Reversible birth-and-death dynamics in continuum: free-energy dissipation and attractor properties

Abstract

We consider continuous-time birth-and-death dynamics in ${\mathbb{R}}^d$ that admit at least one infinite-volume Gibbs point process based on area interactions as a reversible measure. For a large class of starting measures, we show that the specific relative entropy decays along trajectories, and that all possible long-time weak limit points are also Gibbs point processes with respect to the same interaction. Our proof rests on a representation of the entropy dissipation in terms of the Palm version of the propagated measure.