Markov chains in a field of traps (Q1266794)

The paper considers a Markov chain \((X_n)\) with countable state space \(S\). In addition, there is given an i.i.d. sequence of random variables \((T_n)\) the values of which are subsets of \(S\). Moreover, the \(T_n\) depend on some environment which is independent of \((X_n)\). The paper characterizes the situations where (i) \(\pi(x): =P_x[X_n\in T_1\) for some \(n]=1\) for all \(x\) and (ii) \(\widetilde\pi(x): =P_x[X_n\in T_n\) for some \(n]=1\) for all \(x\), and the authors interrelate these two properties. It is easy to see that \(\pi(x)\leq\widetilde\pi(x)\).