The fractional Dodson diffusion equation: a new approach

The aim of this Short Note is to highlight that the 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 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 ≤ 2 and 0 < β ≤ 1. It includes the fractional Brownian motion when 0 < α ≤ 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.