Probability Theory and Related Fields

, Volume 153, Issue 3–4, pp 771–807 | Cite as

The greatest convex minorant of Brownian motion, meander, and bridge

  • Jim Pitman
  • Nathan RossEmail author


This article contains both a point process and a sequential description of the greatest convex minorant of Brownian motion on a finite interval. We use these descriptions to provide new analysis of various features of the convex minorant such as the set of times where the Brownian motion meets its minorant. The equivalence of these descriptions is non-trivial, which leads to many interesting identities between quantities derived from our analysis. The sequential description can be viewed as a Markov chain for which we derive some fundamental properties.


Brownian motion Path decompositions Convex minorant 

Mathematics Subject Classification (2000)

60J65 60J05 60E99 


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  1. 1.
    Abramowitz, M., Stegun, I. (eds.).: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1992). Reprint of the 1972 editionGoogle Scholar
  2. 2.
    Abramson, J., Pitman, J.: Concave majorants of random walks and related Poisson processes. Combin. Probab. Comput. (2010, to appear).
  3. 3.
    Balabdaoui F., Pitman J.: The distribution of the maximal difference between Brownian bridge and its concave majorant. Bernoulli 17(1), 466–483 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bass, R.F.: Markov processes and convex minorants. In: Seminar on Probability, XVIII. Lecture Notes in Math., vol. 1059, pp. 29–41. Springer, Berlin (1984)Google Scholar
  5. 5.
    Bertoin J.: Lévy Processes. Cambridge Tracts in Mathematics, vol. 121. Cambridge University Press, Cambridge (1996)Google Scholar
  6. 6.
    Bertoin J.: The convex minorant of the Cauchy process. Electron. Commun. Probab. 5, 51–55 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bertoin J., Chaumont L., Pitman J.: Path transformations of first passage bridges. Electron. Commun. Probab. 8, 155–166 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Blumenthal R.M.: Weak convergence to Brownian excursion. Ann. Probab. 11(3), 798–800 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Carolan, C., Dykstra, R.: Characterization of the least concave majorant of Brownian motion, conditional on a vertex point, with application to construction. Ann. Inst. Statist. Math. 55(3), 487–497 (2003). doi: 10.1007/BF02517802.
  10. 10.
    Chaumont L., Uribe Bravo G.: Markovian bridges: weak continuity and pathwise constructions. Ann. Probab. 39(2), 609–647 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Çinlar E.: Sunset over Brownistan. Stoch. Process. Appl. 40(1), 45–53 (1992)zbMATHCrossRefGoogle Scholar
  12. 12.
    Denisov I.V.: Random walk and the Wiener process considered from a maximum point. Teor. Veroyatnost. i Primenen. 28(4), 785–788 (1983)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dufresne, D.: On the stochastic equation \({\mathcal{L}(X)={\mathcal{L}}[B(X+C)]}\) and a property of gamma distributions. Bernoulli 2(3), 287–291 (1996). doi: 10.2307/3318525. Google Scholar
  14. 14.
    Fitzsimmons, P., Pitman, J., Yor, M.: Markovian bridges: construction, palm interpretation, and splicing. In: Çinlar, E., Chung, K., Sharpe, M. (eds.) Seminar on Stochastic Processes, 1992, pp. 101–134. Birkhäuser, Boston (1993).
  15. 15.
    Fitzsimmons, P.J.: Another look at Williams’ decomposition theorem. In: Seminar on Stochastic Processes, 1985 (Gainesville, Fla., 1985). Progr. Probab. Statist., vol. 12, pp. 79–85. Birkhäuser Boston, Boston (1986)Google Scholar
  16. 16.
    Freedman D.: Brownian Motion and Diffusion. Holden-Day, San Francisco (1971)zbMATHGoogle Scholar
  17. 17.
    Goldie C.M.: Records, permutations and greatest convex minorants. Math. Proc. Cambridge Philos. Soc. 106(1), 169–177 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Greenwood P., Pitman J.: Fluctuation identities for Lévy processes and splitting at the maximum. Adv. Appl. Probab. 12, 893–902 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Groeneboom P.: The concave majorant of Brownian motion. Ann. Probab. 11(4), 1016–1027 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Imhof J.P.: Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21(3), 500–510 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Le Gall, J.F.: Une approche élémentaire des théorèmes de décomposition de Williams. In: Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math., vol. 1204, pp. 447–464. Springer, Berlin (1986)Google Scholar
  22. 22.
    Meyn, S.P., Tweedie, R.L.: Markov chains and stochastic stability. In: Communications and Control Engineering Series. Springer, London (1993)Google Scholar
  23. 23.
    Nagasawa, M.: Stochastic processes in quantum physics. In: Monographs in Mathematics, vol. 94. Birkhäuser Verlag, Basel (2000)Google Scholar
  24. 24.
    Pitman J.: One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. Appl. Probab. 7, 511–526 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Pitman, J.: Remarks on the convex minorant of Brownian motion. In: Seminar on Stochastic Processes, 1982, pp. 219–227. Birkhäuser, Boston (1983)Google Scholar
  26. 26.
    Pitman, J.: Combinatorial stochastic processes. In: Lecture Notes in Mathematics, vol. 1875. Springer-Verlag, Berlin (2006). Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean PicardGoogle Scholar
  27. 27.
    Pitman, J., Uribe Bravo, G.: The convex minorant of a Lévy process. Ann. Probab. (2010, to appear).
  28. 28.
    Suidan T.M.: Convex minorants of random walks and Brownian motion. Teor. Veroyatnost. i Primenen. 46(3), 498–512 (2001)MathSciNetGoogle Scholar
  29. 29.
    Williams D.: Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. Lond. Math. Soc. 3(28), 738–768 (1974)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.University of California, BerkeleyBerkeleyUSA

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