Coupling and Harnack inequalities for Sierpinski carpets

Abstract

Uniform Harnack inequalities for harmonic functions on the pre- and graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in ${\mathbb{R}}^d$, $d\ge 3$, estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities.

5 pages