# Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion

## TL;DR

This paper investigates the properties of tempered fractional Brownian motion and Langevin equations, revealing transitions between localization, ballistic, and anomalous diffusion behaviors over different time scales.

## Contribution

It introduces the tempered fractional Langevin equation (tfLe), analyzing its diffusion transitions and ergodic properties, extending understanding of tempered fractional stochastic processes.

## Key findings

- Mean squared displacement transitions from ballistic to anomalous and back to ballistic diffusion.
- Overdamped tfLe transitions from subdiffusive to ballistic behavior.
- Harmonic potential influences the diffusion transition dynamics.

## Abstract

This paper further discusses the tempered fractional Brownian motion, 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 tempered fractional Brownian motion displays localization diffusion for the long time limit and for the short time its mean squared displacement has the asymptotic form $t^{2H}$, we show that the asymptotic form of the mean squared displacement of the tfLe transits from $t^2$ (ballistic diffusion for short time) to $t^{2-2H}$, and then to $t^2$ (again ballistic diffusion for long time). On the other hand, the overdamped tfLe has the transition of the diffusion type from $t^{2-2H}$ to $t^2$ (ballistic diffusion). The tfLe with harmonic potential is also considered.