TRANSITION FROM GAUSSIAN TO NON-GAUSSIAN FLUCTUATIONS FOR MEAN-FIELD DIFFUSIONS IN SPATIAL INTERACTION

• 成果类型: Article
• 署名作者: Lucon, Eric; Stannat, Wilhelm
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); Technical University of Berlin
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1194
• 发表日期: 2016
• 页码: 3840-3909
• 关键词: mckean-vlasov limit; propagation; CONVERGENCE; chaos; MODEL; Synchronization; approximation; equilibrium; STABILITY; networks

摘要

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 vertical bar x(i) - x(j)vertical bar(-alpha) for alpha is an element of [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 is an element of [0, 1/2), the fluctuations are Gaussian, governed by a linear SPDE, with scaling root N whereas the fluctuations are deterministic with scaling N1-alpha in the case alpha is an element of (1/2, 1).