Decomposition at the maximum for excursions and bridges of one-dimensional diffusions

  • Jim Pitman
  • Marc Yor


In his fundamental paper [25], Itô showed how to construct a Poisson point process of excursions of a strong Markov process X over time intervals when X is away from a recurrent point a of its statespace. The point process is parameterized by the local time process of X at a. Each point of the excursion process is a path in a suitable space of possible excursions of X,starting at a at time 0, and returning to a for the first time at some strictly positive time ζ, called the lifetime of the excursion. The intensity measure of the Poisson process of excursions is a σ-finite measure Λ on the space of excursions, known as Itô’s excursion law. Accounts of Itô’s theory of excursions can now be found in several textbooks [48, 46, 10]. His theory has also been generalized to excursions of Markov processes away from a set of states [34, 19, 10] and to excursions of stationary, not necessarily Markovian processes [38].


Brownian Motion Path Space Brownian Bridge Bessel Process Path Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Aldous and J. Pitman. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms, 5: 487–512, 1994.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    D.J. Aldous. The continuum random tree I. Ann. Probab., 19: 1–28, 1991.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    D.J. Aldous. The continuum random tree II: an overview. In M.T. Barlow and N.H. Bingham, editors, Stochastic Analysis, pages 23–70. Cambridge University Press, 1991.CrossRefGoogle Scholar
  4. 4.
    D.J. Aldous. The continuum random tree III. Ann. Probab., 21: 248–289, 1993.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    D.J. Aldous. Recursive self-similarity for random trees, random triangulations and Brownian excursion. Ann. Probab., 22: 527–545, 1994.MathSciNetMATHCrossRefGoogle Scholar
  6. S. Asmussen, P. Glynn, and J. Pitman. Discretization error in simulation of one-dimensional reflecting Brownian motion. To appear in Ann. Applied Prob.,1996.Google Scholar
  7. 7.
    Ph. Biane. Decompositions of Brownian trajectories and some applications. In A. Badrikian, P-A Meyer, and J-A Yan, editors, Probability and Statistics; Rencontres Franco-Chinoises en Probabilités et Statistiques; Proceedings of the Wuhan meeting, pages 51–76. World Scientific, 1993.Google Scholar
  8. 8.
    Ph. Biane. Some comments on the paper: “Brownian bridge asymptotics for random mappings” by D. J. Aldous and J. W. Pitman. Random Structures and Algorithms, 5: 513–516, 1994.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Ph. Biane and M. Yor. Valeurs principales associées aux temps locaux Browniens. Bull. Sci. Math. (2), 111: 23–101, 1987.MathSciNetMATHGoogle Scholar
  10. 10.
    R. M. Blumenthal. Excursions of Markov processes. Birkhâuser, 1992.Google Scholar
  11. 11.
    K.L. Chung. Excursions in Brownian motion. Arkiv fur Matematik, 14: 155–177, 1976.MATHCrossRefGoogle Scholar
  12. 12.
    E. Csâki; A. Földes, and P. Salminen. On the joint distribution of the maximum and its location for a linear diffusion. Annales de l’Institut Henri Poincaré, Section B, 23: 179–194, 1987.MATHGoogle Scholar
  13. 13.
    I.V. Denisov. A random walk and a Wiener process near a maximum. Theor. Prob. Appl., 28: 821–824, 1984.MATHCrossRefGoogle Scholar
  14. 14.
    P. Fitzsimmons, J. Pitman, and M. Yor. Markovian bridges: construction, Palm interpretation, and splicing. In Seminar on Stochastic Processes, 1992, pages 101–134. Birkhâuser, Boston, 1993.CrossRefGoogle Scholar
  15. 15.
    P.J. Fitzsimmons. Excursions above the minimum for diffusions. Unpublished manuscript, 1985.Google Scholar
