The M-Wright function in time-fractional diffusion processes: a tutorial survey

In the present review we survey the properties of a transcendental function the Wright type, nowadays known as M-Wright function, entering as a density in a relevant class of self-similar stochastic processes we generally refer to as time-fractional diffusion processes. Indeed, the master equations governing these processes generalize the diffusion equation by means of time-integral operators interpreted as of fractional order. When these generalized diffusion processes are characterized with stationary increments, the M-Wright function is to play the same key role as the Gaussian density in the standard and Brownian motions. Furthermore, these processes provide stochastic suitable for describing phenomena of anomalous diffusion of both slow fast type.