We describe the limit laws, as t → ∞, of a Bessel process ( Rs , s ≦ t ) of dimension d ∈ (0, 2) penalized by an integrable function of its local time Lt 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.
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.
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 M − X 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.
Matsumoto, H. , and Yor, M. , An analogue of Pitman’s 2 M − X 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.
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.
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.
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.
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.
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.
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 ) -.