A simple and general proof for the convergence of Markov processes to their mean-field limits

TL;DR

This paper provides a simple, general proof demonstrating that large stochastic systems modeled by Markov processes converge to their mean-field limits, using only basic mathematical tools.

Contribution

It introduces a straightforward proof technique for mean-field convergence that avoids complex methods like PDEs or operator semigroups.

Findings

Proof applies to a wide class of Markov processes

Convergence established using only Taylor's theorem and basic ODE results

Simplifies the validation of mean-field models in applications

Abstract

Mean-field models approximate large stochastic systems by simpler differential equations that are supposed to approximate the mean of the larger system. It is generally assumed that as the stochastic systems get larger (i.e., more people or particles), they converge to the mean-field models. Mean-field models are common in many fields, but their convergence is rarely proved. The existing approaches rely on operator semigroups, martingales, PDEs, or infinite systems of ODEs. We give a general proof for their convergence using only Taylor's theorem and basic ODE results. We hope this allows applied researchers to routinely show convergence of their mean-field models, putting their work on a stronger foundation.