Geodesic metrics on fractals and applications to heat kernel estimates

Abstract

It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion (the time-changed process) will diffuse according to a different metric D(·,·). In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper, we give a thorough study of his metric on two classes of fractals of primary importance: the nested fractals and the generalized Sierpinski carpets; we assume further that the weight functions g ≔ g_a are generated by “symmetric” weights a. Let $\cal{M}$ be the domain of a such that D_ga defines a metric, and let S be the boundary of $\cal{M}$. One of our main results is that the metrics from g_a satisfy the metric chain condition if and only if a ∈ S. To determine $\cal{M}$ and S, we provide a recursive weight transfer construction on the nested fractals, and a basic symmetric argument on the Sierpinski carpet. As an application, we use the metric chain condition to obtain the lower estimate of the sub-Gaussian heat kernel. This together with the upper estimate obtained by Kigami allows us to have a concrete class of metrics for the time change, and the two-sided sub-Gaussian heat kernel estimate on the fundamental fractals.