Discrete Approximation to Brownian Motion with Darning

TL;DR

This paper introduces a discrete approximation method for Brownian motion with darning (BMD), showing that simple random walks on shrinking lattices converge weakly to BMD, which behaves like standard Brownian motion outside a special 'darning' area.

Contribution

It demonstrates that BMD can be approximated by simple random walks on lattices with decreasing mesh sizes, establishing a weak convergence result.

Findings

Random walks on lattices converge to BMD as mesh size diminishes

BMD travels infinitely fast across the darning area

The approximation holds starting from a single point in the state space

Abstract

Brownian motion with darning (BMD in abbreviation) is introduced and studied in [4] and [5, Chapter 7]. Roughly speaking, BMD travels across the "darning area" at infinite speed, while it behaves like a regular BM outside of this area. In this paper we show that starting from a single point in its state space, BMD is the weak limit of a family of continuous-time simple random walks on square lattices with diminishing mesh sizes. From any vertex in their state spaces, the approximating random walks jump to its nearest neighbors with equal probability after an exponential holding time.