Editing Sierpiński carpet (section)

==Brownian motion on the Sierpiński carpet==
The topic of [[Brownian motion]] on the Sierpiński carpet has attracted interest in recent years.<ref name=barlow>{{Citation|last1=Barlow|first1=Martin|last2=Bass|first2=Richard|title=Brownian motion and harmonic analysis on Sierpiński carpets|url=https://www.math.ubc.ca/~barlow/preprints/61_sch_pub.pdf}}</ref> 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 {{math|{{sqrt|''n''}}}} after {{mvar|n}} steps, but the random walk on the discrete Sierpiński carpet reaches only a mean distance proportional to {{math|{{radic|''n''|''β''}}}} for some {{math|''β'' > 2}}. They also showed that this random walk satisfies stronger [[Large deviations theory|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.