An elementary proof that walk dimension is greater than two for Brownian motion on Sierpi\'{n}ski carpets

TL;DR

This paper provides a simple, self-contained proof that the walk dimension exceeds two for Brownian motion on generalized Sierpiński carpets, using only basic properties of Dirichlet forms and self-similarity.

Contribution

It offers the first elementary proof of the walk dimension being greater than two for these fractals, expanding understanding with minimal assumptions.

Findings

Walk dimension of Brownian motion on Sierpiński carpets is greater than two

Proof relies solely on self-similarity and symmetry of Dirichlet forms

Application to energy measure singularity with respect to self-similar measure

Abstract

We give an elementary self-contained proof of the fact that the walk dimension of the Brownian motion on an arbitrary generalized Sierpi\'{n}ski carpet is greater than two, no proof of which in this generality had been available in the literature. Our proof is based solely on the self-similarity and hypercubic symmetry of the associated Dirichlet form and on several very basic pieces of functional analysis and the theory of regular symmetric Dirichlet forms. We also present an application of this fact to the singularity of the energy measures with respect to the canonical self-similar measure (uniform distribution) in this case, proved first by M. Hino in [Probab. Theory Related Fields 132 (2005), no. 2, 265-290].