Coupling and Harnack inequalities for Sierpiński 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 R d {\mathbb {R}^d} , d ≥ 3 d \geq 3 , estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities.