论文标题
离散近似与布朗运动的近似，有界域中的尺寸变化
Discrete Approximation to Brownian Motion with Varying Dimension in Bounded Domains
论文作者
论文摘要
在本文中，我们研究了[4]中通过在方形晶格上进行连续的时间随机行走引入的与布朗运动的离散近似（缩写为缩写）。 BMVD的状态空间包含$ 2 $维的组件，$ 3 $维的组件和一个加入这两个组件的“牵引点”。这样的状态空间配备了大地距离，在该距离下，BMVD是一个扩散过程。在本文中，我们证明，限制在包含刺激点的有界域上的BMVD是连续时间可逆随机步行的弱极限，并具有指数稳定时间。每次移动，除了在“牵引点”之外，这些随机步行以同样的概率跳到其最近的任何邻居。本文也明确给出了在“牵引点”这样随机步行的行为。
In this paper we study the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) introduced in [4] by continuous time random walks on square lattices. The state space of BMVD contains a $2$-dimensional component, a $3$-dimensional component, and a "darning point" which joins these two components. Such a state space is equipped with the geodesic distance, under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous time reversible random walks with exponential holding times. Upon each move, except at the "darning point", these random walks jump to any of its nearest neighbors with equal probability. The behavior of such a random walk at the "darning point" is also given explicitly in this paper.
