Representation of random walk in fractal space-time

To analyze the anomalous diffusion on a fractal structure with fractal in the time axis, we propose a statistical representation given by a path integral method in arbitrary fractal space-time. Using the method, we can understand easily several properties of the non-Gaussian-type behavior, and a differential equation for the path integral is derived. Finally, to check the validity of this theory, analytical results in this paper are applied to the random walk on the two-dimensional Sierpinski carpet, which agree precisely with numerical results by Monte Carlo simulations in the paper of Fujiwara and Yonezawa [Phys. Rev. E 51 (1995) 2277].