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, 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 < ∞, any Lipschitz mapping from a subset of L p to L q has a Lipschitz extension defined on all of L p .

Embeddings of Discrete Groups and the Speed of Random Walks

“…Theorem 1.1 isolates a geometric property (uniform smoothness) of the target space X which lies at the heart of the phenomenon discovered by Guentner and Kaminker. Our proof is a modification of the martingale method developed by Naor, Peres, Schramm and Sheffield in [28] for estimating the speed of stationary reversible Markov chains in uniformly smooth Banach spaces. This method requires several adaptations in the present setting since the random walk {W t } ∞ t=0 is not stationary-we refer to Section 2 for the details.…”

“…Its proof is a modification of the method that was used in [28] to study the Markov type of uniformly smooth Banach spaces.…”

“…The above corollary about the extension of group actions was previously noted in [6] under the additional restriction that 1 < p 2Z, as a simple corollary of an abstract extension theorem due to Hardin [22] (alternatively this is also a corollary of the classical Plotkin-Rudin theorem [33,35] We shall now pass to the proof of Theorem 2.1. We will use uniform smoothness via the following famous inequality due to Pisier [31] (for the explicit constant below see Theorem 4.2 in [28]). …”

“…Since for p ∈ [1, 2] the metric space L p , x − y p/2 2 embeds isometrically into L 2 (see [42]), it follows that L p has Markov type p with constant 1. For p > 2 it was shown in [28] that L p has Markov type 2 with constants O √ p . We refer to [28] for a computation of the Markov type of various additional classes of metric spaces.…”

A universal Lipschitz extension property of Gromov hyperbolic spaces

“…In the forthcoming formulations, a subset of M is said to be − dense if its distance 2 from each point of M is less than , and − separated if the distance between every two distinct points of the set is more than or equal to . Proposition 2.1 (see [NPSS,Corollary 6.2…”