Heat Kernels and Spectral Asymptotics for some Random Sierpinski Gaskets

Abstract

We discuss two types of randomization for nested fractals based upon the d-dimensional Sierpinski gasket. One type, called homogeneous random fractals, are spatially homogeneous but scale irregular, while the other type, called random recursive fractals are spatially inhomogeneous. We use Dirichlet form techniques to construct Laplace operators on these fractals. The properties of the two types of random fractal differ and we extend and unify previous work to demonstrate that, though the homogeneous random fractals are well behaved in space, the behaviour in time of their on-diagonal heat kernels and their spectral asymptotics is more irregular than that of the random recursive fractals.