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

Abstract : 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 $0\leq \alpha\leq 1$. In a previous work [28], 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 $\alpha\in[0,1/2)$, 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 $\alpha\in(1/2,1)$.
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Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2016, 26 (6), pp.3840-3909. 〈10.1214/16-AAP1194〉
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Eric Luçon, Wilhelm Stannat. Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2016, 26 (6), pp.3840-3909. 〈10.1214/16-AAP1194〉. 〈hal-01112656〉

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