Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

Abstract

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 ℝd 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 ℰμГ on L2(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 ℰμГ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ℰμГ, 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:= C0∞ (ℝd).