Abstract
A study is made of the normalized functionals {____ mathcal M} = M/T^1/2 and {____ mathcal A} = A/T^3/2 associated with one-dimensional first passage Brownian motion with positive initial condition, where M is the maximum value attained and A is the area swept out up to the random time T at which the process first reaches zero. Both {____ mathcal M} and {____ mathcal A} involve two strongly correlated random variables associated with a given Brownian path. Through their study, fresh insights are provided into the fundamental nature of such first passage processes and the underlying correlations. The probability density and the moments of {____ mathcal M} and {____ mathcal A} are calculated exactly and the theoretical results are shown to be in good agreement with those derived from simulations. Intriguingly, there is a precise equivalence in law between the variable {____ mathcal A} and the maximal relative height of the fluctuating interface in the one-dimensional Edwards-Wilkinson model with free boundary conditions. This observation leads to some interesting and still partially unresolved questions.