Large-scale concentration and relaxation for mean-field Langevin particle systems

TL;DR

This paper analyzes the Langevin dynamics of large particle systems with mean-field interactions, establishing new concentration, relaxation, and chaos generation results that improve understanding of their long-term behavior.

Contribution

It introduces novel methods to approximate empirical distributions and proves near-optimal convergence and chaos generation rates for mean-field Langevin systems.

Findings

Established coercive and contractive properties of the modulated free energy functional

Derived near-optimal large-scale concentration and relaxation rates

Proved generation of chaos estimates with optimal particle approximation order

Abstract

We study the Langevin dynamics of diffusive particles with regular pairwise interactions under mean-field scaling. By approximating empirical distributions with conditional distributions, we establish coercive and contractive properties for the modulated free energy functional. These properties yield near-optimal large-scale concentration and relaxation rates for the particle system throughout the subcritical regime. Furthermore, we derive generation of chaos estimates with the optimal order of particle approximation. As a simpler instance, we demonstrate long-time convergence of the independent projection of Langevin dynamics.