The semigroup of the Glauber dynamics of a continuous system of free particles
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We study properties of the semigroup $(e^{-tH})_{t\ge 0}$ on the space $L^ 2(Γ_X,π)$, where $Γ_X$ is the configuration space over a locally compact second countable Hausdorff topological space $X$, $π$ is a Poisson measure on $Γ_X$, 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 $Γ_X$, 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 $γ\inΓ_X$, will never leave $Γ_X$ and has cadlag sample paths in $Γ_X$.