A support property for infinite dimensional interacting diffusion processes

Abstract

The Dirichlet form associated with the intrinsic gradient on Poisson space is known to be quasi-regular on the complete metric space $\ddot\Gamma=$ $\{Z_+$-valued Radon measures on $\IR^d\}$. We show that under mild conditions, the set $\ddot\Gamma\setminus\Gamma$ is $\e$-exceptional, where $\Gamma$ is the space of locally finite configurations in $\IR^d$, that is, measures $\gamma\in\ddot\Gamma$ satisfying $\sup_{x\in\IR^d}\gamma(\{x\})\leq 1$. Thus, the associated diffusion lives on the smaller space $\Gamma$. This result also holds for Gibbs measures with superstable interactions.