Sufficient conditions for regularity, recurrence and ergodicity of Markov processes

Abstract

Two sets of conditions are found on the Q-matrix of an irreducible Markov process on a countably infinite state space which ensure that Q is regular; the first set also implies that the Markov process is ergodic, the second that it is recurrent. If the process is not irreducible, the conditions still imply regularity, and then either non-dissipativity or ‘ultimate recurrence’ (reducible analogues of ergodicity and recurrence) of the process. Conditions sufficient for ergodicity or recurrence of the process in the non-regular and regular case are also given. The results parallel (and use) results for discrete time Markov chains, and the known discrete time recurrence condition is extended to the reducible case. The conditions are illustrated by a competition process example.