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Homogeneous diffusions on the Sierpiński gasket. (English) Zbl 0917.60073
Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 86-107 (1998).
The author constructs a one parameter family of Feller diffusion processes on the (unbounded) Sierpiński gasket which are invariant under some isometries of the fractal but not necessarily scale invariant. The latter property poses a major technical problem because the time scaling of the approximating Markov chains is no more natural. A perturbation result for matrix powers in [the author, J. Theor. Probab. 9, No. 3, 647-658 (1996)] is used to derive convergence results for multi-type branching processes which in turn settle the scaling problem. The construction of the limiting process follows the arguments of Barlow and Perkins. The extreme ends of the above family are the one-dimensional Brownian motion and the so-called Brownian motion on the Sierpinski gasket. For the entire collection see [Zbl 0893.00035].
60J60Diffusion processes
60J80Branching processes
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