Brownian Motion On The Sierpinski Carpet
The topic of Brownian motion on the Sierpinski carpet has attracted interest in recent years. Martin Barlow and Richard Bass have shown that a random walk on the Sierpinski carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to n1/2 after n steps, but the random walk on the discrete Sierpinski carpet reaches only a mean distance proportional to n1/β for some β > 2. 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.
Read more about this topic: Sierpinski Carpet
Famous quotes containing the words motion and/or carpet:
“I have seen in this revolution a circular motion of the sovereign power through two usurpers, father and son, to the late King to this his son. For ... it moved from King Charles I to the Long Parliament; from thence to the Rump; from the Rump to Oliver Cromwell; and then back again from Richard Cromwell to the Rump; then to the Long Parliament; and thence to King Charles, where long may it remain.”
—Thomas Hobbes (1579–1688)
“My veins are filled, once a week with a Neapolitan carpet cleaner distilled from the Adriatic and I am as bald as an egg. However I still get around and am mean to cats.”
—John Cheever (1912–1982)