The Annals of Probability

Martin Capacity for Markov Chains

Itai Benjamini, Robin Pemantle, and Yuval Peres

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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.

Article information

Ann. Probab., Volume 23, Number 3 (1995), 1332-1346.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J65: Brownian motion [See also 58J65] 60J15 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Capacity Markov chain hitting probability Brownian motion tree percolation


Benjamini, Itai; Pemantle, Robin; Peres, Yuval. Martin Capacity for Markov Chains. Ann. Probab. 23 (1995), no. 3, 1332--1346. doi:10.1214/aop/1176988187.

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