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.
"Lyapunov functions for random walks and strings in random environment." Ann. Probab. 26 (4) 1433 - 1445, October 1998. https://doi.org/10.1214/aop/1022855869