Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts
Download PDF
Download PDF
  • Published: 05 December 2007

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

  • Zbigniew Puchała1,2 &
  • Tomasz Rolski1 

Probability Theory and Related Fields volume 142, pages 595–617 (2008)Cite this article

  • 132 Accesses

  • 7 Citations

  • Metrics details

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.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Asmussen S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York

    MATH  Google Scholar 

  2. Biane Ph., Bougerol Ph. and O’Connell N. (2005). Littelmann paths and Brownian paths. Duke Math. J. 130: 127–167

    Article  MATH  MathSciNet  Google Scholar 

  3. Borodin A.N. and Salminen P. (2002). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser Verlag, Basel

    MATH  Google Scholar 

  4. Doumerc Y. and O’Connell N. (2005). Exit problems associated with finite reflection groups. Probab. Theory Relat. Fields 132: 501–538

    Article  MATH  MathSciNet  Google Scholar 

  5. Grabiner D.J. (1999). Brownian motion in a Weyl chamber, non-colliding particles and random matrices. Ann. Inst. H. Poincaré. Probab. Statist. 35: 177–204

    Article  MATH  MathSciNet  Google Scholar 

  6. Itzykson C. and Zuber J.-B. (1980). The planar approximation II. J. Math. Phys. 21: 411–421

    Article  MATH  MathSciNet  Google Scholar 

  7. Karlin, S., McGregor, J.: Coincidence probabilities. Pacific J. Math. 1141–1164 (1959)

  8. Macdonald I.G. (1979). Symmetric Functions and Hall Polynomials. Clarendon, Oxford

    MATH  Google Scholar 

  9. Puchała Z. (2005). A proof of Grabiner theorem on non-colliding particles. Probab. Math. Statist. 25: 129–132

    MATH  MathSciNet  Google Scholar 

  10. Puchała Z. and Rolski T. (2005). The exact asymptotics of the time to collision. Electron. J. Probab. 10: 1359–1380

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, University of Wrocław, pl.Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Zbigniew Puchała & Tomasz Rolski

  2. Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100, Gliwice, Poland

    Zbigniew Puchała

Authors
  1. Zbigniew Puchała
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomasz Rolski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomasz Rolski.

Additional information

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).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received: 03 April 2007

  • Revised: 24 October 2007

  • Published: 05 December 2007

  • Issue Date: November 2008

  • DOI: https://doi.org/10.1007/s00440-007-0116-8

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Brownian motion with drift
  • Collision time
  • Karlin–McGregor formula
  • Stable partition

Mathematics Subject Classification (2000)

  • Primary: 60J65
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature