Weak Convergence of $n$-Particle Systems Using Bilinear Forms

Abstract

The paper is concerned with the weak convergence of $n$-particle processes to deterministic stationary paths as $n\to\infty$. A Mosco type convergence of a class of bilinear forms is introduced. The Mosco type convergence of bilinear forms results in a certain convergence of the resolvents of the $n$-particle systems. Based on this convergence a criterion in order to verify weak convergence of invariant measures is established. Under additional conditions weak convergence of stationary $n$-particle processes to stationary deterministic paths is proved. The method is applied to the particle approximation of a Ginzburg-Landau type diffusion. The present paper is in close relation to the paper L\"obus (2011/2012). Different definitions of bilinear forms and versions of Mosco type convergence are introduced. Both papers demonstrate that the choice of the form and the type of convergence relates to the particular particle system