L\'evy flights and L\'evy walks under stochastic resetting

TL;DR

This paper investigates how stochastic resetting influences the escape kinetics of Levy flights and walks, revealing conditions under which resetting accelerates escape and exploring the relationship between these two processes.

Contribution

It demonstrates that resetting can accelerate escape in Levy processes, clarifies how the effect depends on the stability index, and connects Levy flights and walks under resetting.

Findings

Resetting accelerates escape for Levy processes within certain domain parameters.

The beneficial domain depends on the stability index b1, with narrower domains for heavier tails.

Levy walks can also benefit from resetting even with low coefficient of variation.

Abstract

Stochastic resetting is a protocol of starting anew, which can be used to facilitate the escape kinetics. We demonstrate that restarting can accelerate the escape kinetics from a finite interval restricted by two absorbing boundaries also in the presence of heavy-tailed, L\'evy type, $\alpha$-stable noise. However, the width of the domain where resetting is beneficial depends on the value of the stability index $\alpha$ determining power-law decay of jump length distribution. For heavier (smaller $\alpha$) distributions the domain becomes narrower in comparison to lighter tails. Additionally, we explore connections between L\'evy flights and L\'evy walks in presence of stochastic resetting. First of all, we show that for L\'evy walks, the stochastic resetting can be beneficial also in the domain where coefficient of variation is smaller than 1. Moreover, we demonstrate that in the domain where LW are characterized by a finite mean jump duration/length, with the increasing width of the interval LW start to share similarities with LF under stochastic resetting.