Martingale dimensions for fractals (Q2427054)

About ten years ago \textit{S. Kusuoka} [Publ. Res. Inst. Math. Sci. 25, No.~4, 659--680 (1989; Zbl 0694.60071)] proved that if \({\mathfrak F}\) is the filtration generated by the canonical diffusion the \(d\)-dimensional Sierpiński gasket, there exists one particular martingale additive functional \(M\), such that every \({\mathfrak F}\)-martingale additive functional with finite energy is a stochastic integral of \(M\), and he conjectured that the same should be true for all nested fractals. Using a method different from Kusuoka's, the author now proves the property for a class of fractals including the nested ones.