Fluctuations for spatially extended Hawkes processes

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:130

Article

Chevallier, Julien¹  Ost, Guilherme² 

[1] Univ Grenoble Alpes, LJK, Grenoble INP, CNRS, F-38000 Grenoble, France

[2] Univ Fed Rio de Janeiro, Rio De Janeiro, Brazil

关键词: Hawkes processes;    Central limit theorem;    Neural field equations;    Network of neurons;   

DOI  :  10.1016/j.spa.2020.03.015

来源: Elsevier

【 摘 要 】

In a previous paper Chevallier et al. (2018), it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution u(t, x) of a neural field equation (NFE). The value u(t, x) represents the membrane potential at time t of a typical neuron located in position x, embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean field limit u(t, x). Our first main result is a central limit theorem stating that the spatial distribution associated to these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by u(t, x). To the best of our knowledge, this result appears to be new in the literature. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free