The Annals of Probability

Lyapunov functions for random walks and strings in random environment

Francis Comets, Mikhail Menshikov, and Serguei Popov

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We study two typical examples of countable Markov chains in random environment using the Lyapunov functions method: random walk and random string in random environment. In each case we construct an explicit Lyapunov function. Investigating the behavior of this function, we get the classification for recurrence, transience, ergodicity. We obtain new results for random strings in random environment, though we simply review well-known results for random walks using our approach.

Article information

Ann. Probab., Volume 26, Number 4 (1998), 1433-1445.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Random walk random string Lyapunov function random medium disordered systems product of random matrices Lyapunov exponents


Comets, Francis; Menshikov, Mikhail; Popov, Serguei. Lyapunov functions for random walks and strings in random environment. Ann. Probab. 26 (1998), no. 4, 1433--1445. doi:10.1214/aop/1022855869.

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