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

TL;DR

This paper derives asymptotic formulas for the mean exit time of the fastest among multiple Brownian particles with various initial distributions, revealing new decay laws and including effects of drift, with applications in cell biology.

Contribution

It introduces novel asymptotic formulas for the fastest particle exit times considering extended initial distributions and added drift, extending classical results.

Findings

Continuous algebraic decay law depending on initial distribution tail

Asymptotic formulas derived for 1D and 2D cases

Application to cell biology transduction pathways

Abstract

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.