Generalized Resolvents and Harris Recurrence of Markov Processes

In this paper we consider a φ-irreducible continuous parameter Markov process Φ whose state space is a general topological space. The recurrence and Harris recurrence structure of Φ is developed in terms of generalized forms of resolvent chains, where we allow statemodulated resolvents and embedded chains with arbitrary sampling distributions. We show that the recurrence behavior of such generalized resolvents classifies the behavior of the continuous time process; from this we prove that hitting times on the small sets of a generalized resolvent chain provide criteria for, successively, (i) Harris recurrence ofΦ (ii) the existence of an invariant probability measure π (or positive Harris recurrence of Φ) and (iii) the finiteness of π(f) for arbitrary f .