Feller and ergodic properties of jump–move processes with applications to interacting particle systems

Abstract We consider Markov processes that alternate continuous motions and jumps in a general locally compact Polish space. Starting from a mechanistic construction, a first contribution of this article is to provide conditions on the dynamics so that the associated transition kernel forms a Feller semigroup, and to deduce the corresponding infinitesimal generator. As a second contribution, we investigate the ergodic properties in the special case where the jumps consist of births and deaths, a situation observed in several applications including epidemiology, ecology, and microbiology. Based on a coupling argument, we obtain conditions for convergence to a stationary measure with a geometric rate of convergence. Throughout the article, we illustrate our results using general examples of systems of interacting particles in Rᵈ with births and deaths. We show that in some cases the stationary measure can be made explicit and corresponds to a Gibbs measure on a compact subset of Rᵈ. Our examples include in particular Gibbs measures associated to repulsive Lennard-Jones potentials and to Riesz potentials.