Ballistic L\'evy walk with rests: Escape from a bounded domain

TL;DR

This paper analyzes a Le9vy walk model with rests confined within a bounded domain, deriving first passage time properties and exploring effects of resting times and position-dependent waiting times, validated by simulations.

Contribution

It introduces a Le9vy walk model with finite resting times in a bounded domain and derives analytical expressions for escape times, including effects of resting time and position dependence.

Findings

Mean first passage time is proportional to barrier position.

Long resting times lead to properties similar to Le9vy flights.

Analytical results agree with Monte Carlo simulations.

Abstract

The L\'evy walk process for the lower interval of the time of flight distribution ($\alpha<1$) and with finite resting time between consecutive flights is discussed. The motion is restricted to a region bounded by two absorbing barriers and the escape process is analysed. By means of a Poisson equation, the total density, which includes both flying and resting phase, is derived and the first passage time properties determined: the mean first passage time appears proportional to the barrier position; moreover, the dependence of that quantity on $\alpha$ is established. Two limits emerge from the model: of short waiting time, that corresponds to L\'evy walks without rests, and long waiting time which exhibits properties of a L\'evy flights model. The similar quantities are derived for the case of a position-dependent waiting time. Then the mean first passage time rises with barrier position faster than for L\'evy flights model. The analytical results are compared with Monte Carlo trajectory simulations.