Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion Theory

  • Paavo Salminen
  • Ju-Yi Yen
  • Marc Yor
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)


In this note we present integral representations of the Itô excursion measure associated with a general one-dimensional diffusion X. These representations and identities are natural extensions of the classical ones for reflected Brownian motion, RBM. As is well known, the three-dimensional Bessel process, BES(3), plays a crucial rôle in the analysis of the Brownian excursions. Our main interest is in showing explicitly how certain excursion theoretical formulae associated with the pair (RBM, BES(3)) generalize to pair (X, X ), where X denotes the diffusion obtained from X by conditioning X not to hit 0. We illustrate the results for the pair \((R_{-},R_{+})\) consisting of a recurrent Bessel process with dimension \(d_{-} = 2(1-\alpha ),\) α ∈ (0, 1), and a transient Bessel process with dimension \(d_{+} = 2(1+\alpha )\). Pair (RBM, BES(3)) is, clearly, obtained by choosing \(\alpha = 1/2.\)



Paavo Salminen’s research was funded in part by a grant from Svenska kulturfonden via Stiftelsernas professorspool, Finland.


  1. 1.
    J. Bertoin, Lévy Processes (Cambridge University Press, Cambridge, 1996)zbMATHGoogle Scholar
  2. 2.
    Ph. Biane, M. Yor, Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. (2), 111(1), 23–101 (1987)Google Scholar
  3. 3.
    A.N. Borodin, P. Salminen, Handbook of Brownian Motion-Facts and Formulae. Probability and Its Applications, 2nd edn. (Birkhäuser, Basel, 2002)Google Scholar
  4. 4.
    K.L. Chung, J.B. Walsh, Markov Processes, Brownian Motion, and Time Symmetry, 2nd edn. (Springer, Berlin, 2005)zbMATHCrossRefGoogle Scholar
  5. 5.
    C. Donati-Martin, B. Roynette, P. Vallois, M. Yor, On constants related to the choice of the local time at 0, and the corresponding Itô measure for Bessel processes with dimension \(d = 2(1-\alpha ),\ 0 <\alpha < 1\). Stud. Sci. Math. Hung. 45(2), 207–221 (2008)zbMATHMathSciNetGoogle Scholar
  6. 6.
    P. Fitzsimmons, J. Pitman, M. Yor, Markovian bridges: construction, Palm interpretation, and splicing, in Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992). Progress in Probability, vol. 33 (Birkhäuser, Boston, 1993), pp. 101–134Google Scholar
  7. 7.
    K. Itô, H.P. McKean, Diffusion Processes and Their Sample Paths (Springer, Berlin, 1974)zbMATHGoogle Scholar
  8. 8.
    H. McKean, Excursions of a non-singular diffusion. Z. Wahrsch. verw. Gebiete 1, 230–239 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    J. Pitman, M. Yor, Bessel processes and infinitely divisible laws, in Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Mathematics, vol. 851 (Springer, Berlin, 1981), pp. 285–370Google Scholar
  10. 10.
    J. Pitman, M. Yor, Decomposition at the maximum for excursions and bridges of one-dimensional diffusions, in Itô’s Stochastic Calculus and Probability Theory (Springer, Tokyo, 1996), pp. 293–310Google Scholar
  11. 11.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, 3rd edn. (Springer, Berlin, 1999)zbMATHCrossRefGoogle Scholar
  12. 12.
    P. Salminen, P. Vallois, M. Yor. On the excursion theory for linear diffusions. Jpn. J. Math. 2(1), 97–127 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Williams, Path decompositions and continuity of local time for one-dimensional diffusions. Proc. Lond. Math. Soc. 28, 738–768 (1974)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Faculty of Science and EngineeringÅbo Akademi UniversityÅboFinland
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

Personalised recommendations