Discrete Approximation to Brownian Motion with Darning

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.