On the Law of a Triplet Associated with the Pseudo-Brownian Bridge

  • Mathieu Rosenbaum
  • Marc Yor
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


We identify the distribution of a natural triplet associated with the pseudo-Brownian bridge. In particular, for B a Brownian motion and T 1 its first hitting time of the level one, this remarkable law allows us to understand some properties of the process \((B_{\mathit{uT}_{1}}/\sqrt{T_{1}},\ u \leq 1)\) under uniform random sampling, a study started in (Elie, Rosenbaum, and Yor, On the expectation of normalized Brownian functionals up to first hitting times, Preprint, arXiv:1310.1181, 2013).


Brownian motion Pseudo-Brownian bridge Bessel process Local time Hitting times Scaling Uniform sampling Mellin transform 



We thank the referee for a thorough reading of our paper.


  1. 1.
    P. Biane, J.-F. Le Gall, M. Yor, Un processus qui ressemble au pont brownien. In: Séminaire de Probabilités XXI (Springer, New York, 1987), pp. 270–275Google Scholar
  2. 2.
    R. Elie, M. Rosenbaum, M. Yor, On the expectation of normalized brownian functionals up to first hitting times. arXiv preprint arXiv:1310.1181 (2013)Google Scholar
  3. 3.
    J.W. Pitman, One-dimensional brownian motion and the three-dimensional bessel process. Adv. Appl. Probab. 7, 511–526 (1975)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    J.W. Pitman, Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Probab. 4(11), 1–33 (1999)MathSciNetGoogle Scholar
  5. 5.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, vol. 293 (Springer, New York, 1999)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University Pierre et Marie Curie (Paris 6), LPMAParis cedex 05France

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