Erd\'elyi-Kober Fractional Diffusion

TL;DR

This paper introduces Erdélyi-Kober fractional diffusion as a new class of anomalous diffusion processes driven by fractional integral equations, unifying various known stochastic processes under a common framework.

Contribution

It proposes the concept of Erdélyi-Kober fractional diffusion, connecting generalized grey Brownian motion with fractional integral equations and highlighting its role in modeling anomalous diffusion.

Findings

ggBm models both fast and slow anomalous diffusion

Includes fractional Brownian motion and time-fractional processes as special cases

Mainardi function generalizes Gaussian distribution in this context

Abstract

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 Erd\'elyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as {\it Erd\'elyi-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: $0 < \alpha \le 2$ and $0 < \beta \le 1$. It includes the fractional Brownian motion when $0 < \alpha \le 2$ and $\beta=1$, the time-fractional diffusion stochastic processes when $0 < \alpha=\beta <1$, and the standard Brownian motion when $\alpha=\beta=1$. 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.