Functions of locally bounded variation on Wiener spaces

We introduce the concept of functions of locally bounded variation on abstract Wiener spaces and study their properties. Some nontrivial examples and applications to stochastic analysis are also discussed.Comment: 20 pages, accepted for publicatio

Fractional order Taylor's series and the neo-classical inequality

We prove the neo-classical inequality with the optimal constant, which was conjectured by T. J. Lyons [Rev. Mat. Iberoamericana 14 (1998) 215-310]. For the proof, we introduce the fractional order Taylor's series with residual terms. Their application to a particular function provides an identity that deduces the optimal neo-classical inequality.Comment: 11 pages, 1 figur

Estimates of the local spectral dimension of the Sierpinski gasket

We discuss quantitative estimates of the local spectral dimension of the two-dimensional Sierpinski gasket. The present arguments were inspired by a previous study of the distribution of the Kusuoka measure by R. Bell, C.-W. Ho, and R. S. Strichartz [Energy measures of harmonic functions on the Sierpi\'nski gasket, Indiana Univ. Math. J. 63 (2014), 831-868].Comment: 9 pages, 1 figur

Martingale dimensions for fractals

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].Comment: 22 pages, 1 figur

Estimate of martingale dimension revisited (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

The concept of martingale dimension is defined for symmetric diffusion processes and is interpreted as the multiplicity of filtration. However, if the underlying space is a fractal-like set, then estimating the martingale dimension quantitatively is a difficult problem. To date, the only known nontrivial estimates have been those for canonical diffusions on a class of self-similar fractals. This paper surveys existing results and discusses more-general situations