A criterion on the exponential ergodicity in the Bounded-Lipschitz distance for some piecewise-deterministic Markov processes with random switching

We are concerned with a subclass of piecewise-deterministic Markov processes (PDMPs) that evolve on a Polish space through random jumps at exponentially distributed time intervals, while the behaviour between jumps is governed by a finite collection of flows, selected randomly at the jump times. The main goal of this paper is to provide a set of conditions sufficient for the exponential ergodicity of the corresponding transition semigroup in terms of the bounded Lipschitz distance.