Continuous time random walks and L\'{e}vy walks with stochastic resetting

TL;DR

This paper develops a theoretical framework for intermittent continuous time random walks and Lévy walks with stochastic resetting, revealing how resetting influences their localization and diffusion properties.

Contribution

It introduces a comprehensive theory for CTRW and Lévy walks with stochastic resetting, analyzing their mean squared displacement and probability density functions.

Findings

Resetting localizes CTRW processes.

Lévy walks with resetting diffuse slower.

Resetting significantly alters the asymptotic behavior of the probability density.

Abstract

Intermittent stochastic processes appear in a wide field, such as chemistry, biology, ecology, and computer science. This paper builds up the theory of intermittent continuous time random walk (CTRW) and L\'{e}vy walk, in which the particles are stochastically reset to a given position with a resetting rate $r$. The mean squared displacements of the CTRW and L\'{e}vy walks with stochastic resetting are calculated, uncovering that the stochastic resetting always makes the CTRW process localized and L\'{e}vy walk diffuse slower. The asymptotic behaviors of the probability density function of L\'evy walk with stochastic resetting are carefully analyzed under different scales of $x$, and a striking influence of stochastic resetting is observed.