Brownian Motion and Harmonic Analysis on Sierpinski Carpets

Barlow, Martin T. ; Bass, Richard F.

Canadian journal of mathematics, Tome 51 (1999) no. 4, pp. 673-744

Résumé

We consider a class of fractal subsets of ${{\mathbb{R}}^{d}}$ 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 Poincaré inequalities to this setting.

Détail

DOI : 10.4153/CJM-1999-031-4

Mots-clés : 60J60, 60B05, 60J35, Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutions

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