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 τ for n independent Brownian motions X 1 t ,...,X 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 $${\mathbb{P}}_{\bf x}(\tau > t) = Ch({\bf x})t^{-\alpha} {\rm e}^{-\gamma t}(1 + o(1))$$ as t → ∞ and identify C, h(x), α, γ in terms of the drifts.