ANNALES POLONICI MATHEMATICI 109.3 (2013) Piecewise-deterministic Markov processes by Jolanta Kazak (Katowice) Abstract. Poisson driven stochastic differential equations on a separable Banach space are examined. Some sufficient conditions are given for the asymptotic stability of a Markov operator P corresponding to the change of distribution from jump to jump. We also give criteria for the continuous dependence of the invariant measure for P on the intensity of the Poisson process. 1. Introduction. We will consider the stochastic differential equation of the form (1.1) dξ (t)= a(ξ (t))dt +  Θ σ(ξ (t),θ) N p (Λ ξ (dt), dθ) for t ≥ 0 with the initial condition (1.2) ξ (0) = ξ 0 , where (1.3) Λ ξ (t)= t  0 λ(ξ (s)) ds and (ξ (t)) t≥0 is a stochastic process with values in a separable Banach space X , the functions a and σ are deterministic, and N p is a Poisson random counting measure. In (1.3) the function λ : X → R + , called the intensity of the Poisson process, is bounded and Lipschitzian. The process Λ ξ influences the time at which jumps occur and it depends on the solution ξ of the prob- lem (1.1), (1.2). The process (N p (Λ ξ (t),A)) t≥0 describes the occurrence of jumps. The fact that N p depends on the solution is crucial. The solution ξ is a Markov process which is piecewise-deterministic. It evolves deterministically until a random time (depending on position) when it jumps to a new random state. Such processes feature significantly in con- temporary monographs devoted to Markov processes (see [3]). They have 2010 Mathematics Subject Classification : Primary 37A30; Secondary 93D20. Key words and phrases : Markov operators, asymptotic stability, Poisson driven differential equation. DOI: 10.4064/ap109-3-4 [279] c  Instytut Matematyczny PAN, 2013