Some Brownian functionals and their laws

Annals of Probability

Volumn 25, Issue 3, 1997, Pages 1011-1058

  Donati Martin, C ^a   Yor, M ^a  

^a University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

Author keywords

Generalized Bessel processes; Principal values of local times; Ray Knight theorems; Space time harmonic functions

Indexed keywords

EID: 0031533027     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1024404505     Document Type: Article

References (66)

1. ALILI, L. (1995). Fonctionnelles exponentielles et certaines valeurs principales des temps locaux browniens. Thèse de Doctorat, Univ. Paris VI.
2. ALILI, L. (1997). On some hyperbolic principal values of Brownian local times. Bibl. Rev. Mat. Iberoamericana. To appear.
3. ALILI, L., DONATI-MARTIN, C. and YOR, M. (1997). Une identité en loi remarquable pour l'excursion brownienne normalisée. Bibl. Rev. Mat. Iberoamericana. To appear.
4. BENACHOUR, S., CHASSAING, P., ROYNETTE, B. and VALLOIS, P. (1997). Processus associés à l'équation des milieux poreux. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). To appear.
5. BENJAMINI, I. and LEE, S. (1997). Conditioned diffusions which are Brownian bridges. J. Theoret. Probab. To appear.
6. BIANE, P. (1993). Decomposition of Brownian trajectories and some applications (Notes from lectures given at the Probability Winter School of Wuhan, China, Fall 1990). In Rencontres Franco-chinoises en Probabilités et Statistiques (A. Badrikian, P. A. Meyer and J. A. Yan, eds.) 51-76. World Scientific, Singapore.
7. BIANE, P. and YOR, M. (1987). Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math. 111 23-101.
8. BIANE, P. and YOR, M. (1988). Sur la loi des temps locaux browniens pris en un temps exponentiel. Séminaire de Probabilités XXII. Lecture Notes in Math. 1321 454-466. Springer, Berlin.
9. BREIMAN, L. (1967). First exit times from a square root boundary. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 9-16. Univ. California Press, Berkeley.
10. BROCKHAUS, O. (1993). Sufficient statistics for the Brownian sheet. Séminaire de Probabilités XXVII. Lecture Notes in Math. 1557 44-52. Springer, Berlin.
11. CAIROLI, R. and WALSH, J. B. (1975). Stochastic integrals in the plane. Acta Math. 134 111-183.
12. CHAN, T., DEAN, D. S., JANSONS, K. and ROGERS, L. C. G. (1994). On polymer conformations in elongational flows. Comm. Math. Phys. 160 239-257.
13. CHAN, T. and JANSONS, K. (1996). Semi-flexible polymers in straining flows. J. Statist. Phys. To appear.
14. 2). Proc. Amer. Math. Soc. 100 721-724.
15. DONATI-MARTIN, C. and YOR, M. (1991). Fubini's theorem for double Wiener integrals and the variance of the Brownian path. Ann. Inst. H. Poincaré 27 181-200.
16. DONEY, R., WARREN, J. and YOR, M. (1997). Perturbed Bessel processes. Séminaire de Probabilités XXXII. Lecture Notes in Math. Springer, Berlin. To appear.
17. EISENBAUM, N. (1990). Un théorème de Ray-Knight relatif au supremum des temps locaux browniens. Probab. Theory Related Fields 87 79-95.
18. EMERY, M. and PERKINS, E. (1982). La filtration de B + L. Z. Wahrsch. Verw. Gebiete 59 383-390.
19. FÖLLMER, H. (1990). Martin boundaries on Wiener space. In Diffusion Processes and Related Problems in Analysis (M. Pinsky, ed.) 1 3-16. Birkhäuser, Boston.
20. GETOOR, R. K. and SHARPE, M. J. (1979). Excursions of Brownian motion and Bessel processes. Z. Wahrsch. Verw. Gebiete 47 83-106.
