Stability of piecewise-deterministic Markov processes

We study a form of stability for a general family of non-diffusion Markov processes, known in the literature as piecewise-deterministic Markov process (PDMP). By stability here we mean the existence of an invariant probability measure for the PDMP. It is shown that the existence of such invariant probability measure is equivalent to the existence of a σ-finite invariant measure for a Markov kernel G linked to the resolvent operator U of the PDMP, satisfying a boundedness condition or, equivalently, a Radon-Nikodym derivative. Here we generalize existing results of the literature since we do not require any additional assumptions to establish this equivalence. Moreover, we give sufficient conditions to ensure the existence of such σ-finite measure satisfying the boundedness condition. They are mainly based on a modified Foster-Lyapunov criteria for the case in which the Markov chain generated by G is either recurrent or weak Feller. To emphasize the relevance of our results, three examples are studied in Dufour and Costa (1999) and in particular, we are able to generalize the results obtained by Costa (1990) and Davis (1993) on the capacity expansion model