Glauber dynamics of continuous particle systems

By Yuri Kondratiev, Eugene Lytvynov. 2003

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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 Γ of all locally finite subsets (configurations) in R d , we fix a Gibbs measure μ corresponding to a general pair potential ϕ and activity z > 0 . We consider a Dirichlet form E on L 2 ( Γ , μ ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E . In the case of a positive potential ϕ which satisfies δ : = ∫ R d ( 1 − e − ϕ ( x ) ) z d x 1 , we also prove that the generator H has a spectral gap ⩾ 1 − δ . Furthermore, for any pure Gibbs state μ, we derive a Poincare inequality. The results about the spectral gap and the Poincare inequality are a generalization and a refinement of a recent result from [Ann. Inst. H. Poincare Probab. Statist. 38 (2002) 91–108].

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6c35d7efcebdcf2a9bbe115e1df4db595b5bedaa