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

Abstract

In this note we consider the time of the collision $\tau$ for $n$ independent Brownian motions $X_t^1,...,X_t^n$ 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 $P_x(\tau >t)=Ch(x)t^{-\alpha }{e}^{-\gamma t}(1+o(1))$ as $t\to \infty$ and identify $C,h(x),\alpha ,\gamma$ in terms of the drifts.

arXiv comment20 pages

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