# The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts

## TL;DR

This paper derives the precise asymptotic behavior of the tail distribution for the collision time of multiple independent Brownian particles with different drifts, providing explicit formulas for the asymptotic parameters.

## Contribution

It establishes the exact asymptotics of the collision time tail distribution for Brownian particles with different drifts, including explicit expressions for the constants involved.

## Key findings

- Asymptotic formula for P_x(τ > t) as t→∞
- Explicit expressions for constants C, h(x), α, γ
- Identification of parameters in terms of drifts

## Abstract

In this note we consider the time of the collision $\tau$ for $n$ independent Brownian motions $X^1_t,...,X_t^n$ with drifts $a_1,...,a_n$, each starting from $x=(x_1,...,x_n)$, where $x_1<...<x_n$. We show the exact asymptotics of $P_x(\tau>t) = C h(x)t^{-\alpha}e^{-\gamma t}(1 + o(1))$ as $t\to\infty$ and identify $C,h(x),\alpha,\gamma$ in terms of the drifts.