited to the setting of RCD(K, ∞) spaces. In obtaining (1.1), the key ingredient is the fact that RCD(K, ∞) spaces satisfy a reverse Poincaré inequality of the form…”

“…In Sections 5 and 6, we discuss a number of examples of spaces where these techniques apply, beyond the RCD(K, ∞) setting discussed in [23], and applications to questions such as convergence to equilibrium and quasi-invariance of measures. These examples include Langevin dynamics driven by Lévy processes, semigroups arising in sub-Riemannian geometry, infinite-dimensional spaces modeled on abstract Wiener space, and others.…”

“…In some applications, P will be taken to be a Markov semigroup P t , which may or may not be symmetric with respect to some reference measure m. Our setting is similar to [20]. This is more general than the setting of [23], which only considered the symmetric semigroup P t generated by the Cheeger energy with respect to the given gradient and a reference measure.…”