  16. 16.
    J.-F. Le Gall. Brownian excursions, trees and measure-valued branching processes. Ann. Probab., 19: 1399–1439, 1991.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    J.F. Le Gall. Une approche élémentaire des théorèmes de décomposition de Williams. In Séminaire de Probabilités XX, pages 447–464. Springer, 1986. Lecture Notes in Math. 1204.Google Scholar
  18. 18.
    J.F. Le Gall. The uniform random tree in a brownian excursion. Probab. Th. Rel. Fields, 96: 369–383, 1993.MATHCrossRefGoogle Scholar
  19. 19.
    R.K. Getoor and M.J. Sharpe. Excursions of dual processes. Advances in Mathematics, 45: 259–309, 1982.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    I.I. Gikhman On a nonparametric criterion of homogeneity for k samples. Theory Probab. Appl., 2: 369–373, 1957.CrossRefGoogle Scholar
  21. 21.
    J.-P. Imhof. Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. Journal of Applied Probability, 21: 500–510, 1984.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    J.-P. Imhof. On Brownian bridge and excursion. Studia Sci. Math. Hangar., 20: 1–10, 1985.MathSciNetMATHGoogle Scholar
  23. 23.
    J. P. Imhof. On the range of Brownian motion and its inverse process. Annals of Probability, 13: 1011–1017, 1985.MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    M. E. Ismail and D.H. Kelker. Special functions, Stieltjes transforms, and infinite divisibility. SIAM J. Math. Anal., 10: 884–901, 1979.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    K. Itô. Poisson point processes attached to Markov processes. In Proc. 6th Berk. Symp. Math. Stat. Prob., volume 3, pages 225–240, 1971.Google Scholar
  26. 26.
    K. Itô and H.P. McKean. Diffusion Processes and their Sample Paths. Springer, 1965.MATHGoogle Scholar
  27. 27.
    Th. Jeulin. Temps local et théorie du grossissement: application de la théorie du grossissement à l’étude des temps locaux browniens. In Grossissements de filtrations: exemples et applications. Séminaire de Calcul Stochastique, Paris 1982/83, pages 197–304. Springer-Verlag, 1985. Lecture Notes in Math. 1118.Google Scholar
  28. 28.
    D.P. Kennedy. The distribution of the maximum Brownian excursion. J. Appl. Prob., 13: 371–376, 1976.MATHCrossRefGoogle Scholar
  29. 29.
    J. Kent. Some probabilistic properties of Bessel functions. Annals of Probability, 6: 760–770, 1978.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    J. Kiefer. K-sample analogues of the Kolmogorov-Smirnov and Cramér-von Mises tests. Ann. Math. Stat., 30: 420–447, 1959.MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    F.B. Knight. Characterization of the Lévy measure of inverse local times of gap diffusions. In Seminar on Stochastic Processes, 1981, pages 53–78. Birkhäuser, Boston, 1981.CrossRefGoogle Scholar
  32. 32.
    S. Kotani and S. Watanabe. Krein’s spectral theory of strings and generalized diffusion processes. In Functional Analysis in Markov Processes, pages 235–249. Springer, 1982. Lecture Notes in Math. 923.Google Scholar
  33. 33.
    P. Lévy. Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris, 1965. (first ed. 1948 ).MATHGoogle Scholar
  34. 34.
    B. Maisonneuve. Exit systems. Ann. of Probability, 3: 399–411, 1975.MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    P.W. Millar. Random times and decomposition theorems. In Proc. of Symp. in Pure Mathematics, volume 31, pages 91–103, 1977.MathSciNetGoogle Scholar
  36. 36.
    P.W. Millar. A path decomposition for Markov processes. Ann. Probab., 6: 345–348, 1978.MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    S. A. Molchanov and E. Ostrovski. Symmetric stable processes as traces of degenerate diffusion processes. Theor. Prob. Appl., 14, No. 1: 128–131, 1969.MATHCrossRefGoogle Scholar
