  We derive asymptotic formulas for the mean exit time $\bar{\tau}^{N}$ of the
fastest among $N$ identical independently distributed Brownian particles to an
absorbing boundary for various initial distributions (partially uniformly and
exponentially distributed). Depending on the tail of the initial distribution,
we report here a continuous algebraic decay law for $\bar{\tau}^{N}$, which
differs from the classical Weibull or Gumbell results. We derive asymptotic
formulas in dimension 1 and 2, for half-line and an interval that we compare
with stochastic simulations. We also obtain formulas for an additive constant
drift on the Brownian motion. Finally, we discuss some applications in cell
biology where a molecular transduction pathway involves multiple steps and a
long-tail initial distribution.
