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 $d_H>d_W$.