## Infinite interaction diffusion particles I: Equilibrium process and its scaling limit

Venue: | Forum Math |

Citations: | 4 - 3 self |

### BibTeX

@ARTICLE{Kondratiev_infiniteinteraction,

author = {Yuri Kondratiev and Eugene Lytvynov and Michael Röckner},

title = {Infinite interaction diffusion particles I: Equilibrium process and its scaling limit},

journal = {Forum Math},

year = {},

pages = {9--43}

}

### OpenURL

### Abstract

A stochastic dynamics (X(t))t≥0 of a classical continuous system is a stochastic process which takes values in the space Γ 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 [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form EΓ µ on L2 (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 EΓ µ on the set of smooth cylinder functions. We then show that, for any Dirichlet form EΓ µ, there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], 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 (Rd).