Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction. (English) Zbl 1358.60104

Luçon, Eric; Stannat, Wilhelm

Ann. Appl. Probab. 26, No. 6, 3840-3909 (2016).

Summary: 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 \(| x_{i}-x_{j}|^{-\alpha}\) for \(\alpha \in [0,1)\). In a previous work [the authors, ibid. 24, No. 5, 1946–1993 (2014; Zbl 1309.60096)], 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 \(\alpha \in (\frac{1}{2},1)\).

MSC:

60K35	Interacting random processes; statistical mechanics type models; percolation theory
60J60	Diffusion processes
60G15	Gaussian processes
60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60F05	Central limit and other weak theorems
60G57	Random measures
82C20	Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics

Keywords:

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

Citations:

Zbl 1309.60096