Continuous-time random walks and Lévy walks with stochastic resetting

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 Lévy walk, in which the particles are stochastically reset to a given position at the end of each step of renewal with a resetting rate r, implying that the stochastic resetting process is related to the time of “waiting” for CTRW or “walking” for Lévy walk. The mean-squared displacements of the CTRW and Lévy walk with stochastic resetting are calculated, uncovering that the stochastic resetting always makes the CTRW process localized when the waiting time density is exponential or power law and Lévy walk diffuse more slowly. The asymptotic behaviors of the probability density function of Lévy walk with stochastic resetting are carefully analyzed under different scales of x, and a striking influence of stochastic resetting is observed.