Discrete Approximation to Brownian Motion with Varying Dimension in Bounded Domains

Abstract

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.