  38. 38.
    J. Pitman. Stationary excursions. In Séminaire de Probabilités XXI, pages 289–302. Springer, 1986. Lecture Notes in Math. 1247.Google Scholar
  39. 39.
    J. Pitman. Cyclically stationary Brownian local time proceses. To appear in Probability Theory and Related Fields, 1996.Google Scholar
  40. 40.
    J. Pitman and M. Yor. Bessel processes and infinitely divisible laws. In Stochastic Integrals, pages 285–370. Springer, 1981. Lecture Notes in Math. 851.CrossRefGoogle Scholar
  41. 41.
    J. Pitman and M. Yor. A decomposition of Bessel bridges. Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 59: 425–457, 1982.MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    J. Pitman and M. Yor. Dilatations d’espace-temps, réarrangements des trajectoires browniennes, et quelques extensions d’une identité de Knight. C.R. Acad. Sci. Paris, t. 316, Série I: 723–726, 1993.MathSciNetMATHGoogle Scholar
  43. 43.
    J. Pitman and M. Yor. Some extensions of Knight’s identity for Brownian motion. In preparation, 1995.Google Scholar
  44. 44.
    J. Pitman and M. Yor. Laws of homogeneous functionals of Brownian motion. In preparation, 1996.Google Scholar
  45. 45.
    J. Pitman and M. Yor. Quelques identités en loi pour les processus de Bessel. To appear in: Hommage à P.A. Meyer et J. Neveu, Astérisque, 1996.Google Scholar
  46. 46.
    D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer, Berlin-Heidelberg, 1994. 2nd edition.Google Scholar
  47. 47.
    L. C. G. Rogers. Williams’ characterization of the Brownian excursion law: proof and applications. In Séminaire de Probabilités XV,pages 227–250. Springer-Verlag, 1981. Lecture Notes in Math. 850.Google Scholar
  48. 48.
    L.C.G. Rogers and D. Williams. Diffusions, Markov Processes and Martingales. Wiley, 1987.MATHGoogle Scholar
  49. 49.
    T. Shiga and S. Watanabe. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete, 27: 37–46, 1973.MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    I. Vincze. Einige zweidimensionale Verteilungs-und Grenzverteilungssätze in der Theorie der georneten Stichproben. Magyar Tud. Akad. Mat. Kutató Int. Kôzl., 2: 183–209, 1957.MathSciNetGoogle Scholar
  51. 51.
    G.N. Watson. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge University Press, 1944.MATHGoogle Scholar
  52. 52.
    W. Werner. Quelques propriétés du mouvement brownien plan. Thèse de doctorat, Université Paris V I, March 1993.Google Scholar
  53. 53.
    W. Werner. Sur la forme des composantes connexes du complémentaire de la courbe brownienne plane. Prob. Th. and Rel. Fields, 98: 307–337, 1994.MATHCrossRefGoogle Scholar
  54. 54.
    D. Williams. Decomposing the Brownian path. Bull. Amer. Math. Soc., 76: 87 1873, 1970.Google Scholar
  55. 55.
    D. Williams. Path decomposition and continuity of local time for one dimensional diffusions I. Proc. London Math. Soc. (3), 28: 738–768, 1974.MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    D. Williams. Diffusions, Markov Processes, and Martingales, Vol. 1: Foundations. Wiley, Chichester, New York, 1979.Google Scholar
  57. 57.
    D. Williams. Brownian motion and the Riemann zeta-function. In Disorder in Physical Systems, pages 361–372. Clarendon Press, Oxford, 1990.Google Scholar
  58. 58.
    M. Yor. Some Aspects of Brownian Motion. Lectures in Math., ETH Zürich. Birkhaüser, 1992. Part I: Some Special Functionals.MATHGoogle Scholar

Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • Jim Pitman
    • 1
  • Marc Yor
    • 2
  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA
  2. 2.Laboratoire de ProbabilitésUniversité Pierre et Marie CurieParis Cedex 05France

Personalised recommendations