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

Abstract

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.