An elementary proof of walk dimension being greater than two for Brownian motion on Sierpiński carpets

Naotaka Kajino

Published 2020-05-05Version 1

We give an elementary self-contained proof of the fact that the walk dimension of the Brownian motion on an \emph{arbitrary} generalized Sierpi\'{n}ski carpet is greater than two, no complete proof of which 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 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 symmetric measure in this case, proved first by M.\ Hino in [\emph{Probab.\ Theory Related Fields} \textbf{132} (2005), no.\ 2, 265--290].