Markov Chains in a Field of Traps

Abstract

We consider a Markov chain on a countable state space, on which is placed a random field of traps, and ask whether the chain gets trapped almost surely. We show that the quenched problem (when the traps are fixed) is equivalent to the annealed problem (when the traps are updated each unit of time) and give a criterion for almost sure trapping versus positive probability of nontrapping. The hypotheses on the Markov chain are minimal, and in particular, our results encompass the results of den Hollander, Menshikov and Volkov (1995).