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Fractional kinetic equations driven by Gaussian or infinitely divisible noise. (English) Zbl 1076.60044

This paper investigates a certain type of space- and time-fractional kinetic equation with Gaussian or infinitely divisible noise input. The temporal operator is a sum of fractional derivatives with constant coefficients. The spatial operator is the infinitesimal operator of a Lévy process, in particular the Riesz-Bessel operator. Solutions are provided both for bounded and unbounded spatial domains via the series expansions of the Greens functions. The sample path properties are studied: bounds on the variances of the increments, moduli of continuity, asymptotic temporal spectral density, and to some extend also higher-order properties. In particular, it is shown that long-range dependence may arise in the temporal solution under certain conditions on the spatial operator. Furthermore, it is shown how the studied fractional diffusion processes arise as limits of continuous time random walks. The main emphasis is on equations with Gaussian noise input, but the paper concludes with discussions of general Lévy noise input.

MSC:

60G60 Random fields
60G17 Sample path properties
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
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