Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, 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 describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this 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. 1. Introduction Statistical description of diffusive processes can be performed both at the microscopic and at the macroscopic levels. The microscopic-level description concerns the simulation of the particle trajectories by opportune stochastic models. Instead, the macroscopic-level description requires the derivation of the evolution equation of the probability density function of the particle displacement (i.e., the Master Equation), which is, indeed, connected to the microscopic trajectories. The problem of microscopic and macroscopic descriptions of physical systems and their connection is addressed and discussed in a number of cases by Balescu [1]. The most common examples of this microscopic-to-macroscopic dualism are the Brownian motion process together with the standard diffusion equation and the Ornstein-Uhlenbeck stochastic process with the Fokker-Planck equation (see, e.g., [2, 3]). But the same coupling occurs for several applications of the random walk method at the microscopic level and the resulting macroscopic description provided by the Master Equation for the probability density function [4]. In many diffusive phenomena, the classical flux-gradient relationship does not hold. In these cases anomalous diffusion arises because of the presence of nonlocal and memory effects. In particular, the variance of the spreading particles does no longer grow linearly in time. Anomalous diffusion is referred