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On the occupation times of cones by Brownian motion
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  • Published: September 1995

On the occupation times of cones by Brownian motion

  • Thierry Meyre1 &
  • Wendelin Werner2 

Probability Theory and Related Fields volume 101, pages 409–419 (1995)Cite this article

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  • 8 Citations

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Summary

We study some features concerning the occupation timeA t of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A1<u) asu→0, when the Brownian motion starts at the vertex ofC. We also give the precise integral test, which decides whether a.s., lim inf t→∞ A t/(tf(t))=0 or ∞ for a decreasing functionf.

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References

  1. Azéma, J., Duflo, M., Revuz, D.: Propriétés relatives des processus de Markov récurrnts. Z. Warscheinlichkeitstheor. Verw. Geb.13, 286–314 (1969)

    Google Scholar 

  2. Bertoin, J., Werner, W.: Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process, in Séminaire de Probabilités XXVIII (Lect. Notes Math. 1583) Berlin Heidelberg New York: Springer, 138–152 (1994)

    Google Scholar 

  3. Bingham, N.H., Doney, R.A.: On higher-dimensional analogues of the arc-sine law. J. Appl. Prob.25, 120–131 (1988)

    Google Scholar 

  4. Burhholder, D.L.: Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. Math.26, 182–205 (1977)

    Google Scholar 

  5. De Blassie, R.D.: Exit times from cones in ℝn of Brownian motion. Probab. Theory Relat. Fields74, 1–29 (1987) and79, 95–97 (1988)

    Google Scholar 

  6. Hobson, D.: Asymptotics for an Arcsin Type result. Ann. Inst. Henri Poincaré30, 235–243 (1994)

    Google Scholar 

  7. Lévy, P.: Sur certains processus stochastiques homogènes. Compos. Math.7, 283–339 (1939)

    Google Scholar 

  8. Meyre, T.: Etude asymptotique du temps passé par le mouvement brownien dans un cône. Ann. Inst. Henri Poincaré27, 107–124 (1991)

    Google Scholar 

  9. Mountford, T.S.: Limiting behaviour of the Occupation of wedges by Complex brownian motion. Probab. Theory Relat. Fields84, 55–65 (1990)

    Google Scholar 

  10. Revuz, D., Yor, M.: Continuous Martingales and Brownian motion. Berlin Heidelberg New York: Springer, 1991

    Google Scholar 

  11. Spitzer, F.: Principles of Random walk. Van Nostrand, Princeton, 1964

    Google Scholar 

  12. Yor, M.: Some aspects of Brownian motion, Part I: Some special Functionals. (Lect. Maths.) ETH Zürich. Basel: Birkhäuser 1992

    Google Scholar 

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Author information

Authors and Affiliations

  1. U.F.R. de Mathématiques, Université Paris VII, 2 place Jussieu, F-75251, Paris Cedex 05, France

    Thierry Meyre

  2. Statistical Laboratory, D.P.M.M.S., C.N.R.S. and University of Cambridge, 16 Mill Lane, CB2 1SB, Cambridge, England

    Wendelin Werner

Authors
  1. Thierry Meyre
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  2. Wendelin Werner
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Meyre, T., Werner, W. On the occupation times of cones by Brownian motion. Probab. Th. Rel. Fields 101, 409–419 (1995). https://doi.org/10.1007/BF01200504

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  • Received: 08 July 1994

  • Revised: 21 September 1994

  • Issue Date: September 1995

  • DOI: https://doi.org/10.1007/BF01200504

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Mathematics Subject Classification (1991)

  • 60J65
  • 60G17
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