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  • 1 Université Henri Poincaré Institut de Mathématiques Elie Cartan B.P. 239 F-54506 Vandœuvre-lès-Nancy Cedex France
  • 2 Universités Paris VI et VII Laboratoire de Probabilités et Modèles Aléatoires 4, Place Jussieu — Case 188 F-75252 Paris Cedex 05 France
  • 3 Institut Universitaire de France France
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We describe the limit laws, as t → ∞, of a Bessel process ( R s , st ) of dimension d ∈ (0, 2) penalized by an integrable function of its local time L t at 0, thus extending our previous work of this kind, relative to Brownian motion.

  • Borodin, A. N. and Salminen, P. , Handbook of Brownian motion — facts and formulae , Probability and its Applications. Birkhäuser Verlag, Basel, second edition, 2002. MR 2003g :60001

    Salminen P. , '', in Probability and its Applications , (2002 ) -.

  • Chaumont, L. and Yor, M. , Exercises in probability , volume 13 of Cambridge Series in Statistical and Probabilistic Mathematics , Cambridge University Press, Cambridge, 2003. A guided tour from measure theory to random processes, via conditioning. MR 2004m :60001

    Yor M. , '', in Exercises in probability, volume 13 of Cambridge Series in Statistical and Probabilistic Mathematics , (2003 ) -.

  • Donati-Martin, C., Roynette, B., Vallois, P. and Yor, M. , 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 − α ), 0 < α < 1, to appear in Studia Sci. Math. Hungar. , 45(1) (2008).

  • Getoor, R. K. , The Brownian escape process. Ann. Probab. , 7(5) (1979), 864–867. MR 80h :60102

    Getoor R. K. , 'The Brownian escape process ' (1979 ) 7 Ann. Probab. : 864 -867.

  • Gradinaru, M., Roynette, B., Vallois, P. and Yor, M. , Abel transform and integrals of Bessel local times, Ann. Inst. H. Poincaré Probab. Statist. , 35(4) (1999), 531–572. MR 2000i :60085

    Yor M. , 'Abel transform and integrals of Bessel local times ' (1999 ) 35 Ann. Inst. H. Poincaré Probab. Statist. : 531 -572.

    • Search Google Scholar
  • Jeulin, T. , Semi-martingales et grossissement d’une filtration , volume 833 of Lecture Notes in Mathematics , Springer, Berlin, 1980. MR 82h :60106

    Jeulin T. , '', in Semi-martingales et grossissement d’une filtration, volume 833 of Lecture Notes in Mathematics , (1980 ) -.

  • Jeulin, T. and Yor, M. , eds. , Grossissement de filtrations: exemples et applications (Séminaire de Calcul Stochastique, Paris 1982/83) , volume 1118 of Lecture Notes in Math. , Springer, Berlin, 1985.

    '', in Grossissement de filtrations: exemples et applications (Séminaire de Calcul Stochastique, Paris 1982/83), volume 1118 of Lecture Notes in Math. , (1985 ) -.

  • Kent, J. , Some probabilistic properties of Bessel functions, Ann. Probab. , 6(5) (1978), 760–770. MR 58 #18750

    Kent J. , 'Some probabilistic properties of Bessel functions ' (1978 ) 6 Ann. Probab. : 760 -770.

    • Search Google Scholar
  • Lebedev, N. N. , Special functions and their applications , Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication. MR 50 #2568

    Lebedev N. N. , '', in Special functions and their applications , (1972 ) -.

  • Lukacs, E. , Characteristic functions , Hafner Publishing Co., New York, 1970. Second edition, revised and enlarged. MR 49 #11595

    Lukacs E. , '', in Characteristic functions , (1970 ) -.

  • Mansuy, R. , and Yor, M. , Random Times and Enlargements of Filtrations in a Brownian Setting , volume 1873 of Lecture Notes in Math. , Springer, Berlin, 2006. MR 2007a :60005

    Yor M. , '', in Random Times and Enlargements of Filtrations in a Brownian Setting, volume 1873 of Lecture Notes in Math. , (2006 ) -.

