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 $\mathcal{E}_{π_σ}^Γ$ of the Poisson measure $π_σ$ in terms of the extrinsic one $\mathcal{E}_{π_σ,H_σ^{X}}^{P}$. More precisely, replacing $π_σ$ by a Gibbs measure $μ$ on the configuration space $Γ_{X}$ we derive a relation between the intrinsic pre-Dirichlet form $\mathcal{E}_μ^Γ$ of the measure $μ$ and the extrinsic one $\mathcal{E}_{μ, H_σ^{X}}^{P}$. As a consequence we prove the closability of $\mathcal{E}_μ^Γ$ on $L^{2}(Γ_{X},μ)$ under very general assumptions on the interaction potential of the Gibbs measures $μ$.