The M-Wright function as a generalization of the Gaussian density for fractional diffusion processes

Abstract

The leading role of a special function of the Wright-type, referred to as M-Wright or Mainardi function, within a parametric class of self-similar stochastic processes with stationary increments, is surveyed. This class of processes, known as generalized grey Brownian motion, provides models for both fast and slow anomalous diffusion. In view of a subordination-type formula involving M-Wright functions, these processes emerge to have all finite moments and be uniquely defined by their mean and auto-covariance structure like Gaussian processes. The corresponding master equation is shown to be a fractional differential equation in the Erdélyi-Kober sense and the diffusive process is named Erdélyi-Kober fractional diffusion. In Appendix, an historical overview on the M-Wright function is reported.MSC 2010 : Primary 33E20; Secondary 26A33, 33E30, 44A35, 60G18, 60G22

Short note on the emergence of fractional kinetics

“…It is also referred to as M-Wright/Mainardi function because it was originally obtained by Mainardi as the fundamental solution of the time-fractional diffusion equation [32]. It has been shown that when such fractional diffusion processes are properly characterized with stationary increments, the M-Wright/Mainardi function plays the same key role as the Gaussian density for the standard and fractional Brownian motions [33,34]. The properties of the corresponding master equation lead to such diffusion processes being named as Erdélyi-Kober fractional diffusion [35,36].…”

Self-similar stochastic models with stationary increments for symmetric space-time fractional diffusion

“…What concerns particle displacement PDFs, they generally are not Gaussian [45] and mainly characterized by the decay of their tails. In particular, the PDF related to the time-fractional diffusion equation is a M-Wright/Mainardi [49], [50] density that is unimodal (diffusive) and bimodal (wave-like behaviour) in subdiffusion and superdiffusion cases, respectively. In subdiffusion cases, tails decay with a stretched exponential that is fatter than the Gaussian and in superdiffusion cases with a stretched exponential that is thinner than the Gaussian.…”

On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

“…In this subsection, we shortly discuss the special functions of the Wright type that appear in these derivations. For more details regarding theory and applications of these special functions, we refer the reader to, for example, [30][31][32][33][34][35][36].…”