Japanese Journal of Mathematics

, Volume 1, Issue 1, pp 263–290 | Cite as

Some penalisations of the Wiener measure

Article

Abstract.

A number of limit laws, which are obtained from various penalisations of the Wiener measure on \(C \left(\mathbb{R}_{+}, \mathbb{R}^{d}\right),\) are shown to exist, and are described thoroughly, with the help of path decompositions.

Keywords and phrases.

Brownian motion Wiener measure 3-dimensional Bessel process path decompositions supremum process Lévy and Pitman theorems age process of excursions Brownian windings 

Mathematics Subject Classification (2000).

60F17 60F99 60G44 60H20 60J60 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. AY.
    J. Azéma and M. Yor, Une solution simple au probléme de Skorokhod, Sém. Prob. XIII, Lecture Notes in Math., 721, Springer, 1979, pp. 90–115.Google Scholar
  2. BSm.
    R. Banuelos and R.G. Smits, Brownian motion in cones, Probab. Theory Related Fields, 108 (1997), 299–319.MathSciNetGoogle Scholar
  3. BGM.
    M. Berger, P. Gauduchon and E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Math., 194, Springer, 1971.Google Scholar
  4. B.
    V. Betz, Existence of Gibbs measures relative to Brownian motion, In: Markov Processes and Related Fields, 2003, 9, pp. 85–102.Google Scholar
  5. BL.
    V. Betz and J. Lörinzci, Uniqueness of Gibbs measures relative to Brownian motion, Ann. Inst. H. Poincaré Probab. Statist., 39 (2003), 877–889.Google Scholar
  6. BS.
    Borodin A., Salminen P.,Handbook of Brownian Motion. Facts and Formulae, Second edition, Birkhäuser, 2002.Google Scholar
  7. E.
    S.F. Edwards, The statistical mechanics of polymers with excluded volume, Proc. Phys. Soc., 85 (1965), 613–614.MATHMathSciNetGoogle Scholar
  8. H.
    van der Hofstadt R., One-Dimensional Random Polymers, Thesis monograph, 1997.Google Scholar
  9. IMK.
    K. Itô and H.P. McKean, Diffusion Processes and Their Sample Paths, Springer, 1965.Google Scholar
  10. L.
    N.N. Lebedev, Special Functions and Their Applications, Dover, 1972.Google Scholar
  11. LG1.
    J.F. Le Gall, Sur le temps local d’intersection du mouvement brownien plan, et la méthode de renormalisation de Varadhan, Sém. Prob. XIX, Lecture Notes in Math., 1123, Springer, 1985, pp. 314–331.Google Scholar
  12. LG2.
    J.F. Le Gall, Some properties of planar Brownian motion, Ecole d’Eté Saint-Flour XX, 1990, Lecture Notes in Math., 1527, Springer, 1992, pp. 112–234.Google Scholar
  13. M.
    P.A. Meyer, Probabilités et Potentiel, Publ. Inst. Rech. Math. Av., Hermann, Paris, 1966.Google Scholar
  14. N.
    J. Najnudel, Pénalisations de l’araignée brownienne (submitted to Ann. Inst. Fourier, 2005).Google Scholar
  15. OS.
    H. Osada and H. Spohn, Gibbs measures relative to Brownian motion, Ann. Probab., 27 (1999), 1183–1207.MathSciNetGoogle Scholar
  16. P.
    J. Pitman, One-dimensional Brownian motion and the three-dimensional Bessel process, Adv. Appl. Probab., 7(1975), 511-526.MATHMathSciNetGoogle Scholar
  17. PY1.
    J. Pitman and M. Yor, Asymptotic laws of planar Brownian motion, Ann. Probab., 14 (1986), 733–779.MathSciNetGoogle Scholar
  18. PY2.
    J. Pitman and M. Yor, Further asymptotic laws of planar Brownian motion, Ann. Probab., 17 (1989), 965–1011.MathSciNetGoogle Scholar
  19. RY.
    D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, third edition, 1999.Google Scholar
  20. R.
    L. Rogers, Characterizing all diffusions with the 2M-X property, Ann. Probab., 9 (1981), 561–672.MATHMathSciNetGoogle Scholar
  21. RVY1.
    B. Roynette, P. Vallois and M. Yor, Limiting laws for long Brownian bridges perturbated by their one-sided maximum III, Periodica Math. Hung., 50 (2005), 247–280.MathSciNetGoogle Scholar
  22. RVY2.
    B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by normalized exponential weights, Stud. Math. Hung., to appear in March 2006.Google Scholar
  23. RVY2’.
    B. Roynette, P. Vallois and M. Yor, Complements to [RVY2], in preparation (2006).Google Scholar
  24. RVY3.
    B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time II, Stud. Math. Hung., to appear in 2006.Google Scholar
  25. RVY4.
    B. Roynette, P. Vallois and M. Yor, Some extensions of Pitman and Ray–Knight theorems for penalized Brownian motions and their local times IV, Stud. Math. Hung., to appear in 2006.Google Scholar
  26. RVY5.
    B. Roynette, P. Vallois and M. Yor, Penalizing a Bes(d) process (0<d<2) with a function of its local time at 0, V, submitted to Stud. Math. Hung. in 2005.Google Scholar
  27. RVY6.
    B. Roynette, P. Vallois and M. Yor, Penalisations of multidimensional Brownian motion, VI, submitted to Stud. Math. Hung. in 2005.Google Scholar
  28. RVY7.
    B. Roynette, P. Vallois and M. Yor, Penalizing a Brownian motion with a function of the lengths of its excursions, VII, in preparation.Google Scholar
  29. S.
    F. Spitzer, Some theorems about 2-dimensional Brownian motion, Trans. Amer. Math. Soc., 87 (1958), 187–197.MATHMathSciNetGoogle Scholar
  30. W1.
    J. Westwater, On Edwards model for long polymer chains, Comm. Math. Phys., 72 (1980), 131–174.CrossRefMATHMathSciNetGoogle Scholar
  31. W2.
    J. Westwater, On Edwards model for long polymer chains, II. The self-consistent potential, Comm. Math. Phys., 79 (1981), 53–73.CrossRefMATHMathSciNetGoogle Scholar
  32. W3.
    J. Westwater, On Edwards model for long polymer chains, In: Trends and Developments in the Eighties, (eds. S. Albeverio and P. Blanchard), Bielefeld Encounters in Math. Phys., 4/5, World Scientific, Singapore, 1984.Google Scholar

Copyright information

© The Mathematical Society of Japan and Springer-Verlag 2006

Authors and Affiliations

  1. 1.Institut Elie CartanUniversité Henri PoincaréVandoeuvre-les-Nancy CedexFrance
  2. 2.Laboratoire de Probabilités et Modèles, AléatoiresUniversités Paris VI et VII, Institut Universitaire de FranceParis Cedex 05France

Personalised recommendations