Markov type constants, flat tori and Wasserstein spaces

Abstract

Let $M_p(X,T)$ denote the Markov type p constant at time T of a metric space X, where $p \ge 1$. We show that $M_p(Y,T) \le M_p(X,T)$ in each of the following cases: (a) X and Y are geodesic spaces and Y is covered by X via a finite-sheeted locally isometric covering, (b) Y is the quotient of X by a finite group of isometries, (c) Y is the $L^p$-Wasserstein space over X. As an application of (a) we show that all compact flat manifolds have Markov type 2 with constant 1. In particular the circle with its intrinsic metric has Markov type 2 with constant 1. This answers the question raised by S.-I. Ohta and M. Pichot. Parts (b) and (c) imply new upper bounds for Markov type constants of the $L^p$-Wasserstein space over ${\mathbb {R}}^d$. These bounds were conjectured by A. Andoni, A. Naor and O. Neiman. They imply certain restrictions on bi-Lipschitz embeddability of snowflakes into such Wasserstein spaces.