Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential

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  • Authors: Alekos Cecchin, Paul Nikolaev
  • Preprint year: 2025
  • First public date: 2025-09-01
  • arXiv: 2509.01266

Abstract

For a system of mean field interacting diffusion on $\mathbb{T}^d$, the empirical measure $μ^N$ converges to the solution $μ$ of the Fokker-Planck equation. Refining this mean field limit as a Central Limit Theorem, the fluctuation process $ρ^N_t= \sqrt{N}( μ^N_t -μ_t)$ convergences to the solution $ρ$ of a linear stochastic PDE on the negative Sobolev space $H^{-λ-2}(\mathbb{T}^d)$. The main result of the paper is to establish a rate for such convergence: we show that $|\mathbb{E}[Φ(ρ_t^N)] – \mathbb{E}[Φ(ρ_t)]| = \mathcal{O}(\tfrac{1}{\sqrt{N}})$, for smooth functions on $H^{-λ-2}(\mathbb{T}^d)$. The strategy relies on studying the generators of the processes $ρ^N$ and $ρ$ on $H^{-λ-2}(\mathbb{T}^d)$, and thus estimating their difference. Among others, this requires to approximate in probability $ρ$ with solutions to stochastic diffential equations on the Hilbert space $H^{-λ-2}(\mathbb{T}^d)$. The flexibility of the approach permits to establish a rate for the fluctuations, not only in case of a regular drift, but also for the the 2D viscous Vortex model, governed by the Biot-Savart kernel, and for the repulsive Coulomb potential.

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BibTeX

@article{arxiv250901266,
  title = {Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential},
  author = {Alekos Cecchin and Paul Nikolaev},
  year = {2025},
  eprint = {2509.01266},
  archivePrefix = {arXiv},
  primaryClass = {math.PR}
}

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