Exponential Ergodicity in the Bounded-Lipschitz Distance for Some Piecewise-Deterministic Markov Processes with Random Switching Between Flows

In this paper, we study a subclass of piecewise-deterministic Markovprocesses with a Polish state space, involving deterministic motion punctuatedby random jumps that occur at exponentially distributed time intervals. Overeach of these intervals, the process follows a flow, selected randomly among afinite set of all possible ones. Our main goal is to provide a set ofverifiable conditions guaranteeing the exponential ergodicity for suchprocesses (in terms of the bounded Lipschitz distance), which would refer onlyto properties of the flows and the transition law of the Markov chain given bythe post-jump locations. Moreover, we establish a simple criterion on theexponential ergodicity for a particular instance of these processes, applicableto certain biological models, where the jumps result from the action of aniterated function system with place-dependent probabilities.