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

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.