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{1176993604,
author = {Lyons, Terry},
title = {A Simple Criterion for Transience of a Reversible Markov Chain},
journal = {Ann. Probab.},
volume = {11},
number = {4},
year = {1983},
pages = { 393-402},
language = {en},
url = {http://dml.mathdoc.fr/item/1176993604}
}
Lyons, Terry. A Simple Criterion for Transience of a Reversible Markov Chain. Ann. Probab., Tome 11 (1983) no. 4, pp. 393-402. http://gdmltest.u-ga.fr/item/1176993604/