Stochastic Mean-Field Limit: Non-Lipschitz Forces \& Swarming

TL;DR

This paper extends the mean-field limit theory for stochastic particle systems with non-Lipschitz interactions, such as those in swarming models, demonstrating convergence to a kinetic PDE despite local Lipschitz conditions.

Contribution

It introduces a novel approach to handle locally Lipschitz interaction potentials in stochastic systems, broadening the applicability of mean-field limit results.

Findings

Proves mean-field convergence for locally Lipschitz interactions

Extends classical theory to non-globally Lipschitz potentials

Applicable to models like Cucker-Smale with noise

Abstract

We consider general stochastic systems of interacting particles with noise which are relevant as models for the collective behavior of animals, and rigorously prove that in the mean-field limit the system is close to the solution of a kinetic PDE. Our aim is to include models widely studied in the literature such as the Cucker-Smale model, adding noise to the behavior of individuals. The difficulty, as compared to the classical case of globally Lipschitz potentials, is that in several models the interaction potential between particles is only locally Lipschitz, the local Lipschitz constant growing to infinity with the size of the region considered. With this in mind, we present an extension of the classical theory for globally Lipschitz interactions, which works for only locally Lipschitz ones.