Stochastic Resetting and Applications

TL;DR

This review explores stochastic processes with resetting, highlighting how resetting induces non-equilibrium states, optimizes target reaching times, and extends to complex systems and non-Poissonian scenarios.

Contribution

It provides a comprehensive overview of resetting in stochastic processes, including generalizations to complex systems and non-Poissonian resetting mechanisms.

Findings

Resetting leads to nontrivial stationary states.

Optimal resetting rate minimizes mean target reaching time.

Extensions include multiparticle, extended systems, and memory-based resetting.

Abstract

In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate $r$, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate $r$. We then generalise to an arbitrary stochastic process (e.g. L\'evy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.