Modelling transport in disordered media via diffusion 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 noninteger dimensions, they are natural models for disordered media, and diffusion on fractals can be used to model transport in disordered media. In particular, such diffusion processes can be used to obtain bounds on the fundemantal 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.