Multiscale Modeling &Simulation, Volume 18, Issue 1, Page 383-414, January 2020.
We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in ${\mathbb R}^d$ and the scaling limit of the associated continuous-time random walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process $\varepsilon^2 Z(t/\varepsilon^2)$ converges to a linear drift $\bar{U}t$ and that $\varepsilon (Z(t/\varepsilon^2)-\bar{U}t/\varepsilon^2)$ converges weakly to a Brownian motion. The diffusivity of the limiting Brownian motion can be computed by solving a set of linear algebra problems. As we allow for jump rates to be irreversible, our framework allows for the modeling of very general forms of interactions such as attraction, repulsion, and bonding. The case of interest to us is that of null drift $\bar{U}=0$ and we provide some sufficient conditions for null drift that include certain symmetries of the graph. We also provide a formal asymptotic derivation of the effective diffusivity in analogy with homogenization theory for PDEs. For the case of reversible jump rates, we derive an equivalent variational formulation. This derivation involves developing notions of gradient for functions on the graph's nodes, divergence for ${\mathbb R}^d$-valued functions on the graph's edges, and a divergence theorem.

中文翻译：

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在$ \ mathbb R ^ d $中的图上随机游走的均质化：一种预测具有精细尺度障碍和相互作用的媒体中宏观尺度扩散性的方法

多尺度建模与仿真，第18卷，第1期，第383-414页，2020年1月。
我们建议在适当定义的图上进行随机游走，以作为在受限环境中对粒子运动进行精细尺度建模的框架，在这种环境中，粒子可能与障碍物相互作用，并且与精细尺度相比，粒子的平均路径长度可能不可忽略。这激发了我们对嵌入在{{\ mathbb R} ^ d $中的周期图，有向图和加权图以及图节点上相关联的连续时间随机游动$ Z（t）$的缩放限制的研究，该图沿着图的边缘具有由边缘权重给出的跳跃率。我们证明了缩放过程$ \ varepsilon ^ 2 Z（t / \ varepsilon ^ 2）$收敛到线性漂移$ \ bar {U} t $，而$ \ varepsilon（Z（t / \ varepsilon ^ 2）- \ bar {U} t / \ varepsilon ^ 2）$微弱地收敛到布朗运动。极限布朗运动的扩散率可以通过求解一组线性代数问题来计算。由于我们允许跳跃率是不可逆的，因此我们的框架可以对非常普遍的相互作用形式进行建模，例如吸引，排斥和键合。我们感兴趣的情况是零漂移$ \ bar {U} = 0 $，我们为零漂移提供了一些充分的条件，其中包括图的某些对称性。我们还提供了与PDE均质化理论相似的有效扩散率的形式渐近派生。对于可逆跳跃率的情况，我们推导了等效的变式公式。此推导涉及为图的节点上的函数开发梯度概念，图的边缘上$ {\ mathbb R} ^ d $值函数的散度以及散度定理。