21. IKEDA, N., KUSUOKA, S. and MANABE, S. (1994). Lévy's stochastic area formula for Gaussian processes. Comm. Pure Appl. Math. 47 329-360.
22. IKEDA, N. and MANABE, S. (1996). Van Vleck-Pauli formula for Wiener integrals and Jacobi fields. In Itô's Stochastic Calculus and Probability Theory (N. Ikeda, S. Watanabe, M. Fukushima and H. Kunita, eds.) 141-156. Springer, New York.
23. ISMAIL, M. E. (1977) Integral representations and complete monotonicity of various quotients of Bessel functions. Canad. J. Math. 29 1198-1207.
24. ISMAIL, M. E. and KELKER, D. H. (1979). Special functions, Stieltjes transforms and infinite divisibility. SIAM J. Math. Anal. 10 884-901.
25. JEANBLANC, M., PITMAN, J. and YOR, M. (1997). The Feynman-Kac formula and decomposition of Brownian paths. In Computational and Applied Mathematics. Birkhäuser, Boston. To appear.
26. JEULIN, T. (1980). Semi-martingales et grossissement d'une filtration. Lecture Notes in Math. 833. Springer, Berlin.
27. JEULIN, T. (1984). Ray-Knight's theorems on Brownian local times and Tanaka formula. In Seminar on Stochastic Processes 1983 131-142. Birkhäuser, Boston.
28. JEULIN, T. (1985). Application de la théorie du grossissement à l'étude des temps locaux browniens. Grossissements de filtrations: Exemples et Applications. Lecture Notes in Math. 1118 197-304. Springer, Berlin.
29. JEULIN, T. and YOR, M. (1981). Sur les distributions de certaines fonctionnelles du mouvement brownien. Séminaire de Probabilités XV. Lecture Notes in Math. 850 210-226. Springer, Berlin.
30. JEULIN, T. and YOR, M. (1990). Filtration des ponts browniens et équations différentielles linéaires. Séminaire de Probabilités XXIV. Lecture Notes in Math. 1426 227-265. Springer, Berlin.
31. KNIGHT, F. B. (1981). Characterization of the Lévy measure of inverse local time of gap diffusions. In Seminar on Stochastic Processes 53-78. Birkhäuser, Boston.
32. KOTANI, S. and WATANABE, S. (1982). Krein's spectral theory of strings and general diffusion processes. Functional Analysis in Markov Processes. Lecture Notes in Math. 923 235-259. Springer.
33. LEBEDEV, N. N. (1972). Special Functions and Their Applications. Dover, New York.
34. LE GALL, J. F. (1985). Sur la mesure de Hausdorff de la courbe brownienne. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123 297-313. Springer, Berlin.
35. LE GALL, J. F. and YOR, M. (1986). Excursions browniennes et carrés de processus de Bessel. C.R. Acad. Sci. Paris Sér. I 303 73-76.
36. LEURIDAN, C. (1995). Les théorèmes de Ray-Knight et la mesure d'Itô pour le mouvement brownien sur le tore ℝ\ℤ. Stochastics Stochastics Rep. 53 109-128.
37. LEURIDAN, C. (1998). Le théorème de Ray-Knight en un temps fixe. Séminaire de Probabilités XXXII. Lecture Notes in Math. Springer, Berlin. To appear.
38. MCGILL, P. (1981). A direct proof of the Ray-Knight theorem. Séminaire de Probabilités XV. Lecture Notes in Math. 850 206-209. Springer, Berlin.
39. MCGILL, P. (1982). Markov properties of diffusion local time: a martingale approach. Adv. in Appl. Probab. 14 789-810.
40. NAGASAWA, M. and DOMENIG, T. (1996). Diffusion processes on an open interval and their time reversal. In Itô's Stochastic Calculus and Probability Theory (N. Ikeda, S. Watanabe, M. Fukushima and H. Kunita, eds.) 261-280. Springer, New York.
41. NEVEU, J. (1968). Processus Aléatoires Gaussiens. Univ. Montreal Press.
