Dirichlet forms on unconstrained Sierpinski carpets in ${\mathbb{R}}^3$

Abstract

We prove the existence of a strongly local, regular, self-similar Dirichlet form with a sub-Gaussian heat kernel estimate on an unconstrained Sierpinski carpet in ${\mathbb{R}}^3$. In the setting under consideration, the walk dimension $d_W$ and the Hausdorff dimension $d_H$ always satisfy the inequality that $d_H>d_W$.

44 pages, 1 figure