Upper estimate of martingale dimension for self-similar fractals

TL;DR

This paper investigates the upper bounds of martingale dimension for diffusion processes on self-similar fractals, establishing that it equals one for certain fractals and is bounded by spectral dimension for others.

Contribution

It provides new upper estimates of martingale dimension for diffusions on self-similar fractals, including exact and bounded cases.

Findings

Martingale dimension equals 1 on post-critically finite self-similar sets.

Martingale dimension is dominated by spectral dimension on Sierpinski carpets.

The approach applies general strategies to self-similar Dirichlet forms.

Abstract

We study upper estimates of the martingale dimension $d_m$ of diffusion processes associated with strong local Dirichlet forms. By applying a general strategy to self-similar Dirichlet forms on self-similar fractals, we prove that $d_m=1$ for natural diffusions on post-critically finite self-similar sets and that $d_m$ is dominated by the spectral dimension for the Brownian motion on Sierpinski carpets.