Diffusion processes on fractal fields: Heat kernel estimates and large deviations

A fractal field is a collection of fractals with, in general, different Hausdorff dimensions, embedded in ℝ². We will construct diffusion processes on such fields which behave as Brownian motion in ℝ² outside the fractals and as the appropriate fractal diffusion within each fractal component of the field. We will discuss the properties of the diffusion process in the case where the fractal components tile ℝ². By working in a suitable shortest path metric we will establish heat kernel bounds and large deviation estimates which determine the trajectories followed by the diffusion over short times.