Geodesic metrics on fractals and applications to heat kernel estimates

• Published in: Science China. Mathematics 2023-05, Vol.66 (5), p.907-934
• Main Authors: Gu, Qingsong, Lau, Ka-Sing, Qiu, Hua, Ruan, Huo-Jun
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Brownian motion; Chains; Fractals; Kernels; Mathematics; Mathematics and Statistics; Self-similarity; Weighting functions
• ISSN: 1674-7283; 1869-1862
• DOI: 10.1007/s11425-021-1989-3

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 M be the domain of a such that D ga defines a metric, and let S be the boundary of 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 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.