Rigorous mean-field limit and cross diffusion

TL;DR

This paper rigorously derives a cross-diffusion system as the mean-field limit of a stochastic multi-species particle system, showing existence and error estimates for the resulting equations.

Contribution

It provides a rigorous two-step derivation of local cross-diffusion equations from a stochastic particle model, including existence proofs and error bounds.

Findings

Mean-field limit established for multi-species stochastic systems

Derived cross-diffusion equations from nonlocal to local interactions

Proved global existence for small initial data and provided error estimates

Abstract

The mean-field limit in a weakly interacting stochastic many-particle system for multiple population species in the whole space is proved. The limiting system consists of cross-diffusion equations, modeling the segregation of populations. The mean-field limit is performed in two steps: First, the many-particle system leads in the large population limit to an intermediate nonlocal diffusion system. The local cross-diffusion system is then obtained from the nonlocal system when the interaction potentials approach the Dirac delta distribution. The global existence of the limiting and the intermediate diffusion systems is shown for small initial data, and an error estimate is given.