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.

中文翻译：

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随机重置到随机位置的布朗搜索器的首次通过时间

我们研究了原点周围随机分布的重置点对在一维移动的布朗搜索器的平均首次通过时间的影响。我们将搜索效率与复位到原点对应的搜索效率进行比较，发现后者的平均首次通过时间可以大于或小于分布式情况，具体取决于复位点是对称分布还是非对称分布。特别是，我们证明了存在最佳重置率，如果目标位于该区间之外，则该最优重置率可以将分布式重置的平均首次通过时间最小化到有限区间。当目标位置属于重置区间或​​者无限大时，则不存在最佳重置率，但存在使平均首次通过时间最小化的最佳重置区间宽度或重置特征尺度。我们还表明，在重置点上平均的首次通过密度仅取决于其一阶矩。因此，存在一个等价点，使得重置到该点的第一通道问题在统计上与分布式重置的情况等价。我们通过分析这些案例的首次通过时间的波动来结束我们的研究。我们所有的分析结果都通过数值模拟得到验证。