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Ergodic properties of anomalous diffusion processes. (English) Zbl 1227.82058

Summary: In this paper we study ergodic properties of some classes of anomalous diffusion processes. Using the recently developed measure of dependence called the correlation cascade, we derive a generalization of the classical Khinchin theorem. This result allows us to determine ergodic properties of Lévy-driven stochastic processes. Moreover, we analyze the asymptotic behavior of two different fractional Ornstein-Uhlenbeck processes, both originating from subdiffusive dynamics. We show that only one of them is ergodic.

MSC:

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
26A33 Fractional derivatives and integrals
60J60 Diffusion processes
37A25 Ergodicity, mixing, rates of mixing
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