Brownian motion and harmonic analysis on Sierpinski carpets. (English) Zbl 0945.60071

Barlow, Martin T.; Bass, Richard F.

Can. J. Math. 51, No. 4, 673-744 (1999).

This long paper is a major contribution to the analysis on fractals. A class of fractals called generalized Sierpiński carpets is considered, which provide a reasonably simple but general family of infinitely ramified fractals, which may have spectral dimension \(>2\). Earlier papers of the authors like [Ann. Inst. Henri Poincaré, Probab. Stat. 25, No. 3, 225-257 (1989; Zbl 0691.60070)] were restricted to a two-dimensional situation. The starting point of the analysis is a uniform Harnack inequality. The proof uses a new probabilistic coupling technique, which is of independent interest also in a non-fractal context. Then a locally isotropic diffusion (Brownian motion) on the fractal is constructed and used to derive sharp estimates for the heat kernel. Some classical Sobolev and Poincaré inequalities are extended to the present setting, and basic properties of the Brownian motion, like recurrence and transience, moduli of continuity, Hausdorff dimension of the range and existence of self-intersections are obtained.

Reviewer: Peter Mörters (Kaiserslautern)

MSC:

60J60	Diffusion processes
60B05	Probability measures on topological spaces
60J35	Transition functions, generators and resolvents

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