TL;DR
This paper proves that the martingale dimension for certain self-similar fractals, including nested fractals, is always one, confirming a longstanding conjecture in the field.
Contribution
It establishes that the martingale dimension for canonical diffusions on a broad class of self-similar fractals is always one, resolving a conjecture by S. Kusuoka.
Findings
Martingale dimension is always one for the studied fractals.
Confirms Kusuoka's conjecture on martingale dimensions.
Applicable to nested and self-similar 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].
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