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

Discrete Approximation to Brownian Motion with Darning