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

Abstract

A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $Γ$ of all locally finite subsets (configurations) in $\Bbb R$ 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, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal E}_μ^Γ$ on $L^2(Γ;μ)$, 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 ${\cal E}_μ^Γ$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\cal E}_μ^Γ$, there exists a diffusion process that is properly associated with it. Finally, 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 $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of ${\cal D}{:=}C_0^\infty({\Bbb R})$.