Quantitative mean-field limit for interacting branching diffusions

TL;DR

This paper proves an explicit convergence rate for mean-field interacting branching diffusions towards nonlinear equations, using a novel coupling method based on optimal transport, extending propagation of chaos techniques.

Contribution

Introduces a new coupling approach for binary branching diffusions using optimal transport, providing sharp convergence rates and extending chaos propagation methods to branching populations.

Findings

Established explicit convergence rates in dual bounded-Lipschitz distance.

Extended propagation of chaos techniques to stochastic branching systems.

Demonstrated the method's effectiveness even in simple binary branching cases.

Abstract

We establish an explicit rate of convergence for some systems of mean-field interacting diffusions with logistic binary branching towards the solutions of nonlinear evolution equations with non-local self-diffusion and logistic mass growth, shown to describe their large population limits in arXiv:1303.3939. The proof relies on a novel coupling argument for binary branching diffusions based on optimal transport, which allows us to sharply mimic the trajectory of the interacting binary branching population by certain system of independent particles with suitably distributed random space-time births. We are thus able to derive an optimal convergence rate, in the dual bounded-Lipschitz distance on finite measures, for the empirical measure of the population, from the convergence rate in 2-Wasserstein distance of empirical distributions of i.i.d. samples. Our approach and results extend techniques and ideas on propagation of chaos from kinetic models to stochastic systems of interacting branching populations, and appear to be new in this setting, even in the simple case of pure binary branching diffusions.