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

In this work, we investigate the extreme value statistics of a one-dimensional Brownian motion (with the diffusion constant D) during a time interval 0,t in the presence of a reflective boundary at the origin, when starting from a positive position x0. We first obtain the distribution P(M|x0,t) of the maximum displacement M and its expectation 〈M〉. In the short-time limit, i.e., t≪td where td=x02/D is the diffusion time from the starting position x0 to the reflective boundary at the origin, the particle behaves like a free Brownian motion without any boundaries. In the long-time limit, t≫td, 〈M〉 grows with t as 〈M〉∼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 then obtain the joint distribution P(M,tm|x0,t) of M and the time tm at which this maximum is achieved, from which the marginal distribution P(tm|x0,t) of tm is also obtained. For t≪td, P(tm|x0,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 tm, P(tm|x0,t) deviates from the U-shaped distribution and becomes asymmetric with respect to t/2. Moreover, we compute the expectation 〈tm〉 of tm, and find that 〈tm〉/t is an increasing function of t. In two limiting cases, 〈tm〉/t→1/2 for t≪td and 〈tm〉/t→(1+2G)/4≈0.708 for t≫td, where G≈0.916 is the Catalan’s constant. Finally, we analytically compute the statistics of the last time tℓ the particle crosses the starting position x0 and the occupation time to spent above x0. We find that 〈tℓ〉/t→1/2 in the short-time and long-time limits, and reaches its maximum at an intermediate value of t. The fraction of the occupation time 〈to〉/t is a monotonic function of t, and tends towards 1 in the long-time limit. All the theoretical results are validated by numerical simulations.