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A stochastic dynamics ( X ( t )) t ≥0 of a classical continuous system is a stochastic process which takes values in the space Г of all locally finite subsets (configurations) in ℝ d and which has a Gibbs measure μ as an invariant measure. We assume that μ corresponds to a symmetric pair potential φ( x − y ). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics—the so-called gradient stochastic dynamics, or interacting Brownian particles—has been investigated. By using the theory of Dirichlet forms from Ma Z.-M., Röckner M.: An Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin 1992, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ℰ μ Г on L 2 (Г; μ), and under general conditions on the potential φ, prove its closability. For a potential φ having a “weak” singularity at zero, we also write down an explicit form of the generator of ℰ μ Г on the set of smooth cylinder functions. We then show that, for any Dirichlet form ℰ μ Г , there exists a diffusion process that is properly associated with it. Finally, in a way parallel to Grothaus M., Kondratiev Yu. G., Lytvynov E., Röckner M.: Scaling limit of stochastic dynamics in classical continuous systems. Ann. Prob. 31 (2003), 1494-1532 , we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C(0, ∞), 𝒟′), where 𝒟′ is the dual space of 𝒟:= C 0 ∞ (ℝ d ).
Forum Mathematicum – de Gruyter
Published: Jan 26, 2006
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