Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces
Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces
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光滑巴纳赫空间和格罗莫夫双曲度量空间中的马尔可夫链
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发表时间：
2004
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通讯作者：
S. Sheffield
中科院分区：
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作者：
A. Naor;Y. Peres;O. Schramm;S. Sheffield
度量空间X具有马尔可夫型2，如果对于任意可逆的非稳态马尔可夫链fZtg（其中Z0根据平稳分布选择）和任意从状态空间到X的映射f，从f（Z0）到f（Zt）的距离Dt满足E（d2）•k2te （d2）对于某些K = K(X) 2)具有马尔可夫型2；这证明了鲍尔的一个猜想。我们还证明了树、双曲群和缩负曲率的单连通黎曼流形具有马尔可夫2型。我们的结果被用于解决关于Lipschitz扩展和嵌入的几个猜想。特别地，我们回答了Johnson和Lindenstrauss在1982年提出的一个问题，证明了对于1 < q < 2 < p < 1，从Lp的一个子集到Lq的任何Lipschitz映射在所有Lp上都有一个Lipschitz扩展。
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.