General approach to stochastic resetting

TL;DR

This paper investigates how stochastic resetting influences diffusion and subdiffusion processes, revealing conditions for steady states, the form of probability distributions, and relaxation behaviors under various resetting protocols.

Contribution

It provides a general framework for understanding stochastic resetting effects on diffusion and subdiffusion, including explicit solutions and conditions for steady states.

Findings

MSD relaxes to a constant with finite mean and variance of reset times

PDF exhibits a cusp independent of reset time distribution details

Steady state can exist for subdiffusion with only finite mean reset times

Abstract

We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that MSD relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the PDF of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H-function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean.