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

## TL;DR

This paper studies reversible birth-and-death processes in continuous space, demonstrating entropy decay and convergence to Gibbs measures, revealing attractor properties and energy dissipation mechanisms.

## Contribution

It introduces a framework for analyzing entropy dissipation and long-term behavior of birth-and-death dynamics with Gibbs measures in continuum.

## Key findings

- Entropy decreases along trajectories for a broad class of initial measures.
- Long-time limits of the process are Gibbs point processes with the same interaction.
- The proof uses a novel representation of entropy dissipation via Palm measures.

## 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.