Stochastic resetting prevails over sharp restart for broad target distributions

TL;DR

This paper investigates how stochastic resetting outperforms sharp restart in minimizing search times for targets with broad, heavy-tailed distributions, introducing the concept of conjugate target distributions to optimize resetting protocols.

Contribution

It introduces the notion of conjugate target distributions to a given waiting time distribution and derives explicit expressions for diffusive searches in arbitrary dimensions.

Findings

Stochastic resetting is more effective than sharp restart for heavy-tailed target distributions.

Explicit conjugate target distributions are derived for diffusive processes.

Resetting strategies can be optimized based on target distribution tails.

Abstract

Resetting has been shown to reduce the completion time for a stochastic process, such as the first passage time for a diffusive searcher to find a target. The time between two consecutive resetting events is drawn from a waiting time distribution $\psi(t)$, which defines the resetting protocol. Previously, it has been shown that deterministic resetting process with a constant time period, referred to as sharp restart, can minimize the mean first passage time to a fixed target. Here we consider the more realistic problem of a target positioned at a random distance $R$ from the resetting site, selected from a given target distribution $P_T(R)$. We introduce the notion of a conjugate target distribution to a given waiting time distribution. The conjugate target distribution, $P_T^*(R)$, is that $P_T(R)$ for which $\psi(t)$ extremizes the mean time to locate the target. In the case of diffusion we derive an explicit expression for $P^*_T(R)$ conjugate to a given $\psi(t)$ which holds in arbitrary spatial dimension. Our results show that stochastic resetting prevails over sharp restart for target distributions with exponential or heavier tails.