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Fluctuations for spatially extended Hawkes processes. (English) Zbl 1454.60062

Summary: In a previous paper [the first author et al., Stochastic Processes Appl. 129, No. 1, 1–27 (2019; Zbl 1404.60069)], 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.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G57 Random measures
60K35 Interacting random processes; statistical mechanics type models; percolation theory

Citations:

Zbl 1404.60069
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References:

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