Existence Of The Invariant Measure Of Piecewise Determinisitic Markov Processes

In this paper, we study a form of stability for a general family of nondif-fusion 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 is equivalent to the existence of a cr梖inite invariant measure for a Markov kernel G lined to the resolvent operater U of the PDMP, satisfying a boundedness condition : irS(E) < oo. Here, since we don't require any additional assumptions to establish this equivalence, we generalize existing results of Costa (1990), Davis (1993),Costa and Du-four(1999) and Liu Guoxin (1998) in the literature . They are mainly based on a modified Foster-Lyapunov criterion for the case in which the Markov chain above in either recurrent or weak Feller .