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Fluctuations for spatially extended Hawkes processes
نوسانات مربوط به فرآیندهای توسعه یافته فضایی گسترده هاوکس-2020 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 |
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