Normalized functionals of first passage Brownian motion and a curious connection with the maximal relative height of fluctuating interfaces

摘要

A study is made of the normalized functionals M equivalent to M/T-1/2 and A equivalent to 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. and. 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. and. 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. 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.