First-passage time of a Brownian searcher with stochastic resetting to random positions

TL;DR

This paper investigates how stochastic resetting to random positions affects the mean first passage time of a one-dimensional Brownian searcher, revealing optimal reset strategies and equivalences with fixed resetting points.

Contribution

It introduces a comprehensive analysis of first-passage times with distributed resetting, identifying optimal reset rates and intervals, and establishes a link between resetting distributions and fixed points.

Findings

Existence of an optimal reset rate for finite interval targets.

No optimal reset rate when the target is inside the resetting interval or at infinity.

First-passage density depends only on the first moment of the resetting distribution.

Abstract

We study the effect of a resetting point randomly distributed around the origin on the mean first passage time of a Brownian searcher moving in one dimension. We compare the search efficiency with that corresponding to reset to the origin and find that the mean first passage time of the latter can be larger or smaller than the distributed case, depending on whether the resetting points are symmetrically or asymmetrically distributed. In particular, we prove the existence of an optimal reset rate that minimizes the mean first-passage time for distributed resetting to a finite interval if the target is located outside this interval. When the target position belongs to the resetting interval or it is infinite then no optimal reset rate exists, but there is an optimal resetting interval width or resetting characteristic scale which minimizes the mean first-passage time. We also show that the first-passage density averaged over the resetting points depends on its first moment only. As a consequence, there is an equivalent point such that the first-passage problem with resetting to that point is statistically equivalent to the case of distributed resetting. We end our study by analyzing the fluctuations of the first-passage times for these cases. All our analytical results are verified through numerical simulations.