Geodesic metrics on fractals and applications to heat kernel estimates  

作　　者：Qingsong Gu Ka-Sing Lau Hua Qiu Huo-Jun Ruan 

机构地区：[1]Department of Mathematics,Nanjing University,Nanjing 210093,China [2]Department of Mathematics,The Chinese University of Hong Kong,Hong Kong,China [3]Department of Mathematics,University of Pittsburgh,Pittsburgh,PA 15217,USA [4]School of Mathematical Sciences,Zhejiang University,Hangzhou 310027,China

出　　处：《Science China Mathematics》2023年第5期907-934,共28页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.12101303 and 12171354);supported by National Natural Science Foundation of China(Grant No.12071213);supported by National Natural Science Foundation of China(Grant No.11771391);supported by the Hong Kong Research Grant Council;the Natural Science Foundation of Jiangsu Province in China(Grant No.BK20211142);Zhejiang Provincial National Science Foundation of China(Grant No.LY22A010023);the Fundamental Research Funds for the Central Universities of China(Grant No.2021FZZX001-01)。

摘　　要：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 further assume that the weight functions g := ga are generated by“symmetric” weights a. Let M be the domain of a such that Dgadefines a metric, and let S be the boundary of M. One of our main results is that the metrics from ga 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.

关 键 词：Brownian motion heat kernel metric chain condition nested fractal quasisymmetry resistance metric Sierpinski carpet weight function 

分 类 号：O189.11[理学—数学]