Asymptotics of extreme statistics of escape time in 1,2 and 3-dimensional diffusions

TL;DR

This paper derives asymptotic laws for the distribution of the earliest and second earliest arrival times of many Brownian particles to a small target in 1, 2, and 3 dimensions, with applications to cellular biochemical activation.

Contribution

It provides new asymptotic expressions for the distribution of extreme first and second arrival times in multi-dimensional diffusion, validated by simulations.

Findings

Asymptotic laws accurately describe extreme arrival times in various dimensions.

Results applicable to biochemical pathway activation in cells.

Stochastic simulations confirm theoretical predictions.

Abstract

The first of $N$ identical independently distributed (i.i.d.) Brownian trajectories that arrives to a small target, sets the time scale of activation, which in general is much faster than the arrival to the target of only a single trajectory. Analytical asymptotic expressions for the minimal time is notoriously difficult to compute in general geometries. We derive here asymptotic laws for the probability density function of the first and second arrival times of a large number of i.i.d. Brownian trajectories to a small target in 1,2, and 3 dimensions and study their range of validity by stochastic simulations. The results are applied to activation of biochemical pathways in cellular transduction.