In a previous paper, 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 with 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.

中文翻译：

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空间扩展霍克斯过程的波动

在之前的一篇论文中，已经表明空间扩展霍克斯过程的平均场极限被表征为神经场方程 (NFE) 的唯一解 $u(t,x)$。值 $u(t,x)$ 表示位于 $x$ 位置的典型神经元在时间 $t$ 的膜电位，嵌入无限的神经元网络中。在本文中，我们通过研究这种随机系统围绕其平均场极限 $u(t,x)$ 的波动来补充这一结果。我们的第一个主要结果是一个中心极限定理，说明与这些波动相关的空间分布收敛到一些由高斯噪声驱动的随机微分方程的唯一解。在我们的第二个主要结果中，我们表明，这个随机微分方程的解可以很好地近似于满足 $u(t,x)$ 的神经场方程的随机版本。据我们所知，这个结果在文献中似乎是新的。