Maximum of N Independent Brownian Walkers till the First Exit From the Half Space

TL;DR

This paper analyzes the maximum distance reached by N independent Brownian searchers before one finds a target at the origin, deriving the distribution's power-law tail and connecting it to exit probabilities in hypercubes.

Contribution

It provides an exact solution for the distribution of the maximum visited distance and the exit probability for multiple Brownian particles, extending understanding of search processes and boundary value problems.

Findings

Power-law tail in the maximum distance distribution: P_N(m|x) ~ B_N (x_1...x_N)/m^{N+1}.

All moments up to order N-1 are finite; higher moments diverge.

Analytical results match Monte Carlo simulations perfectly.

Abstract

We consider the one-dimensional target search process that involves an immobile target located at the origin and $N$ searchers performing independent Brownian motions starting at the initial positions $\vec x = (x_1,x_2,..., x_N)$ all on the positive half space. The process stops when the target is first found by one of the searchers. We compute the probability distribution of the maximum distance $m$ visited by the searchers till the stopping time and show that it has a power law tail: $P_N(m|\vec x)\sim B_N (x_1x_2... x_N)/m^{N+1}$ for large $m$. Thus all moments of $m$ up to the order $(N-1)$ are finite, while the higher moments diverge. The prefactor $B_N$ increases with $N$ faster than exponentially. Our solution gives the exit probability of a set of $N$ particles from a box $[0,L]$ through the left boundary. Incidentally, it also provides an exact solution of the Laplace's equation in an $N$-dimensional hypercube with some prescribed boundary conditions. The analytical results are in excellent agreement with Monte Carlo simulations.