Extremal statistics for a resetting Brownian motion before its first-passage time

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 $\left[0, M \right] $ from the origin, we obtain the distribution $P_r(M|x_0)$ of the maximum displacement $M$ and thus gives the expected value $\langle M \rangle$ of $M$ as functions of the resetting rate $r$ and $x_0$. We find that $\langle M \rangle$ decreases monotonically as $r$ increases, and tends to $2 x_0$ as $r \to \infty$. In the opposite limit, $\langle M \rangle$ diverges logarithmically as $r \to 0$. 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 Lapalce domain by using a path decomposition technique, from which the expected value $\langle t_m \rangle$ of $t_m$ is obtained explicitly. Interestingly, $\langle t_m \rangle$ shows a nonmonotonic dependence on $r$, and attains its minimum at an optimal $r^{*} \approx 2.71691 D/x_0^2$, where $D$ is the diffusion coefficient. Finally, we perform extensive simulations to validate our theoretical results.