Contraction and regularizing properties of heat flows in metric measure spaces

Abstract

We illustrate some novel contraction and regularizing properties of the Heat flow in metric-measure spaces that emphasize an interplay between Hellinger-Kakutani, Kantorovich-Wasserstein and Hellinger-Kantorvich distances. Contraction properties of Hellinger-Kakutani distances and general Csiszár divergences hold in arbitrary metric-measure spaces and do not require assumptions on the linearity of the flow.When weaker transport distances are involved, we will show that contraction and regularizing effects rely on the dual formulations of the distances and are strictly related to lower Ricci curvature bounds in the setting of RCD(K, ∞) metric measure spaces. Contents2010 Mathematics Subject Classification. Primary: 49Q20, 47D07; Secondary: 30L99.

Transportation inequalities for Markov kernels and their applications

“…Our setting is similar to [25]. This is more general than the setting of [28], which only considered the symmetric semigroup P t generated by the Cheeger energy with respect to the given gradient and a given reference measure.…”

“…In this section, we consider the family of Hellinger-Kantorovich distances studied in [26,27,28]. We focus on the so-called dynamic dual formulation of these distances, in which they may be defined as the supremum of a difference of integrals over a class of subsolutions of a Hamilton-Jacobi-type equation in time and space variables.…”

“…Item 2 holds because ϕ ∈ A ca,cb if and only if cϕ ∈ A a,b . For item 3, in the notation of [28,Eq. (39)] (see also [27,Section 8.4]), we have H K 2 α = W α/4,1 , and the general statement follows using item 2.…”

“…(39)] (see also [27,Section 8.4]), we have H K 2 α = W α/4,1 , and the general statement follows using item 2. Item 4 can be found as Proposition 2.10 of [28], but goes back at least as far as [30,Section 3]; see also other references in [28]. Item 5 is almost Proposition 2.8 of [28], but there is a slight difference because our class of functions A 0,1 is required to be Lipschitz in space, whereas [28,Eq.…”

“…The goal of this paper is to build upon recent results of G. Luise and G. Savaré [28] 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 H K α introduced in [27].…”