Extremal statistics for first-passage trajectories of drifted Brownian motion under stochastic resetting

Abstract

We study the extreme value statistics of first-passage trajectories generating from a one-dimensional drifted Brownian motion subject to stochastic resetting to the starting point with a constant rate $r$. Each stochastic trajectory starts from a positive position $x_0$ and terminates whenever the particle hits the origin for the first time. \textcolor{blue}{We obtain the exact expression for the marginal distribution $P_r(M|x_0)$ of the maximum displacement $M$}. We find that stochastic resetting has a profound impact on $P_r(M|x_0)$ and the expected value $\langle M \rangle$ of $M$. Depending on the drift velocity $v$, $\langle M \rangle$ shows three distinct trends of change with $r$. For $v \geq 0$, $\langle M \rangle$ decreases monotonically with $r$, and tends to $2x_0$ as $r \to \infty$. For $v_c<v<0$, $\langle M \rangle$ shows a nonmonotonic dependence on $r$, in which a minimum $\langle M \rangle$ exists for an intermediate level of $r$. For $v\leq v_c$, $\langle M \rangle$ increases monotonically with $r$. Moreover, by deriving the propagator and using path decomposition technique, we obtain in the Laplace domain the joint distribution of $M$ and the time $t_m$ at which the maximum $M$ is reached. Interestingly, the dependence of the expected value $\langle t_m \rangle$ of $t_m$ on $r$ is either monotonic or nonmonotonic, depending on the value of $v$. For $v>v_m$, there is a nonzero resetting rate at which $\langle t_m \rangle$ attains its minimum. Otherwise, $\langle t_m \rangle$ increases monotonically with $r$. We provide an analytical determination of two critical values of $v$, $v_c\approx -1.69415 D/x_0$ and $v_m\approx -1.66102 D/x_0$, where $D$ is the diffusion constant. Finally, numerical simulations are performed to support our theoretical results.