TRANSITION FROM GAUSSIAN TO NON-GAUSSIAN FLUCTUATIONS FOR MEAN-FIELD DIFFUSIONS IN SPATIAL INTERACTION
TRANSITION FROM GAUSSIAN TO NON-GAUSSIAN FLUCTUATIONS FOR MEAN-FIELD DIFFUSIONS IN SPATIAL INTERACTION
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DOI：
10.1214/16-aap1194
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发表时间：
2016-12-01
影响因子：
1.8
通讯作者：
Stannat, Wilhelm
中科院分区：
文献类型：
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作者：
Lucon, Eric;Stannat, Wilhelm
我们考虑空间约束内的 N 个无序平均场相互作用扩散系统：每个粒子 theta(i) 附着在周期晶格的一个位点 x(i) 上，粒子 theta(i) 和 theta(j) 之间的相互作用随着垂直条 x(i) - x(j) 的垂直条 (-alpha) 的增加而减小，其中 alpha 是 [0, 1) 的元素。在之前的作品中[安。应用。很可能。 24 (2014) 1946-1993]，结果表明，粒子的经验测量在大群体中收敛于 McKean-Vlasov 型非线性偏微分方程的解。本文的目的是研究与这种收敛相关的波动。我们特别展示了缩放和波动性质中的相变：当 alpha 是 [0, 1/2) 的元素时，波动是高斯的，由线性 SPDE 控制，缩放根为 N，而在 alpha 是 (1/2, 1) 的元素的情况下，波动是缩放 N1-alpha 确定的。
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).