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

TL;DR

This paper establishes a new approach using Mosco type convergence of bilinear forms to analyze the weak convergence of large particle systems to deterministic paths, with applications to Ginzburg-Landau diffusions.

Contribution

It introduces a novel Mosco type convergence framework for bilinear forms that links to the weak convergence of invariant measures in particle systems.

Findings

Proves weak convergence of stationary particle processes to deterministic paths.

Develops a criterion for verifying weak convergence of invariant measures.

Applies the method to particle approximation of Ginzburg-Landau type diffusion.

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.