Propagation of chaos for a class of first order models with singular mean field interactions

TL;DR

This paper proves the convergence of large particle systems with singular mean field interactions to a macroscopic limit, using gradient flow theory in Wasserstein spaces, applicable to both stochastic and deterministic models.

Contribution

It establishes the existence of a macroscopic limit for singular mean field interacting particles, extending previous results to a broader class of potentials with quasi-convexity.

Findings

Convergence results for particle systems with singular interactions

Applicability to both stochastic and deterministic models

Use of gradient flow theory in Wasserstein spaces

Abstract

Dynamical systems of N particles in \R^{D} interacting by a singular pair potential of mean field type are considered. The systems are assumed to be of gradient type and the existence of a macroscopic limit in the many particle limit is established for a large class of singular interaction potentials in the stochastic as well as the deterministic settings. The main assumption on the potentials is an appropriate notion of quasi-convexity. When D=1 the convergence result is sharp when applied to strongly singular repulsive interactions and for a general dimension D the result applies to attractive interactions with Lipschitz singular interaction potentials, leading to stochastic particle solutions to the corresponding macroscopic aggregation equations. The proof uses the theory of gradient flows in Wasserstein spaces of Ambrosio-Gigli-Savaree.