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

TL;DR

This paper studies jump-move Markov processes in Polish spaces, establishing conditions for Feller properties and ergodicity, with applications to interacting particle systems including Gibbs measures with specific potentials.

Contribution

It provides new conditions for Feller semigroup formation and ergodic convergence in jump-move processes, especially for systems with births and deaths.

Findings

Conditions for Feller semigroup and generator derivation.

Ergodic convergence with geometric rate under certain conditions.

Explicit stationary Gibbs measures for specific potentials.

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. In 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 the convergence to a stationary measure with a geometric rate of convergence. Throughout the article, we illustrate our results by general examples of systems of interacting particles in R d 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 R d. Our examples include in particular Gibbs measure associated to repulsive Lennard-Jones potentials and to Riesz potentials.