A class of self-similar stochastic processes with stationary increments to model anomalous diffusion in physics. (English) Zbl 1173.26005

The present paper expresses the motivation of the grey noise theory [cf. W. R. Schneider, World Scientific, 676–681 (1990)], which leads to a class of self similar stochastic processes. The authors extend the grey Brownian motion (which is the fundamental solution of the time-fractional diffusion equation) to a class called generalized grey Brownian motion. In the first section, the mathematical construction of the problem is described; then it is shown that this particular class is made up by H-ssi processes, which contain either Gaussian or non-Gaussian processes. In the final section of the paper it is shown how the partial integro-differential equation of fractional type is used to described the time evolution of the marginal density function. The list of references is exhaustive for a better understanding.


26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
44A10 Laplace transform
60G18 Self-similar stochastic processes
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