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Two-particle anomalous diffusion: probability density functions and self-similar stochastic processes. (English) Zbl 1339.60116

Summary: Two-particle dispersion is investigated in the context of anomalous diffusion. Two different modelling approaches related to time subordination are considered and unified in the framework of self-similar stochastic processes. By assuming a single-particle fractional Brownian motion and that the two-particle correlation function decreases in time with a power law, the particle relative separation density is computed for the cases with time sub-ordination directed by a unilateral \(M\)-Wright density and by an extremal Lévy stable density. Looking for advisable mathematical properties (for instance, the stationarity of the increments), the corresponding self-similar stochastic processes are represented in terms of fractional Brownian motions with stochastic variance, whose profile is modelled by using the \(M\)-Wright density or the Lévy stable density.

MSC:

60J60 Diffusion processes
60G18 Self-similar stochastic processes
60G22 Fractional processes, including fractional Brownian motion
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
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