On the first hitting time of a high-dimensional orthant

TL;DR

This paper investigates the tail behavior of the first time multiple independent Brownian particles all become negative, extending understanding from low to high dimensions using spectral geometry techniques.

Contribution

It provides new asymptotic estimates for the principal eigenvalue of the complement of a high-dimensional orthant, advancing the analysis of hitting times in high dimensions.

Findings

Derived asymptotic estimates for the principal eigenvalue in high dimensions

Analyzed tail asymptotics of hitting times as the number of particles grows

Extended spectral geometry methods to high-dimensional orthant problems

Abstract

We consider a collection of independent standard Brownian particles (or random walks), starting from a configuration where at least one particle is positive, and study the first time they all become negative. This is clearly equivalent to studying the first hitting time from the negative orthant or the first exit time from the complement of the negative orthant. While it turns out to be possible to compute the distribution of these hitting times for one and two particles, the distribution (and even its tail asymptotics) is not known in closed form for three or more particles. In this paper we study the tail asymptotics of the distribution as the number of particles tends to infinity. Our main techniques come from spectral geometry: we prove new asymptotic estimates for the principal eigenvalue of the complement of a high-dimensional orthant, which we believe are of independent interest.