Propagation of Chaos for Singular Interactions via Regular Drivers

TL;DR

This paper develops a new framework using regular driver functions to prove propagation of chaos in particle systems with singular, density-dependent interactions, overcoming classical analytical challenges.

Contribution

It introduces an implicit dynamics approach with a regular driver to handle singular interactions, establishing uniform bounds and convergence to McKean-Vlasov SDEs.

Findings

Established uniform $L^ abla$ bounds for densities.

Proved convergence to singular McKean-Vlasov SDEs.

Provided a constructive method for analyzing singular systems.

Abstract

We introduce a framework to prove propagation of chaos for interacting particle systems with singular, density-dependent interactions, a classical challenge in mean-field theory. Our approach is to define the dynamics implicitly via a regular driver function. This regular driver is engineered to generate a singular effective interaction, yet its underlying regularity provides the necessary analytical control. Our main result establishes propagation of chaos under a key dissipation condition. The proof hinges on deriving a priori bounds, uniform in a regularization parameter, for the densities of the associated non-linear Fokker-Planck equations. Specifically, we establish uniform bounds in $L^\infty([0,T]; H^1(\R) \cap L^\infty(\R))$. These are established via an energy method for the excess mass to secure the $L^\infty$ bound, and a novel energy estimate for the $H^1$ norm that critically leverages the stability condition. These bounds provide the compactness necessary to pass to the singular limit, showing convergence to a McKean-Vlasov SDE with a singular, density-dependent diffusion coefficient. This work opens a new path to analyzing singular systems and provides a constructive theory for a class of interactions that present significant challenges to classical techniques.