42. NUALART, D. (1981). Martingales à variation indépendante du chemin. Processus aléatoires à deux indices. Lecture Notes in Math. 863 128-148. Springer, Berlin.
43. O'CONNELL, N. (1993). The genealogy of branching processes. Ph.D. dissertation, Univ. California, Berkeley.
44. PERKINS, E. (1982). Local time is a semimartingale. Z. Wahrsch. Verw. Gebiete 60 79-117.
45. PITMAN, J. (1996). Cyclically stationary Brownian local time processes. Probab. Theory Related Fields 106 299-329.
46. PITMAN, J. and YOR, M. (1981). Bessel processes and infinitely divisible laws. Stochastic Integrals. Lecture Notes in Math. 851 285-370. Springer, Berlin.
47. PITMAN, J. and YOR, M. (1982). Sur une décomposition des ponts de Bessel. Functional Analysis in Markov Processes. Lecture Notes in Math. 923 276-285. Springer, Berlin.
48. PITMAN, J. and YOR, M. (1996). Quelques identités en loi pour les processus de Bessel. Astérisque 236 249-276.
49. RAUSCHER, B. (1997). Some remarks on Pitman's theorem. Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655 266-271. Springer, Berlin.
50. REVUZ, D. and YOR, M. (1994). Continuous Martingales and Brownian Motion, 2nd ed. Springer, Berlin.
51. ROGERS, L. C. G. and WILLIAMS, D. (1987). Markov Processes and Martingales 2. Itô Calculus. Wiley, New York.
52. RUIZ DE CHAVEZ, J. (1984). Le théorème de Paul Lévy pour des mesures signées. Séminaire de Probabilités XVIII. Lecture Notes in Math. 1059 245-255. Springer, Berlin.
53. SAISHO, Y. and TANEMURA, H. (1990). Pitman type theorem for one-dimensional diffusion processes. Tokyo J. Math. 18 429-440.
54. SHEPPARD, P. (1987). On the Ray-Knight Markov property of local times. J. London Math. Soc. 31 377-384.
55. TAKAOKA, K. (1997). On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman's theorem. Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655 256-265. Springer, Berlin.
56. VALLOIS, P. (1991). Une extension des théorèmes de Ray-Knight sur les temps locaux browniens. Probab. Theory Related Fields 88 443-482.
57. VAN DER HOFSTAD, R., DEN HOLLANDER, F. and KÖNIG, W. (1997). Central limit theorem for the Edwards model. Ann. Probab. To appear.
58. VERSHIK, A. and YOR, M. (1997). Multiplicativité du processus gamma et étude asymptotique des lois stables d'indice α, lorsque α tend vers 0. Preprint 289, Lab. Probabilités, Univ. Paris VI.
59. WIDDER, D. (1975). The Heat Equation. Academic Press, New York.
60. WILLIAMS, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusions. I. Proc. London Math. Soc. 28 738-768.
61. WONG, E. and ZAKAI, M. (1974). Martingales and stochastic integrals for processes with a multi-dimensional parameter. Z. Wahrsch. Verw. Gebiete 29 109-122.
62. YAMADA, T. (1996). Principal values of Brownian local times and their related topics. In Itô's Stochastic Calculus and Probability Theory (N. Ikeda, S. Watanabe, M. Fukushima and H. Kunita, eds.) 413-422. Springer, New York.
63. YOR, M. (1991). Une explication du théorème de Ciesielski-Taylor. Ann. Inst. H. Poincaré 27 201-213.
64. YOR, M. (1992). Tsirel'son's equation in discrete time. Probab. Theory Related Fields 91 135-152.
65. YOR, M. (1992). Some Aspects of Brownian Motion. I. Some Special Functionals. Lectures Math. ETH Zürich. Birkhäuser, Basel.
66. YOR, M. (1997). Some Aspects of Brownian Motion. II. Some recent martingale problems. Lectures Math. ETH Zürich. Birkhäuser, Basel.