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

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≥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 Rd. The proof relies on an adaptation of an argument of Carlen–Loss (Duke Math. J. 81 (1995) 135–157) to show a decay rate of the solution to the limiting equation, and on an improvement of the modulated-energy method developed in (SIAM J. Math. Anal. 48 (2016) 2269–2300; Duke Math. J. 169 (2020) 2887–2935; Nguyen, Rosenzweig and Serfaty (2021)), making it so that all prefactors in the time derivative of the modulated energy are controlled by a decaying bound on the limiting solution.