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

The recent development of analysis on fractal spaces is physically motivated by the study of diffusion in disordered media. The natural questions that arise concern the existence and uniqueness of a suitable Laplace operator, and the behaviour of the associated heat semigroup, on a space which is fractal. The classes of fractals for which these questions were ®rst answered were classes of exactly self-similar fractals, with strong spatial symmetry, such as nested or af®ne nested fractals (see, for example, [2, 8]). The existence of a Laplacian and estimates on the heat kernel were obtained by considering the associated diffusion process and using the symmetry of the space. The uniqueness of the Laplacian for nested and af®ne nested fractals has recently been solved through consideration of their Dirichlet forms [23]. In [15] the framework of post critically ®nite (which we abbreviate to p.c.f.) self-similar sets was introduced in order to capture the notion of exactly selfsimilar ®nitely rami®ed fractals as used in the physics literature. Finitely rami®ed fractals have the property that the intersection of any connected subset of the fractal with the rest of the set should occur only at a ®nite number of points. This makes these structures much easier to analyse than in®nitely rami®ed sets such as the Sierpinski carpet [5]. The p.c.f. self-similar sets do not have spatial symmetry in general and have provided a mathematical test bed for analysis on fractals. In this paper we will obtain uniform short time estimates on the heat kernel associated with a natural Laplacian on the fractal. A Laplacian can be constructed on a p.c.f. self-similar fractal as a limit of discrete Laplacians on graph approximations to the fractal based on the rami®cation points. We construct these operators via their Dirichlet forms, which can be set on any L-space with a full measure, n. In [23], criteria are given which give a partial answer to the existence and uniqueness of Laplace type operators on p.c.f. self-similar sets. In a series of papers [16, 17, 18, 20, 25, 7] some of the interesting spectral properties of p.c.f. sets have been elucidated. It has been shown that there can exist localized eigenfunctions if there is a high degree of symmetry in the set and corresponding Laplace operator. This corresponds to oscillation in the leading order term for the asymptotics of the eigenvalue counting function. Such behaviour occurs for nested fractals [7] and p.c.f. sets with strong harmonic