Global-in-time mean-field convergence for singular Riesz-type diffusive flows

TL;DR

This paper proves global-in-time mean-field convergence for particle systems with singular Riesz-type interactions and noise, using a modulated-energy approach, applicable to both conservative and gradient flows in high dimensions.

Contribution

It provides the first global-in-time convergence result for singular Riesz-type flows in both conservative and gradient cases on , extending previous local or less general results.

Findings

Established quantitative convergence rates to the limiting PDE

Proved decay of solutions to the limiting equation in a singular setting

Extended modulated-energy methods to handle singular interactions with noise

Abstract

We consider the mean-field limit of systems of particles with singular interactions of the type $-\log|x|$ or $|x|^{-s}$, with $0< s<d-2$, and with an additive noise in dimensions $d \geq 3$. We use a modulated-energy approach to prove a quantitative convergence rate to the solution of the corresponding limiting PDE. When $s>0$, the convergence is global in time, and it is the first such result valid for both conservative and gradient flows in a singular setting on $\mathbb{R}^d$. The proof relies on an adaptation of an argument of Carlen-Loss to show a decay rate of the solution to the limiting equation, and on an improvement of the modulated-energy method developed in arXiv:1508.03377, arXiv:1803.08345, arXiv:2107.02592 making it so that all prefactors in the time derivative of the modulated energy are controlled by a decaying bound on the limiting solution.