On a relation between intrinsic and extrinsic Dirichlet forms for interacting particle systems

Abstract

In this paper we extend the result obtained in \cite{AKR97} (see also \cite {AKR96}) on the representation of the intrinsic pre- Dirichlet form ${\mathcal{E}}_{{\pi }_{\sigma }}^{\Gamma }$ of the Poisson measure ${\pi }_{\sigma }$ in terms of the extrinsic one ${\mathcal{E}}_{{\pi }_{\sigma },H_{\sigma }^X}^P$. More precisely, replacing ${\pi }_{\sigma }$ by a Gibbs measure $\mu$ on the configuration space ${\Gamma }_X$ we derive a relation between the intrinsic pre-Dirichlet form ${\mathcal{E}}_{\mu }^{\Gamma }$ of the measure $\mu$ and the extrinsic one ${\mathcal{E}}_{\mu ,H_{\sigma }^X}^P$. As a consequence we prove the closability of ${\mathcal{E}}_{\mu }^{\Gamma }$ on $L^2({\Gamma }_X,\mu )$ under very general assumptions on the interaction potential of the Gibbs measures $\mu$.

22 pages