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December 2016 Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction
Eric Luçon, Wilhelm Stannat
Ann. Appl. Probab. 26(6): 3840-3909 (December 2016). DOI: 10.1214/16-AAP1194


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 $\alpha\in(\frac{1}{2},1)$.


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Eric Luçon. Wilhelm Stannat. "Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction." Ann. Appl. Probab. 26 (6) 3840 - 3909, December 2016.


Received: 1 February 2015; Revised: 1 December 2015; Published: December 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1358.60104
MathSciNet: MR3582819
Digital Object Identifier: 10.1214/16-AAP1194

Primary: 60F05 , 60G57
Secondary: 60H15 , 82C20 , 92B25

Keywords: Fluctuations , Kuramoto model , neuronal models , spatially-extended particle systems , Stochastic partial differential equations , weakly interacting diffusions , weighted empirical measures

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.26 • No. 6 • December 2016
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