Upper estimate of martingale dimension for self-similar fractals

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.