Asymptotic densities of ballistic L\'evy walks

TL;DR

This paper introduces an analytical approach to determine the asymptotic density profiles of ballistic Lévy walks, highlighting how different step execution methods and variable speeds influence the resulting propagators.

Contribution

It presents a novel analytical method for analyzing the asymptotic densities of coupled continuous time random walks in the ballistic regime, emphasizing the impact of step execution and speed variability.

Findings

Different step execution scenarios significantly affect propagators.

Variable speeds during steps are reflected in the asymptotic density.

The method distinguishes ballistic Lévy walks from standard non-ballistic walks.

Abstract

We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard non-ballistic random walks.