Asymptotics for the fastest among N stochastic particles: role of an extended initial distribution and an additional drift component

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