Power laws, random walks, and fractional diffusion processes as well scaled refinement limits

General Invited Lecture Some types of anomalous diffusion can be modelled by generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is provided by the integral equation of the so-called continuous time random walk (CTRW), which in essence describes a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps are asymptotically governed by power laws with negative exponents related to the orders of the fractional derivatives. Further we give CTRW approximations to distributed order time-fractional diffusion, using linear combinations of individually scaled waiting time densities governed by different power laws.