Copyright © 2010 Francesco Mainardi et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as -Wright function, entering as a probability density in a
relevant class of self-similar stochastic processes that we generally refer to as time-fractional
diffusion processes. Indeed, the master equations governing these processes
generalize the standard diffusion equation by means of time-integral operators interpreted
as derivatives of fractional order. When these generalized diffusion processes are properly
characterized with stationary increments, the -Wright function is shown to play the
same key role as the Gaussian density in the standard and fractional Brownian motions.
Furthermore, these processes provide stochastic models suitable for describing phenomena
of anomalous diffusion of both slow and fast types.