Extremal statistics for a resetting Brownian motion before its first-passage time
Wusong Guo, Hao Yan, Hanshuang Chen
Abstract
We study the extreme value statistics of a one-dimensional resetting Brownian motion (RBM) till its first passage through the origin starting from the position ${x}_{0}$ $(>0)$. By deriving the exit probability of RBM in an interval $[0,M]$ from the origin, we obtain the distribution ${P}_{r}(M|{x}_{0})$ of the maximum displacement $M$ and thus gives the expected value $\ensuremath{\langle}M\ensuremath{\rangle}$ of $M$ as functions of the resetting rate $r$ and ${x}_{0}$. We find that $\ensuremath{\langle}M\ensuremath{\rangle}$ decreases monotonically as $r$ increases, and tends to $2{x}_{0}$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$. In the opposite limit, $\ensuremath{\langle}M\ensuremath{\rangle}$ diverges logarithmically as $r\ensuremath{\rightarrow}0$. Moreover, we derive the propagator of RBM in the Laplace domain in the presence of both absorbing ends, and then leads to the joint distribution ${P}_{r}(M,{t}_{m}|{x}_{0})$ of $M$ and the time ${t}_{m}$ at which this maximum is achieved in the Laplace domain by using a path decomposition technique, from which the expected value $\ensuremath{\langle}{t}_{m}\ensuremath{\rangle}$ of ${t}_{m}$ is obtained explicitly. Interestingly, $\ensuremath{\langle}{t}_{m}\ensuremath{\rangle}$ shows a nonmonotonic dependence on $r$, and attains its minimum at an optimal ${r}^{*}\ensuremath{\approx}2.71691D/{x}_{0}^{2}$, where $D$ is the diffusion coefficient. Finally, we perform extensive simulations to validate our theoretical results.