Glauber dynamics of continuous particle systems

Abstract

This paper is devoted to the construction and study of an equilibrium Glauber-type dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space $\Gamma$ of all locally finite subsets (configurations) in ${\Bbb R}^d$, we fix a Gibbs measure $\mu$ corresponding to a general pair potential $\phi$ and activity $z>0$. We consider a Dirichlet form $ \cal E$ on $L^2(\Gamma,\mu)$ which corresponds to the generator $H$ of the Glauber dynamics. We prove the existence of a Markov process $\bf M$ on $\Gamma$ that is properly associated with $\cal E$. In the case of a positive potential $\phi$ which satisfies $\delta{:=}\int_{{\Bbb R}^d}(1-e^{-\phi(x)}) z dx<1$, we also prove that the generator $H$ has a spectral gap $\ge1-\delta$. Furthermore, for any pure Gibbs state $\mu$, we derive a Poincar\'e inequality. The results about the spectral gap and the Poincar\'e inequality are a generalization and a refinement of a recent result by L. Bertini, N. Cancrini, and F. Cesi.