First-passage and extreme value statistics for overdamped Brownian motion in a linear potential

We investigate the first-passage properties and extreme-value statistics of an overdamped Brownian particle confined by an external linear potential V(x) = N|x-x(0)|, where N > 0 is the strength of the potential and x(0) > 0 is the position of the lowest point of the potential, which coincides with the starting position of the particle. The Brownian motion terminates whenever the particle passes through the origin at a random time t(f) . Our study reveals that the mean first-passage time (t(f) ) exhibits a nonmonotonic behavior with respect to N, with a unique minimum occurring at an optimal value of mu similar or equal to 1.24468D/x(0) , where D is the diffusion constant of the Brownian particle. Moreover, we examine the distribution P(M|x(0)) of the maximum displacement M during the first-passage process, as well as the statistics of the time tm at which M is reached. Intriguingly, there exists another optimal N similar or equal to 1.24011D/x(0) that minimizes the mean time (t(m)). All our analytical findings are corroborated through numerical simulations.