Langevin dynamics for L\'evy walk with memory

TL;DR

This paper develops a Langevin framework for Lévy walks with memory effects modeled by correlated waiting times, revealing how memory influences diffusion, aging, and ergodicity in stochastic processes.

Contribution

It introduces a Langevin description for Lévy walks with correlated waiting times, analyzing the impact of memory on diffusion and nonstationary behaviors.

Findings

Memory correlation suppresses diffusion speed.

Friction leads to sub-ballistic superdiffusion.

Stronger correlations slow down diffusion.

Abstract

Memory effects, sometimes, can not be neglected. In the framework of continuous time random walk, memory effect is modeled by the correlated waiting times. In this paper, we derive the two-point probability distribution of the stochastic process with correlated increments as well as the one of its inverse process, and present the Langevin description of L\'evy walk with memory, i.e., correlated waiting times. Based on the built Langevin picture, the properties of aging and nonstationary are discussed. The Langevin system exhibits sub-ballistic superdiffusion if the friction force is involved, while it displays super-ballistic diffusion or hyperdiffusion if there is no friction. It is discovered that the correlation of waiting times suppresses the diffusion behavior whether there is friction or not, and the stronger the correlation of waiting times becomes, the slower the diffusion is. In particular, the correlation function, correlation coefficient, ergodicity, and scaling property of the corresponding stochastic process are also investigated.