Stationarity and uniform in time convergence for the graphon particle system

TL;DR

This paper analyzes the long-term behavior of heterogeneously interacting diffusive particle systems with graphon-based interactions, establishing ergodicity, uniform convergence, and Euler approximation rates.

Contribution

It introduces a framework for studying long-time dynamics of graphon-based particle systems, proving exponential ergodicity and uniform-in-time convergence results.

Findings

Exponential ergodicity of the systems under convexity assumptions

Uniform-in-time law of large numbers for marginal distributions

Explicit convergence rates for Euler approximation

Abstract

We consider the long time behavior of heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear diffusions whose probability distributions are fully coupled. Under suitable assumptions, including a certain convexity condition, we show the exponential ergodicity for both systems, establish the uniform-in-time law of large numbers for marginal distributions as the number of particles increases, and introduce the uniform-in-time Euler approximation. The precise rate of convergence of the Euler approximation is provided.