Abstract
In this note we consider the time of the collision τ for n independent Brownian motions X 1 t ,...,X n t 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.
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This work was partially supported by a Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Programme: Programme HANAP under contract number MTKD-CT- 2004-13389, MNiSW Grants N201 049 31/3997 (2007) and N519 012 31/1957 (2006–2009), and MNiSW Grant N N201 4079 (2007–2009).
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Puchała, Z., Rolski, T. The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts. Probab. Theory Relat. Fields 142, 595–617 (2008). https://doi.org/10.1007/s00440-007-0116-8
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DOI: https://doi.org/10.1007/s00440-007-0116-8