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# Brownian Motion and Harmonic Analysis on Sierpinski Carpets

Published:1999-08-01
Printed: Aug 1999
• Martin T. Barlow
• Richard F. Bass
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## Abstract

We consider a class of fractal subsets of $\R^d$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincar\'e inequalities to this setting.
 Keywords: Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutions
 MSC Classifications: 60J60 - Diffusion processes [See also 58J65] 60B05 - Probability measures on topological spaces 60J35 - Transition functions, generators and resolvents [See also 47D03, 47D07]