The topic of Brownian motion on the Sierpiński carpet has attracted interest in recent years. Martin Barlow and Richard Bass have shown that a random walk on the Sierpiński carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to after steps, but the random walk on the discrete Sierpiński carpet reaches only a mean distance proportional to for some . They also showed that this random walk satisfies stronger large deviation inequalities (so called "sub-Gaussian inequalities") and that it satisfies the elliptic Harnack inequality without satisfying the parabolic one. The existence of such an example was an open problem for many years.
