Stability and ergodicity of piecewise deterministic Markov processes

The main goal of this paper is to establish some equivalence results on stability, recurrence between a piecewise deterministic Markov process (PDMP for short) {X(t)} and an embedded discrete-time Markov chain {¿_n} generated by a Markov kernel G that can be explicitly characterized in terms of the three local characteristics of the PDMP contrary to the resolvent kernel. First we establish some important results characterizing {¿_n} as a sampling of the PDMP {X(t)} and deriving a connection between the probability of the first return time to a set for the discrete-time Markov chains generated by G and the resolvent kernel R of the PDMP. From these results we obtain equivalence results regarding recurrence and positive recurrence between {X(t)} and {¿_n}.