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arXiv · 2605.31207

Cover time statistics of one-dimensional Brownian motion under stochastic resetting

Anirban Ghosh·Sanjib Sabhapandit
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Abstract

We investigate the effect of stochastic resetting on the statistics of the cover time of a one-dimensional Brownian motion with a diffusion constant $D$, confined to a finite interval of length $L$. The cover time $t_c$, defined as the minimum time required for the particle to visit every point of the interval at least once, exhibits a non-monotonic dependence on the scaled reset rate $\rho = rL^2/4D$. The scaled mean cover time $4D\langle t_c\rangle/L^2$ initially decreases with increasing $\rho$, attains a minimum at an optimal value $\rho^*$, and then increases with $\rho$, indicating an optimal resetting rate that minimizes the search duration. Furthermore, we derive an exact analytical expression for the cover time distribution, including its asymptotic limits, which agree well with numerical simulations. These results demonstrate that stochastic resetting serves as an efficient mechanism for optimizing cover-time processes in confined geometries.

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Anirban Ghosh, Sanjib Sabhapandit. 2026-05-29. Cover time statistics of one-dimensional Brownian motion under stochastic resetting. https://arxiv.org/abs/2605.31207

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