A diffusion defined on a fractal state space

Abstract

In the plane, we define a fractal known as the Vicsek snowflake in terms of a family of affine contractions in R 2. We show that the Vicsek snowflake is a nested fractal in the sense of Lindstrøm (1990). We define random walks on the Vicsek snowflake and explicitly find an invariant probability for random walk. From this invariant probability, we construct a Brownian motion on the Vicsek snowflake. We show that this Brownian motion is the unique diffusion limit under weak convergence of rescaled random walks with any probability parameter. We show that Brownian motion on the Vicsek snowflake has a scaling property reminiscent of Brownian motion in R 1. Using a coupling argument, we show that our Brownian motion has transition densities with respect to Hausdorff measure on the snowflake.