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Generalized fractional master equation for self-similar stochastic processes modelling anomalous diffusion. (English) Zbl 1260.60163
Summary: 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.

60J60Diffusion processes
60G18Self-similar processes
60G22Fractional processes, including fractional Brownian motion
Full Text: DOI
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