In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of $\beta$-amyloid peptide (A$\beta$) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of A$\beta$ in the monomeric form at the level of neuronal membranes.

中文翻译：

---

阿尔茨海默病的数学模型：一种通过 Smoluchowski 方程随机均质化的方法

在本说明中，我们应用随机均质化理论来寻找具有非齐次 Neumann 边界条件的一组 Smoluchowski 凝结扩散方程的解的渐近行为。该系统旨在模拟 $\beta$-淀粉样肽 (A$\beta$) 在脑组织中的聚集和扩散，这一过程与阿尔茨海默病的发展相关。与我们之前工作中使用的方法相反，在本文中，我们通过假设神经元空间分布的随机模型来解释大脑细胞结构的非周期性。此外，我们考虑了淀粉样蛋白聚集体的非周期性随机扩散系数和在神经元膜水平上以单体形式随机产生的 A$\beta$。