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.

49 pages, 7 figures; minor revision with adding a reference

References in corpus (2)

• Stability of parabolic Harnack inequalities on metric measure spaces
• Martingale dimensions for fractals

Cited by in corpus (5)

• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Construction of $p$-energy and associated energy measures on Sierpiński carpets
• Backward problems for stochastic differential equations on the Sierpinski gasket
• Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket
• Estimates of the local spectral dimension of the Sierpinski gasket