 03334 Yu. Kondratiev, E.Lytvynov
 Glauber dynamics of continuous particle systems
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Jul 16, 03

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Abstract. This paper is devoted to the construction and study of
an equilibrium Glaubertype 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 $\R^d$, 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
$\delta{:=}\int_{\R^d}(1e^{\phi(x)})\, z\, dx<1$, 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 of L. Bertini, N. Cancrini, and F. Cesi.
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