Fluctuations for spatially extended Hawkes processes

Autor(es)

Chevallier, Julien ^[1] ; Ost, Guilherme ^[2]

Afiliação do(s) autor(es)

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

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

• Tipo de documento: Artigo Científico
• Fonte: Stochastic Processes and their Applications; v. 130, n. 9, p. 5510-5542, SEP 2020.
• Citações Web of Science: 0

Resumo

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. (AU)

• Processo FAPESP: 13/07699-0 - Centro de Pesquisa, Inovação e Difusão em Neuromatemática - NeuroMat
• Beneficiário: Oswaldo Baffa Filho
• Modalidade de apoio: Auxílio à Pesquisa - Centros de Pesquisa, Inovação e Difusão - CEPIDs