Discrete Approximation to Brownian Motion with Varying Dimension in Bounded Domains

TL;DR

This paper establishes that a discrete random walk on a lattice approximates a complex Brownian motion with varying dimensions and a special joining point within bounded domains, providing a rigorous weak convergence proof.

Contribution

It introduces a discrete approximation scheme for Brownian motion with varying dimension and a darning point, proving weak convergence of the random walks to the continuous process.

Findings

Weak convergence of random walks to BMVD in bounded domains

Explicit description of the walk's behavior at the darning point

Construction of reversible random walks with exponential holding times

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.