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 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, 60B05, 60J35 show english descriptions Diffusion processes [See also 58J65]
Probability measures on topological spaces
Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60 - Diffusion processes [See also 58J65]
60B05 - Probability measures on topological spaces
60J35 - Transition functions, generators and resolvents [See also 47D03, 47D07]

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