The Annals of Applied Probability

Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction

Eric Luçon and Wilhelm Stannat

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We consider a system of $N$ disordered mean-field interacting diffusions within spatial constraints: each particle $\theta_{i}$ is attached to one site $x_{i}$ of a periodic lattice and the interaction between particles $\theta_{i}$ and $\theta_{j}$ decreases as $\vert x_{i}-x_{j}\vert^{-\alpha}$ for $\alpha\in[0,1)$. In a previous work [Ann. Appl. Probab. 24 (2014) 1946–1993], it was shown that the empirical measure of the particles converges in large population to the solution of a nonlinear partial differential equation of McKean–Vlasov type. The purpose of the present paper is to study the fluctuations associated to this convergence. We exhibit in particular a phase transition in the scaling and in the nature of the fluctuations: when $\alpha\in[0,\frac{1}{2})$, the fluctuations are Gaussian, governed by a linear SPDE, with scaling $\sqrt{N}$ whereas the fluctuations are deterministic with scaling $N^{1-\alpha}$ in the case $\alpha\in(\frac{1}{2},1)$.

Article information

Ann. Appl. Probab., Volume 26, Number 6 (2016), 3840-3909.

Received: February 2015
Revised: December 2015
First available in Project Euclid: 15 December 2016

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60G57: Random measures
Secondary: 60H15: Stochastic partial differential equations [See also 35R60] 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs 92B25: Biological rhythms and synchronization

Weakly interacting diffusions spatially-extended particle systems weighted empirical measures fluctuations Kuramoto model neuronal models stochastic partial differential equations


Luçon, Eric; Stannat, Wilhelm. Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction. Ann. Appl. Probab. 26 (2016), no. 6, 3840--3909. doi:10.1214/16-AAP1194.

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