Coupling and Harnack inequalities for Sierpiński carpets

Barlow, Martin T.; Bass, Richard F.

doi:10.1090/S0273-0979-1993-00424-5

Coupling and Harnack inequalities for Sierpiński carpets
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by Martin T. Barlow and Richard F. Bass PDF

Bull. Amer. Math. Soc. 29 (1993), 208-212 Request permission

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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