April 2023 Global-in-time mean-field convergence for singular Riesz-type diffusive flows
Matthew Rosenzweig, Sylvia Serfaty
Author Affiliations +
Ann. Appl. Probab. 33(2): 954-998 (April 2023). DOI: 10.1214/22-AAP1833


We consider the mean-field limit of systems of particles with singular interactions of the type log|x| or |x|s, with 0<s<d2, and with an additive noise in dimensions d3. 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.

Funding Statement

The first author was supported by the Simons Foundation through the Simons Collaboration on Wave Turbulence and by NSF Grant DMS-2052651.
The second author was supported by NSF Grant DMS-2000205 and by the Simons Foundation through the Simons Investigator program.


The authors thank Elias Hess–Childs for his careful reading of the manuscript. The second author thanks Eric Vanden–Eijnden for helpful comments.


Download Citation

Matthew Rosenzweig. Sylvia Serfaty. "Global-in-time mean-field convergence for singular Riesz-type diffusive flows." Ann. Appl. Probab. 33 (2) 954 - 998, April 2023. https://doi.org/10.1214/22-AAP1833


Received: 1 August 2021; Revised: 1 March 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 1516.35414
MathSciNet: MR4564418
Digital Object Identifier: 10.1214/22-AAP1833

Primary: 35Q35 , 35Q70 , 60H30

Keywords: McKean–Vlasov , Mean-field limit , Riesz potentials , uniform-in-time convergence

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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