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..0. We first obtain the distribution P (M|x(0),t) of the maximum displacement M and its expectation < M > In the short-time limit, i.e., t << t(d) where t(f) =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 >> t(d,) < M > grows with t as < M > similar to root t, which is similar to the free Brownian motion, but the prefactor is pi/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,t(m) |x(0),t) of M and the time t(m) at which this maximum is achieved, from which the marginal distribution P (M|x(0),t) of t(m) is also obtained. For t << t(d) , P(t(m)|x(0),t) looks like a U-shaped attributed to the arcsine law of free Brownian motion. For.. 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 < t(m)>/t of t(m) , and find that < t(m)>/t is an increasing function of t. In two limiting cases, < t(m)>/t -> 1/2 for t << t(d) and < t(m)>/t -> (1 + 2G)/4 approximate to 0.708 for t >> t(d), where G approximate to 0.916 is the Catalan's constant. Finally, we analytically compute the statistics of the last time t(l) the particle crosses the starting position.. 0 and the occupation time t(0) spent above x(0). We find that < t(l)>/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. < t(0)>/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.