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

We study properties of the semigroup (e −tH )t≥0 on the space L 2 ( X,�), where X is the configuration space over a locally compact second countable Hausdorff topological space X, � is a Poisson measure on X, and H is the generator of the Glauber dynamics. We explicitly construct the corresponding Markov semigroup of kernels (Pt)t≥0 and, using it, we prove the main results of the paper: the Feller property of the semigroup (Pt)t≥0 with respect to the vague topology on the configuration space X, and the ergodic property of (Pt)t≥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 ∈ X, will never leave X and has cadlag sample paths in X. 2000 AMS Mathematics Subject Classification: 60K35, 60J75, 60J80, 82C21