Open Access
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

Abstract

We consider a system of N disordered mean-field interacting diffusions within spatial constraints: each particle θi is attached to one site xi of a periodic lattice and the interaction between particles θi and θj decreases as |xixj|α for α[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 α[0,12), the fluctuations are Gaussian, governed by a linear SPDE, with scaling N whereas the fluctuations are deterministic with scaling N1α in the case α(12,1).

Citation

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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. https://doi.org/10.1214/16-AAP1194

Information

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

Subjects:
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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