L\'{e}vy-walk-like Langevin dynamics

Abstract

Continuous time random walks and Langevin equations are two classes of stochastic models for describing the dynamics of particles in the natural world. While some of the processes can be conveniently characterized by both of them, more often one model has significant advantages (or has to be used) compared with the other one. In this paper, we consider the weakly damped Langevin system coupled with a new subordinator|$\alpha$-dependent subordinator with $1<\alpha<2$. We pay attention to the diffusion behaviour of the stochastic process described by this coupled Langevin system, and find the super-ballistic diffusion phenomena for the system with an unconfined potential on velocity but sub-ballistic superdiffusion phenomenon with a confined potential, which is like L\'{e}vy walk for long times. One can further note that the two-point distribution of inverse subordinator affects mean square displacement of this coupled weakly damped Langevin system in essential.