 Fluctuations for spatially extended Hawkes processesJulien Chevallier and Guilherme Ost Stochastic Processes and their Applications, 2020, vol. 130, issue 9, 5510-5542 Abstract: 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. Keywords: Hawkes processes; Central limit theorem; Neural field equations; Network of neurons (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:130:y:2020:i:9:p:5510-5542 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spa.2020.03.015 Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
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