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The \(M\)-Wright function in time-fractional diffusion processes: a tutorial survey. (English) Zbl 1222.60060
Summary: We survey the properties of a transcendental function of the Wright type, nowadays known as \(M\)-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 \(M\)-Wright function is shown to play the same key role as the Gaussian density in the standard and the fractional Brownian motion. Furthermore, these processes provide stochastic models suitable for describing phenomena of anomalous diffusion of both slow and fast type.

MSC:
60J60 Diffusion processes
26A33 Fractional derivatives and integrals
60G17 Sample path properties
35R11 Fractional partial differential equations
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