Stability of Piecewise-Deterministic Markov Processes

In this paper, we study a form of stability for a general family of nondiffusion 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 an invariant probability measure is equivalent to the existence of a $\sigma$-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--Nikodým derivative. Here we generalize existing results of the literature [O. Costa, J. Appl. Prob., 27, (1990), pp. 60--73; M. Davis, Markov Models and Optimization, Chapman and Hall, 1993] since we do not require any additional assumptions to establish this equivalence. Moreover, we give sufficient conditions to ensure the existence of such a $\sigma$-finite measure satisfying the boundedness condition. They are mainly based on a modified Foster--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 results, we study three examples and in particular, we are able to generalize the results obtained by Costa and Davis on the capacity expansion model.