The Annals of Probability

A Simple Criterion for Transience of a Reversible Markov Chain

Terry Lyons

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An old argument of Royden and Tsuji is modified to give a necessary and sufficient condition for a reversible countable state Markov chain to be transient. This Royden criterion is quite convenient and can, on occasion, be used as a substitute for the criterion of Nash-Williams [6]. The result we give here yields a very simple proof that the Nash-Williams criterion implies recurrence. The Royden criterion also yields as a trivial corollary that a recurrent reversible random walk on a state space $X$ remains recurrent when it is constrained to run on a subset $X'$ of $X$. An apparently weaker criterion for transience is also given. As an application, we discuss the transience of a random walk on a horn shaped subset of $\mathbb{Z}^d$.

Article information

Ann. Probab. Volume 11, Number 2 (1983), 393-402.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 31C25: Dirichlet spaces 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14]

Recurrence transience Markov chain symmetric Markov chain energy


Lyons, Terry. A Simple Criterion for Transience of a Reversible Markov Chain. Ann. Probab. 11 (1983), no. 2, 393--402. doi:10.1214/aop/1176993604.

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