  We study extreme-value statistics of Brownian trajectories in one dimension.
We define the maximum as the largest position to date and compare maxima of two
particles undergoing independent Brownian motion. We focus on the probability
P(t) that the two maxima remain ordered up to time t, and find the algebraic
decay P ~ t^(-beta) with exponent beta=1/4. When the two particles have
diffusion constants D1 and D2, the exponent depends on the mobilities,
beta=(1/pi)arctan[sqrt(D2/D1)]. We also use numerical simulations to
investigate maxima of multiple particles in one dimension and the largest
extension of particles in higher dimensions.
