Extremal statistics for a one-dimensional Brownian motion with a reflective boundary

Abstract

We investigate the extreme value statistics of a one-dimensional Brownian motion (with the diffusion constant $D$) during a time interval $\left[0, t \right]$ in the presence of a reflective boundary at the origin, starting from a positive position $x_0$. By deriving the survival probability of the Brownian particle without hitting an absorbing boundary at $x=M$, we obtain the distribution $P(M|x_0,t)$ of the maximum displacement $M$ and its expectation $\langle M \rangle$. In the short-time limit, i.e., $t \ll t_d$ where $t_d=x_0^2/D$ is the diffusion time from the starting position $x_0$ to the reflective boundary at the origin, the particle behaves like a free Brownian motion without any boundaries. In the long-time limit, $t \gg t_d$, $\langle M \rangle$ grows with $t$ as $\langle M \rangle \sim \sqrt{t}$, which is similar to the free Brownian motion, but the prefactor is $π/2$ times of the free Brownian motion, embodying the effect of the reflective boundary. By solving the propagator and using a path decomposition technique, we obtain the joint distribution $P(M,t_m|x_0,t)$ of $M$ and the time $t_m$ at which this maximum is achieved, from which the marginal distribution $P(t_m|x_0,t)$ is also obtained. For $t \ll t_d$, $P(t_m|x_0,t)$ looks like a U-shaped attributed to the arcsine law of free Brownian motion. For $t$ equal to or larger than order of magnitude of $t_m$, $P(t_m|x_0,t)$ deviates from the U-shaped distribution and becomes asymmetric with respect to $t/2$. Moreover, we compute the expectation $\langle t_m \rangle$ of $t_m$, and find that $\langle t_m \rangle/t$ is an increasing function of $t$. In two limiting cases, $\langle t_m \rangle/t \to 1/2$ for $t \ll t_d$ and $\langle t_m \rangle/t \to (1+2G)/4 \approx 0.708$ for $t \gg t_d$, where $G\approx0.916$ is the Catalan's constant. All the theoretical results are validated by numerical simulations.