Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones

TL;DR

This paper rigorously derives a mean-field limit for a particle system modeling collective behavior with boundary constraints and discontinuous interactions, extending previous results to more complex kernels.

Contribution

It extends the propagation of chaos results to systems with discontinuous kernels and no-flux boundary conditions, providing quantitative convergence estimates.

Findings

Established convergence of empirical measures to weak solutions of the continuity equation.

Extended previous models to include discontinuous interaction kernels.

Provided quantitative estimates of convergence in law.

Abstract

We consider a $N$-particle interacting particle system with the vision geometrical constraints and reflected noises, proposed as a model for collective behavior of individuals. We rigorously derive a continuity-type of mean-field equation with discontinuous kernels and the normal reflecting boundary conditions from that stochastic particle system as the number of particles $N$ goes to infinity. More precisely, we provide a quantitative estimate of the convergence in law of the empirical measure associated to the particle system to a probability measure which possesses a density which is a weak solution to the continuity equation. This extends previous results on an interacting particle system with bounded and Lipschitz continuous drift terms and normal reflecting boundary conditions by Sznitman[J. Funct. Anal., 56, (1984), 311--336] to that one with discontinuous kernels.