Generalized fractional master equation for self-similar stochastic processes modelling anomalous diu000busion

摘要

The Master Equation approach to model anomalous diffusion is considered. In particular, the formulation is extended to the time-stretching generalization on the basis of the superposition mechanism of processes with different diffusion coefficients distributed according to a spectrum function. When this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi– Kober fractional diffusion that is the Master Equation of the generalized grey Brownian motion (ggBm). The generalized grey Brownian motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of selfsimilar processes with stationary increments and 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 diu000busion stochastic processes when 0<α=β< 1, and the standard Brownian motion when 0<α =β<1. In the ggBm framework, the M-Wright function (known also as 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.