Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
The main aim of the paper is the analysis of the diffusive behavior described by a Langevin equation with generalized Gaussian noise. The basic motivation for considering generalized versions of Brownian motion is the description of general diffusion behavior which might reproduce the motion of free or harmonically bound particles in complex environments. In particular, the paper discusses the recently introduced tempered fractional Brownian motion [\textit{M. M. Meerschaert} and \textit{F. Sabzikar}, Stat. Probab. Lett. 83, No. 10, 2269--2275 (2013; Zbl 1287.60050)] and considers the corresponding Fokker-Planck equation, obtained by derivation of the Gaussian probability density function. Non-Markovianity of the process is implied by a time-dependent diffusion constant. The authors also study ergodicity of the process by means of a suitable indicator. The formal derivative of the tempered fractional Brownian motion defines the tempered fractional Gaussian noise, which turns out to be a stationary Gaussian process. The authors further introduce a generalized Langevin equation driven by tempered fractional Gaussian noise, which they term tempered fractional Langevin equation, considering both the case of a free particle and the presence of a harmonic potential. The friction kernel considered in the generalized Langevin, at variance with the previous work [\textit{T. Sandev}, Mathematics 5, No. 4, Paper No. 66, 11 p. (2017; Zbl 1395.82195)], is the autocorrelation function of the tempered fractional Gaussian noise, so as to comply with the second fluctuation-dissipation theorem, which fixes the relation between noise and friction in generalized Langevin equations. Short and long time behavior of the mean square displacement is studied in detail by means of the Laplace transform, showing an important dependence on the parameter which characterises the tempered noise. The theoretical analysis is supported by numerical studies.
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