Scaling Limit of Moderately Interacting Particle Systems with Singular Interaction and Environmental Noise

TL;DR

This paper studies the behavior of particle systems with singular interactions and environmental noise, proving convergence of their empirical measures to solutions of nonlinear Fokker-Planck equations, applicable to specific kernels.

Contribution

It establishes convergence results for mollified empirical measures of particle systems with singular kernels and environmental noise, including Biot-Savart and Poisson kernels.

Findings

Mollified empirical measures converge strongly to solutions of nonlinear Fokker-Planck equations.

Results apply to Biot-Savart and repulsive Poisson kernels.

Convergence holds for local solutions of the equations.

Abstract

We consider moderately interacting particle systems with singular interaction kernel and environmental noise. It is shown that the mollified empirical measures converge in strong norms to the unique (local) solutions of nonlinear Fokker-Planck equations. The approach works for the Biot-Savart and repulsive Poisson kernels.