Uniqueness of Brownian motion on Sierpiński carpets

Martin T. Barlow; Richard F. Bass; Takashi Kumagai; Alexander Teplyaev

doi:10.4171/JEMS/211

Uniqueness of Brownian motion on Sierpiński carpets

Martin T. Barlow, Richard F. Bass, Takashi Kumagai, Alexander Teplyaev

Research output: Contribution to journal › Article › peer-review

65 Citations (Scopus)

Abstract

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpiński carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

Original language	English
Pages (from-to)	655-701
Number of pages	47
Journal	Journal of the European Mathematical Society
Volume	12
Issue number	3
DOIs	https://doi.org/10.4171/JEMS/211
Publication status	Published - 2010
Externally published	Yes

Keywords

Brownian motion
Diffusions
Dirichlet forms
Fractals
Sierpiński carpet
Uniqueness

ASJC Scopus subject areas

Mathematics(all)
Applied Mathematics

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10.4171/JEMS/211

Cite this

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abstract = "We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpi{\'n}ski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.",

keywords = "Brownian motion, Diffusions, Dirichlet forms, Fractals, Sierpi{\'n}ski carpet, Uniqueness",

author = "Barlow, {Martin T.} and Bass, {Richard F.} and Takashi Kumagai and Alexander Teplyaev",

year = "2010",

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AU - Barlow, Martin T.

AU - Bass, Richard F.

AU - Kumagai, Takashi

AU - Teplyaev, Alexander

PY - 2010

Y1 - 2010

N2 - We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpiński carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

AB - We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpiński carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

KW - Brownian motion

KW - Diffusions

KW - Dirichlet forms

KW - Fractals

KW - Sierpiński carpet

KW - Uniqueness

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Uniqueness of Brownian motion on Sierpiński carpets

Abstract

Keywords

ASJC Scopus subject areas

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