Martingale dimensions for fractals

Abstract

We prove that the martingale dimensions for canonical diffusion processes on a class of self-similar sets including nested fractals are always one. This provides an affirmative answer to the conjecture of S. Kusuoka [Publ. Res. Inst. Math. Sci. 25 (1989) 659--680].

22 pages, 1 figure

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• Measurable Riemannian structures associated with strong local Dirichlet forms
• Backward problems for stochastic differential equations on the Sierpinski gasket
• Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces
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• Sobolev inequalities on product Sierpinski spaces
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• Minkowski measurability results for self-similar tilings and fractals with monophase generators
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