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Coupling and Harnack inequalities for Sierpiński carpets

Authors: Martin T. Barlow and Richard F. Bass
Journal: Bull. Amer. Math. Soc. 29 (1993), 208-212
MSC (2000): Primary 60B99; Secondary 28A80, 60J35
MathSciNet review: 1215306
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Abstract: Uniform Harnack inequalities for harmonic functions on the pre-and graphical Sierpinski carpets are proved using a probabilistic coupling argument. Various results follow from this, including the construction of Brownian motion on Sierpinski carpets embedded in $ {\mathbb{R}^d}$, $ d \geq 3$, estimates on the fundamental solution of the heat equation, and Sobolev and Poincaré inequalities.

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Keywords: Harnack inequality, Sierpinski carpets, self-similar, fractals, Brownian motion, heat equation, transition densities, Poincaré inequality, Sobolev inequality, spectral dimension, electrical resistance
Article copyright: © Copyright 1993 American Mathematical Society

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