  • Matsumoto, H. , and Yor, M. , An analogue of Pitman’s 2 MX theorem for exponential Wiener functionals, I, A time-inversion approach, Nagoya Math. J. , 159 (2000), 125–166. MR 2001j :60145

    Yor M. , 'An analogue of Pitman’s 2M − X theorem for exponential Wiener functionals, I, A time-inversion approach ' (2000 ) 159 Nagoya Math. J. : 125 -166.

    • Search Google Scholar
  • Matsumoto, H. , and Yor, M. , An analogue of Pitman’s 2 MX theorem for exponential Wiener functionals, II, The role of the generalized inverse Gaussian laws, Nagoya Math. J. , 162 (2001), 65–86. MR 2002d :60070

    Yor M. , 'An analogue of Pitman’s 2M − X theorem for exponential Wiener functionals, II, The role of the generalized inverse Gaussian laws ' (2001 ) 162 Nagoya Math. J. : 65 -86.

    • Search Google Scholar
  • Pitman, J. and Yor, M. , Bessel processes and infinitely divisible laws, in: Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980) , volume 851 of Lecture Notes in Math. , pages 285–370. Springer, Berlin, 1981. MR 82j :60149

    Yor M. , '', in Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), volume 851 of Lecture Notes in Math. , (1981 ) -.

  • Pitman, J. W. , One-dimensional Brownian motion and the three-dimensional Bessel process, Advances in Appl. Probability , 7(3) (1975), 511–526. MR 51 #11677

    Pitman J. W. , 'One-dimensional Brownian motion and the three-dimensional Bessel process ' (1975 ) 7 Advances in Appl. Probability : 511 -526.

    • Search Google Scholar
  • Revuz, D. and Yor, M. , Continuous martingales and Brownian motion , volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] , Springer-Verlag, Berlin, third edition, 1999. MR 2000h :60050

    Yor M. , '', in Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] , (1999 ) -.

  • Roynette, B., Vallois, P. and Yor, M. , Pénalisations et extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère, Séminaire de Probabilités, XXXIX (P. A. Meyer, in memoriam) , 305–336. Lecture Notes in Math., Volume 1874, Springer, Berlin, 2006. MR 2276902

    Yor M. , '', in Séminaire de Probabilités, XXXIX (P. A. Meyer, in memoriam) , (2006 ) -.

  • Roynette, B., Vallois, P. , and Yor, M. , Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III, Period. Math. Hungar. , 50(1–2) (2005), 247–280. MR 2006m :60119

    Yor M. , 'Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III ' (2005 ) 50 Period. Math. Hungar. : 247 -280.

    • Search Google Scholar
  • Roynette, B., Vallois, P. and Yor, M. , Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II, Studia Sci. Math. Hungar. , 43 (2006), 295–360. MR 2253307

    Yor M. , 'Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time, II ' (2006 ) 43 Studia Sci. Math. Hungar. : 295 -360.

    • Search Google Scholar
  • Roynette, B., Vallois, P. and Yor, M. , Some extensions of Pitman’s and Ray-Knight’s theorems for penalized Brownian motions and their local times, IV, Studia Sci. Math. Hungar. , 44(4) (2007), 469–516.

    Yor M. , 'Some extensions of Pitman’s and Ray-Knight’s theorems for penalized Brownian motions and their local times, IV ' (2007 ) 44 Studia Sci. Math. Hungar. : 469 -516.

    • Search Google Scholar
  • Watanabe, S. , On time inversion of one-dimensional diffusion processes, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , 31 (1974/75), 115–124. MR 51 #1983

    Watanabe S. , 'On time inversion of one-dimensional diffusion processes ' (1974 ) 31 Z. Wahrscheinlichkeitstheorie und Verw. Gebiete : 115 -124.

    • Search Google Scholar
  • Yor, M. , Grossissement de filtrations et absolue continuité de noyaux, in: Grossissement de filtrations: exemples et applications (Séminaire de Calcul Stochastique, Paris 1982/83) , volume 1118 of Lecture Notes in Math. , pages 6–14. Springer, Berlin, 1985.

    Yor M. , '', in Grossissement de filtrations: exemples et applications (Séminaire de Calcul Stochastique, Paris 1982/83), volume 1118 of Lecture Notes in Math. , (1985 ) -.

  • Yor, M. , Some aspects of Brownian motion. Part II , Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1997. Some recent martingale problems. MR 98e :60140

    Yor M. , '', in Some aspects of Brownian motion. Part II , (1997 ) -.