## 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.

Original language | English (US) |
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Pages (from-to) | 754-798 |

Number of pages | 45 |

Journal | Annals of Applied Probability |

Volume | 33 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2023 |

## Keywords

- McKean-Vlasov
- Mean-field limit
- Riesz potentials
- uniform-in-time convergence

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty