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

A metric space X has Markov-type 2 if for any reversible finite-state Markov chain {Z,} (with Z₀ chosen according to the stationary distribution) and any map f from the state space to X, the distance D _t from f(Z₀) to f(Z_t) satisfies E(D _t²) ≤ K² t E(D₁²) for some K = K(X) < ∞. 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 < ∞, any Lipschitz mapping from a subset of L_p to L_q has a Lipschitz extension defined on all of L_p.