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July, 1995 Martin Capacity for Markov Chains
Itai Benjamini, Robin Pemantle, Yuval Peres
Ann. Probab. 23(3): 1332-1346 (July, 1995). DOI: 10.1214/aop/1176988187


The probability that a transient Markov chain, or a Brownian path, will ever visit a given set $\Lambda$ is classically estimated using the capacity of $\Lambda$ with respect to the Green kernel $G(x, y)$. We show that replacing the Green kernel by the Martin kernel $G(x, y)/G(0, y)$ yields improved estimates, which are exact up to a factor of 2. These estimates are applied to random walks on lattices and also to explain a connection found by Lyons between capacity and percolation on trees.


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Itai Benjamini. Robin Pemantle. Yuval Peres. "Martin Capacity for Markov Chains." Ann. Probab. 23 (3) 1332 - 1346, July, 1995.


Published: July, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0840.60068
MathSciNet: MR1349175
Digital Object Identifier: 10.1214/aop/1176988187

Primary: 60J45
Secondary: 60J10 , 60J15 , 60J65 , 60K35

Keywords: Brownian motion , capacity , hitting probability , Markov chain , percolation , tree

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 3 • July, 1995
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