Brownian motion on the Sierpinski carpet

TL;DR

This paper proves the uniqueness of the Brownian motion law and Laplacian on generalized Sierpinski carpets by establishing a unique local regular Dirichlet form invariant under the carpet's symmetries.

Contribution

It demonstrates the uniqueness of the Dirichlet form, Brownian motion law, and Laplacian on generalized Sierpinski carpets, extending understanding of stochastic processes on fractals.

Findings

Unique local regular Dirichlet form exists on the carpet

Brownian motion law is uniquely determined

Laplacian is well defined on the fractal

Abstract

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.