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].
"Martingale dimensions for fractals." Ann. Probab. 36 (3) 971 - 991, May 2008. https://doi.org/10.1214/07-AOP349