Transport properties of random walks under stochastic non-instantaneous resetting

TL;DR

This paper investigates how non-instantaneous stochastic resetting affects the transport properties of random walks, revealing conditions under which the mean square displacement can increase, decrease, or stabilize, and identifying regimes of stochastic localization.

Contribution

It introduces a two-state model with non-instantaneous resetting combining exploration and ballistic return, analyzing its effects on transport properties for exponential and Pareto resetting PDFs.

Findings

Stationary distribution and MSD depend on the propagator and resetting PDF.

MSD can increase, decrease, or stay constant with returning velocity.

Pareto resetting can induce sub-diffusive behavior and stochastic localization.

Abstract

Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce non-instantaneous resetting as a two-state model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport sub-diffusive and even reaching a stationary MSD., i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.