Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces

A metric space X has Markov type 2, if for any reversible flnite-state Markov chain fZtg (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfles E(D 2) • K 2 tE(D 2) for some K = K(X) 2) has Markov type 2; this proves a conjecture of Ball. 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 1982, by showing that for 1 < q < 2 < p < 1, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension deflned on all of Lp.

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