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.