On singularity of energy measures on self-similar sets
type: Item
instance of: scholarly article
title: On singularity of energy measures on self-similar sets
zbMATH Open document ID: 1097.31005
author: Masanori Hino
publication date: 2005-06-06T00:00:00Z
review text: For the classical Brownian motion in \(\mathbb{R}^d\), the associated Dirichlet form \((\mathcal{E}, \mathcal{F})\) on \(L^2(\mathbb{R}^d, dx)\) is given by \[ \mathcal{E}(f,g)=\frac{1}{2} \int_{\mathbb{R}^d} \langle \nabla f, \nabla g\rangle \,dx, \quad f,g \in \mathcal{F}=H^1(\mathbb{R}^d). \] Hence, the corresponding energy measure \(\mu_{\langle f \rangle}:=| \nabla f| ^2 \,dx\), \(f \in \mathcal{F}\), is absolutely continuous with respect to the Lebesgue measure on \(\mathbb{R}^d\). In general, this is no longer true if the underlying space is a fractal. The paper provides a general criterion for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type volume measures including the Hausdorff measure. Roughly spoken, the energy measure is singular to the canonical volume measure (i.e. the Hausdorff measure) if the elliptic Harnack inequality holds and the walk dimension is greater than 2. A special feature of the paper is that the considerations include infinitely ramified fractals. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on two-dimensional Sierpinski carpets.
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60G30
zbMATH DE Number: 2173777
zbMATH Keywords: self-similar set
zbMATH Keywords: Dirichlet form
zbMATH Keywords: energy measure
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: walk dimension
zbMATH Keywords: Sierpiński carpet
reviewed by: Uta Freiberg
MaRDI profile type: Publication
full work available at URL: s00440-004-0396-1
OpenAlex ID: W2038415495
cites work: Q4218973
cites work: Q4453420
cites work: The construction of Brownian motion on the Sierpinski carpet
cites work: On the resistance of the Sierpiński carpet
cites work: Transition densities for Brownian motion on the Sierpinski carpet
cites work: Brownian Motion and Harmonic Analysis on Sierpinski Carpets
cites work: Q4946126
cites work: Stability of parabolic Harnack inequalities on metric measure spaces
cites work: What is not in the domain of the Laplacian on Sierpinski gasket type fractals
cites work: Transition density estimates for Brownian motion on affine nested fractals
cites work: Dirichlet forms and symmetric Markov processes
cites work: Transition density estimates for diffusion processes on homogeneous random Sierpiński carpets
cites work: Small-time Gaussian behavior of symmetric diffusion semigroups
cites work: Harmonic Calculus on P.C.F. Self-Similar Sets
cites work: Q4500513
cites work: Analysis on fractals
cites work: Brownian motion penetrating fractals. An application of the trace theorem of Besov spaces
cites work: Dirichlet forms of fractals and products of random matrices
cites work: Dirichlet forms on fractals: Poincaré constant and resistance
cites work: Mesures associées à une forme de Dirichlet. Applications
cites work: Brownian motion on nested fractals
cites work: Resistance in higher-dimensional Sierpiński carpets
cites work: Composite media and asymptotic Dirichlet forms
cites work: On function spaces related to fractional diffusions on d-sets
cites work: Q3909512
DOI: 10.1007/S00440-004-0396-1
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 2173777