Brownian motion on the golden ratio Sierpinski gasket

TL;DR

This paper constructs a unique, self-similar Dirichlet form on the golden ratio Sierpinski gasket, enabling the analysis of Brownian motion with sub-Gaussian heat kernel estimates on this fractal without finitely ramified structure.

Contribution

It introduces a novel Dirichlet form on the golden ratio Sierpinski gasket, extending analysis techniques to a complex fractal without finitely ramified cells.

Findings

Existence of a self-similar Dirichlet form on the gasket

The associated process satisfies sub-Gaussian heat kernel estimates

The Dirichlet form is unique and decimation invariant

Abstract

We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without any finitely ramified cell structure, via a study on the trace of electrical networks on an infinite graph. The Dirichlet form is self-similar in the sense of an infinite iterated function system, and is decimation invariant with respect to a graph-directed construction. A theorem of uniqueness is also provided. Lastly, the associated process satisfies the two-sided sub-Gaussian heat kernel estimate.