The Lévy walk with rests under stochastic resetting

Abstract

Abstract The Lévy walk with rests (LWR) model is a typical two-state stochastic process that has been widely and successfully adopted in the study of intermittent stochastic phenomena in physical and biological systems. Stochastic processes under resetting provide treatable and interesting schemes to study foraging and search strategies. In this manuscript, we focus on the anomalous diffusive behavior of the LWR under stochastic resetting. We consider both the case of instantaneous resetting, in which the particle stochastically returns to a given position immediately, and the case of noninstantaneous resetting, in which the particle returns to a given position with a finite velocity. The anomalous diffusive behaviors are analyzed and discussed by calculating the mean squared displacement analytically and numerically. Results reveal that the stochastic resetting can not only hinder the diffusion, where the diffusion evolves toward a saturation state, but also enhances it, where as compared with the LWR without resetting, the diffusion exponent surprisingly increases. As far as we know, the enhancement effect caused by stochastic resetting has not yet been reported. In addition, the resetting time probability density function (PDF) of the instantaneous resetting and the return time PDF of the noninstantaneous resetting are studied. Results reveal that the resetting time PDF could follow a power law provided that the sojourn time PDF is power-law distributed and the sojourn time with a heavier tail plays a decisive role in determining the resetting time PDF, whereas the shape of the return time PDF is determined by not only by the sojourn time PDF, but also by the return manner.