Upper estimate of martingale dimension for self-similar fractals

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.