相互作用Hawkes过程振荡系统中级联扩散的大偏差
Large deviations for cascades of diffusions arising in oscillating systems of interacting Hawkes processes
摘要：
我们考虑Ditlevsen和Loecherach（2017）中引入的相互作用Hawkes过程的振荡系统，以建模相互作用神经元的多类系统及其强度过程的扩散近似。这种扩散，它结合了记忆项定义的霍克斯过程的动态，是次椭圆。它由一个由$2-$维布朗运动驱动的高维微分方程链给出。我们研究大人口，即，小噪声限制的不变测度，我们建立了一个大偏差结果的精神Freidlin和Wentzell。
英文摘要：
We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Loecherbach (2017) to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high dimensional chain of differential equations driven by $2-$dimensional Brownian motion. We study the large-population-, i.e., small noise-limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell.
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