Method of Lyapunov functions for analysis of absorption and explosion in Markov chains

The main goal of this paper is to derive some sufficient conditions for testing the absorption, explosion, and nonexplosion of time-inhomogeneous Markov chains with a countable state space. The method of Lyapunov functions is used for this purpose. Several theorems concerned with such sufficient conditions are proved for a general class of Markov chains. Then they are applied to some problems in time-inhomogeneous birth-death processes and branching processes.