Fractional diffusion and medium heterogeneity: the case of the continuous time random walk

In this contribution, we show that fractional diffusion emerges from a simple Markovian Gaussian random walk when the medium displays a power-law heterogeneity. Within the framework of the continuous-time random walk, the heterogeneity of the medium is represented by the selection, at any jump, of a different time-scale for an exponential survival probability. The resulting process is a non-Markovian non-Gaussian random walk. In particular, for a power-law distribution of the time scales, the resulting random walk corresponds to a time-fractional diffusion process. We relate the power-law of the medium heterogeneity to the fractional-order of the diffusion. This relation provides an interpretation and an estimation of the fractional order of derivation in terms of environmental heterogeneity. The results are supported by simulations.