Heat Kernel Asymptotics for the Measurable Riemannian Structure on the Sierpinski Gasket

For the measurable Riemannian structure on the Sierpinski gasket introduced by Kigami, various short time asymptotics of the associated heat kernel are established, including Varadhan’s asymptotic relation, some sharp one-dimensional asymptotics at vertices, and a non-integer-dimensional on-diagonal...

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Bibliographic Details
Published in:Potential analysis 2012-01, Vol.36 (1), p.67-115
Main Author: Kajino, Naotaka
Format: Article
Language:English
Subjects:
Functional Analysis
Geometry
Mathematics
Mathematics and Statistics
Potential Theory
Probability Theory and Stochastic Processes
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Summary:For the measurable Riemannian structure on the Sierpinski gasket introduced by Kigami, various short time asymptotics of the associated heat kernel are established, including Varadhan’s asymptotic relation, some sharp one-dimensional asymptotics at vertices, and a non-integer-dimensional on-diagonal behavior at almost every point. Moreover, it is also proved that the asymptotic order of the eigenvalues of the corresponding Laplacian is given by the Hausdorff and box-counting dimensions of the space.
ISSN:0926-2601
1572-929X
DOI:10.1007/s11118-011-9221-5