Graphon particle systems with common noise

TL;DR

This paper analyzes a nonlinear graphon particle system influenced by individual and shared noise, establishing a law of large numbers for empirical and interaction measures using advanced probabilistic techniques.

Contribution

It introduces a framework for graphon-based particle systems with common noise and proves convergence results in a non-Markovian setting.

Findings

Law of large numbers for empirical measures

Convergence of interaction measures under graphon interactions

Use of generalized Wasserstein metrics for non-Markovian systems

Abstract

We study a nonlinear graphon particle system driven by both idiosyncratic and common noise, where interactions are governed by a graphon and represented as positive finite measures. Each particle evolves via a McKean-Vlasov-type SDE with graphon-weighted conditional laws. We prove a law of large numbers for the empirical and interaction measures, using generalized Wasserstein metrics and weak convergence techniques suited for the non-Markovian structure induced by common noise.