Modelling Transport in Disordered Media via Diiusion on Fractals

When viewed at an appropriate scale, a disordered medium can behave as if it is strictly less than three dimensional. As fractals typically have non-integer dimensions, they are natural models for disordered media, and diiusion on fractals can be used to model transport in disordered media. In particular, such diiusion processes can be used to obtain bounds on the fundamental solution to the heat equation on a fractal. In this paper we review the work in this area and describe how bounds on branching processes lead to bounds on heat kernels.