Martingale Dimensions of Diffusion Processes on Fractal-Like Spaces

The concept of martingale dimension has been defined for diffusion processes and more general stochastic processes, and is interpreted as the multiplicity of the associated filtration. This corresponds to the number of independent noises that the process possesses. Determining the martingale dimension is a difficult problem for diffusion processes on singular spaces such as fractal sets. The relationship between the martingale dimension and other kinds of dimensions is also not fully understood. In this paper, we give a survey of developments for this topic and discuss recent research on the case of diffusion processes on metric measure spaces.