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 $⟨M⟩$ of $M$ as functions of the resetting rate $r$ and $x_0$. We find that $⟨M⟩$ decreases monotonically as $r$ increases, and tends to $2x_0$ as $r\to \infty$. In the opposite limit, $⟨M⟩$ 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 $⟨t_m⟩$ of $t_m$ is obtained explicitly. Interestingly, $⟨t_m⟩$ shows a nonmonotonic dependence on $r$, and attains its minimum at an optimal $r^{\ast }\approx 2.71691D/x_0^2$, where $D$ is the diffusion coefficient. Finally, we perform extensive simulations to validate our theoretical results.

arXiv commenttwo-column 9 pages, 4 figures

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