Interacting Brownian motions with measurable potentials
The author considers the infinitely dimensional diffusion process \[ dX^i_t = d B^i_t - \sum_{j=1, j\neq i}^{\infty} \tfrac{1}{2}\bigtriangledown \Phi (X_t^i - X_t^i) dt, \;\;1\leq i \leq {+\infty}, \] where the measurable function \(\Phi \) is the interacting potential, which is dominated by a function with some regularity properties. On the set of all locally finite configurations on \({\mathbb R^d}\), denoted by \(\Theta\), a Dirichlet form \(E\) and the set \(D\) of functions of \(L^2(\Theta, \mu)\) for which the Dirichlet form is finite, are considered, where \(\mu\) is a canonical (Gibbs) measure associated with \(\Phi\). The main result states that \((E, D)\) is closable on \(L^2(\Theta,\mu)\).
scientific article; zbMATH DE number 1192510