Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces

Abstract

A metric space $X$ has Markov-type $2$ if for any reversible finite-state Markov chain $\{Z_t\}$ (with $Z_0$ chosen according to the stationary distribution) and any map $f$ from the state space to $X$, the distance $D_t$ from $f(Z_0)$ to $f(Z_t)$ satisfies $\mathbb{E}(D_t^2)\le K^{2}t\mathbb{E}(D_1^2)$ for some $K=K(X)<\infty$. This notion is due to K.Ball [2], who showed its importance for the Lipschitz extension problem. Until now, however, only Hilbert space (and metric spaces that embed bi-Lipschitzly into it) was known to have Markov-type 2. We show that every Banach space with modulus of smoothness of power-type $2$ (in particular, $L_p$ for $p>2$) has Markov-type $2$; this proves a conjecture of Ball (see [2, Section 6]). We also show that trees, hyperbolic groups, and simply connected Riemannian manifolds of pinched negative curvature have Markov-type $2$. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in [28, Section 2] by showing that for $1<q<2<p<\infty$, any Lipschitz mapping from a subset of $L_p$ to $L_q$ has a Lipschitz extension defined on all of $L_p$