Asymptotically one-dimensional diffusion on the Sierpiński gasket and multi-type branching processes with varying environment
type: Item
instance of: scholarly article
title: Asymptotically one-dimensional diffusion on the Sierpiński gasket and multi-type branching processes with varying environment
zbMATH Open document ID: 1003.60071
DOI: 10.1023/A:1014858709284
published in: Journal of Theoretical Probability
publication date: 2002-08-20T00:00:00Z
review text: Asymptotically one-dimensional processes on fractals were introduced by \textit{K. Hattori}, \textit{T. Hattori} and \textit{H. Watanabe} [Probab. Theory Relat. Fields 100, No. 1, 85-116 (1994; Zbl 0808.60067)]. For the Sierpiński gasket it was shown that there is a natural class of diffusions on this fractal, not equivalent in Euclidean space, which have rather different local properties to the Brownian motion. Due to homogenization effects they behave globally like the Brownian motion, yet locally they have a preferred direction of motion. \textit{B. M. Hambly} and \textit{T. Kumagai} [Potential Anal. 8, No. 4, 359-397 (1998; Zbl 0910.60062)] have given an alternative construction of the diffusion process using Dirichlet forms, extended the class of fractals considered to some nested fractals and obtained some new results. The aim of the reviewed paper is to investigate these problems using branching processes techniques. On the first step two side estimations for the left tail of the limiting distribution for a supercritical multi-type branching process with varying environment are obtained. Then the spectral dimension of the diffusion processes are calculated and short time heat kernel estimates in the Euclidean metric are obtained.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 1784210
zbMATH Keywords: diffusion processes
zbMATH Keywords: Sierpiński gasket
zbMATH Keywords: spectral dimension
zbMATH Keywords: heat kernel
zbMATH Keywords: multi-type branching processes
zbMATH Keywords: varying environment
zbMATH Keywords: Dirichlet forms
author: Ben M. Hambly
author: Owen D. Jones
reviewed by: Valentin Topchii
MaRDI profile type: Publication
full work available at URL: a:1014858709284
OpenAlex ID: W1552614571
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description: scientific article; zbMATH DE number 1784210