Transition Density Estimates for Diffusion Processes on Post Critically Finite Self-Similar Fractals

Abstract

The framework of post critically finite (p.c.f) self-similar fractals was introduced to capture the idea of a finitely ramified fractal, that is, a connected fractal set where any component can be disconnected by the removal of a finite number of points. These ramification points provide a sequence of graphs which approximate the fractal and allow a Laplace operator to be constructed as a suitable limit of discrete graph Laplacians. In this paper we obtain estimates on the heat kernel associated with the Laplacian on the fractal which are best possible up to constants. These are short time estimates for the Laplacian with respect to a natural measure and expressed in terms of an effective resistance metric. Previous results on fractals with spatial symmetry have obtained heat kernel estimates of a non-Gaussian form but which are of Aronson type. By considering a range of examples which are not spatially symmetric, we show that uniform Aronson type estimates do not hold in general on fractals.