Erdélyi-Kober Fractional Diffusion

The aim of this Short Note is to highlight that the {\it generalized grey Brownian motion} (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as {\it Erdélyi-Kober fractional diffusion}. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: $0 < α\le 2$ and $0 < β\le 1$. It includes the fractional Brownian motion when $0 < α\le 2$ and $β=1$, the time-fractional diffusion stochastic processes when $0 < α=β<1$, and the standard Brownian motion when $α=β=1$. In the ggBm framework, the Mainardi function emerges as a natural generalization of the Gaussian distribution recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

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