We derive asymptotic formulas for the mean exit time 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 , which differs from the classical Weibull or Gumbel 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.

The extreme narrow escape theory describes the statistical properties of the fastest among many identical stochastic particles to escape from a narrow window. We study here the arrival of the fastest particle when a killing term is added inside a one dimensional interval. Killing represents a degradation that leads to removal of the moving particles with a given probability. Using the time dependent flux for the solution of the diffusion equation, we compute asymptotically the mean time for the fastest to escape alive. We also study the role of several killing distributions on the mean extreme time for the fastest and compare the results with Brownian simulations. Finally, we discuss some possible applications to cell biology.

No abstract

No abstract