Erdélyi-Kober Fractional Diffusion

Details Erdélyi-Kober Fractional Diffusion Erdélyi-Kober Fractional Diffusion

2011-12-05

arxiv.org/abs/1112.0890v1

Fractional Calculus and Applied Analysis 15, 117--127 (2012)

Physics

Mathematical Physics

Accepted for publication in Fractional Calculus and Applied Analysis

Scientific paper

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.

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