Brownian motion on a random recursive Sierpinski gasket

Abstract

We introduce a random recursive fractal based on the Sierpinski gasket and construct a diffusion upon the fractal via a Dirichlet form. This form and its symmetrizing measure are determined by the electrical resistance of the fractal. The effective resistance provides a metric with which to discuss the properties of the fractal and the diffusion. The main result is to obtain uniform upper and lower bounds for the transition density of the Brownian motion on the fractal in terms of this metric. The bounds are not tight as there are logarithmic corrections due to the randomness in the environment, and the behavior of the shortest paths in the effective resistance metric is not well understood. The results are deduced from the study of a suitable general branching process.