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 $\mu$.