We study the asymptotic behavior of multiscale stochastic spatial gene networks. Multiscaling takes into account the difference of abundance between molecules, and captures the dynamic of rare species at a mesoscopic level. We introduce an assumption of spatial correlations for reactions involving rare species and a new law of large numbers is obtained. According to the scales, the whole system splits into two parts with different but coupled dynamics. The high scale component converges to the usual spatial model which is the solution of a partial differential equation, whereas the low scale component converges to the usual homogeneous model which is the solution of an ordinary differential equation. Comparisons are made in the supremum norm.

中文翻译：

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多尺度随机空间基因网络的最高范数中的大数定律

我们研究了多尺度随机空间基因网络的渐近行为。多尺度考虑了分子之间丰度的差异，并在介观水平上捕捉了稀有物种的动态。我们引入了涉及稀有物种的反应的空间相关性假设，并获得了新的大数定律。根据尺度，整个系统分为两部分，具有不同但耦合的动态。高尺度分量收敛于通常的空间模型，即偏微分方程的解，而低尺度分量收敛于通常的齐次模型，即常微分方程的解。比较是在最高范数中进行的。