Uniformly integrable potential operators and the existence of quasi-stationary distributions

Abstract

We consider irreducible Markov processes having lifetimes with finite means. We show-using the theory developed in [8]-that the condition of uniform integrability of the potential operators implies the existence of quasi-stationary distributions. We also show that this condition (weaker than the usually assumed compactness of operators) is not necessary for the existence of quasi-stationary distributions. As an auxiliary result we prove the existence in this context of a probability excessive irreducibility measure.