Brownian motion penetrating fractals. An application of the trace theorem of Besov spaces
type: Item
instance of: scholarly article
title: Brownian motion penetrating fractals. An application of the trace theorem of Besov spaces
zbMATH Open document ID: 0960.31005
author: Takashi Kumagai
published in: Journal of Functional Analysis
publication date: 2001-05-17T00:00:00Z
review text: Let \(F\) be a closed connected set in \(\mathbb{R}^n\) and \(\mu\) a positive Borel measure on \(F\). Let \(({\mathcal E}, D({\mathcal E}))\) be a local regular Dirichlet form on \(L^2(F, \mu)\) so that \(D({\mathcal E})\) is contained in a Lipschitz or Besov space. The first main result of this really interesting paper is that under some assumptions on \(\mu\) there is a local regular Dirichlet form \((\widetilde{{\mathcal E}}, D(\widetilde{{\mathcal E}}))\) on \(L^2(\mathbb{R}^n, \mu+ \lambda)\), where \(\lambda\) denotes Lebesgue measure, whose restriction to \(F\) is \(({\mathcal E}, D({\mathcal E}))\) and which is the classical Dirichlet form on \(\mathbb{R}^n\setminus F\) associated to the Laplacian. So, by Fukushima's theory of regular Dirichlet forms there exists a corresponding diffusion process on \(\mathbb{R}^n\). The second main result of the paper under consideration states that under some assumptions on the order of the Lipschitz resp. Besov space in relation to properties of \(\mu\), the Dirichlet form \((\widetilde{{\mathcal E}}, D(\widetilde{{\mathcal E}}))\) is irreducible and the diffusion penetrates \(F\), provided the Newtonian 1-capacity of \(F\) is strictly positive. Applications include the cases where \(F\) is a nested fractal or the Sierpinski carpet.
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 1416440
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: diffusion processes
zbMATH Keywords: fractals
zbMATH Keywords: Besov spaces
zbMATH Keywords: capacity
MaRDI profile type: Publication
full work available at URL: jfan.1999.3500
OpenAlex ID: W206532914
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DOI: 10.1006/JFAN.1999.3500
description: scientific article; zbMATH DE number 1416440