SUBORDINATION IN FRACTIONAL DIFFUSION PROCESSES VIA CONTINUOUS TIME RANDOM WALK

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. This formula allows us to treat the CTRW as a discrete-space discrete-time random walk that in the continuum limit tends towards a generalized diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Plots of simulations for some case-studies are given in order to display the sample paths for the fractional diffusion processes, generally non Markovian, that are obtained by the composition of two Markovian processes.