Two-sided sub-Gaussian estimates of heat kernels on intervals for self-similar measures with overlaps

Qingsong Gu The Chinese University of Hong Kong Jiaxin Hu Tsinghua University Sze-Man Ngai Hunan Normal University and Georgia Southern University

Publications of CMSA of Harvard mathscidoc:1611.38001

2016.11
We obtain two-sided sub-Gaussian estimates of heat kernels for strongly local Dirichlet forms on intervals, equipped with self-similar measures generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. We first give a framework for heat kernel estimates on intervals, and then consider examples of self-similar measures to illustrate this phenomenon. These examples include the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. The walk dimensions obtained under these new metrics are strictly greater than 2 and are closely related to the spectral dimension of fractal Laplacians.
Heat kernel, sub-Gaussian, self-similar measure, second-order
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@inproceedings{qingsong2016two-sided,
  title={Two-sided sub-Gaussian estimates of heat kernels on intervals for self-similar measures with overlaps},
  author={Qingsong Gu, Jiaxin Hu, and Sze-Man Ngai},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161110065554269738610},
  year={2016},
}
Qingsong Gu, Jiaxin Hu, and Sze-Man Ngai. Two-sided sub-Gaussian estimates of heat kernels on intervals for self-similar measures with overlaps. 2016. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161110065554269738610.
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