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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.01266 |
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| _version_ | 1866916928109412352 |
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| author | Cecchin, Alekos Nikolaev, Paul |
| author_facet | Cecchin, Alekos Nikolaev, Paul |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential Cecchin, Alekos Nikolaev, Paul Probability 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. |
| title | Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential |
| topic | Probability |
| url | https://arxiv.org/abs/2509.01266 |