The semigroup of the Glauber dynamics of a continuous system of free particles

Abstract

We study properties of the semigroup $(e^{-tH}{)}_{t\ge 0}$ on the space $L^2({\Gamma }_X,\pi )$, where ${\Gamma }_X$ is the configuration space over a locally compact second countable Hausdorff topological space $X$, $\pi$ is a Poisson measure on ${\Gamma }_X$, and $H$ is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov semigroup of kernels $(P_t{)}_{t\ge 0}$ and, using it, we prove the main results of the paper: the Feller property of the semigroup $(P_t{)}_{t\ge 0}$ with respect to the vague topology on the configuration space ${\Gamma }_X$, and the ergodic property of $(P_t{)}_{t\ge 0}$. Following an idea of D. Surgailis, we also give a direct construction of the Glauber dynamics of a continuous infinite system of free particles. The main point here is that this process can start in every $\gamma \in {\Gamma }_X$, will never leave ${\Gamma }_X$ and has cadlag sample paths in ${\Gamma }_X$.