Stability of piecewise-deterministic Markov processes

In this paper, we study a form of stability for a general family of nondiff usion Markov processes known in the literature as piecewise-deterministic M arkov process (PDMP). By stability here we mean the existence of an invaria nt probability measure for the PDMP. It is shown that the existence of such an invariant probability measure is equivalent to the existence of a sigma -finite invariant measure for a Markov kernel G linked to the resolvent ope rator U of the PDMP, satisfying a boundedness condition or, equivalently, a Radon-Nikodym derivative. Here we generalize existing results of the liter ature [O. Costa, J. Appl. Prob., 27, (1990), pp. 60-73; M. Davis, Markov Mo dels and Optimization, Chapman and Hall, 1993] since we do not require any additional assumptions to establish this equivalence. Moreover, we give suf ficient conditions to ensure the existence of such a sigma-finite measure s atisfying the boundedness condition. They are mainly based on a modified Fo ster-Lyapunov criterion for the case in which the Markov chain generated by G is either recurrent or weak Feller. To emphasize the relevance of our re sults, we study three examples and in particular, we are able to generalize the results obtained by Costa and Davis on the capacity expansion model.