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

We consider the mean-field limit of systems of particles with singular interactions of the type − log |x| or |x|, 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 R. The proof relies on an adaptation of an argument of Carlen-Loss [CL95] to show a decay rate of the solution to the limiting equation, and on an improvement of the modulated-energy method developed in [Due16, Ser20, NRS21], making it so that all prefactors in the time derivative of the modulated energy are controlled by a decaying bound on the limiting solution.

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