Localization And Ballistic Diffusion For The Tempered Fractional Brownian-langevin Motion

This paper discusses the tempered fractional Brownian motion(tfBm),its ergodicity,and the derivation of the corresponding Fokker–Planck equation.Then we introduce the generalized Langevin equation with the tempered fractional Gaussian noise for a free particle,called tempered fractional Langevin equation(tfLe).While the tfBm displays localization diffusion for the long time limit and for the short time its mean squared displacement(MSD)is the power law function of with exponent 2,where is the Hurst exponent.We show that the asymptotic form of the MSD of the tfLe transits from ballistic diffusion for short time to ballistic diffusion for long time.On the other hand,the diffusion type of the overdamped tfLe is considered.The tfLe with harmonic potential is also considered.The reason why we do not consider the asymptotic form of its mean square displacement in the case of harmonic potential is because in this case,the mean square displacement is a bounded constant.