The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest

Annals of Probability

Volumn 27, Issue 1, 1999, Pages 261-283

  Pitman, Jim ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Bessel process; Brownian meander; Galton Watson branching process; Local time; Random mapping; Ray Knight theorems

Indexed keywords

EID: 0033243136     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1022677262     Document Type: Article

References (79)

1. ALDOUS, D. and PITMAN, J. (1994). Brownian bridge asymptotics for random mappings. Random Structures and Algorithms 5 487-512.
2. ALDOUS, D. J. (1991). The continuum random tree I. Ann. Probab. 19 1-28.
3. ALDOUS, D. J. (1991). The continuum random tree II: an overview. In Stochastic Analysis (M. T. Barlow and N. H. Bingham, eds.) 23-70. Cambridge Univ. Press.
4. ALDOUS, D. J. (1993). The continuum random tree III. Ann. Probab. 21 248-289.
5. BERTOIN, J. and PITMAN, J. (1994). Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 147-166.
6. BIANE, P. (1986). Relations entre pont et excursion du mouvement brownien réel. Ann. Inst. H. Poincaré 22 1-7.
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. BLUMENTHAL, R. M. (1983). Weak convergence to Brownian excursion. Ann. Probab. 11 798-800.
10. BORODIN, A. N. (1986). On the character of convergence to Brownian local time I. Probab. Theory Related Fields 72 231-250.
11. BORODIN, A. N. (1986). On the character of convergence to Brownian local time II. Probab. Theory Related Fields 72 251-277.
12. BORODIN, A. N. (1989). Brownian local time. Uspekhi Mat. Nauk. 44 7-48.
13. CARMONA, P., PETIT, F. and YOR, M. (1994). Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion. Probab. Theory Related Fields 100 1-29.
14. CHAUMONT, L. (1997). An extension of Vervaat's transformation and its consequences. Prépublication 402, Laboratoire de Probabilités, Univ. Paris VI.
15. CHUNG, K. L. (1996). Excursions in Brownian motion. Arkiv für Matematik 14 155-177.
16. DRMOTA, M. and GITTENBERGER, B. (1997). On the profile of random trees. Random Structures Algorithms 10 421-451.
17. DRMOTA, M. and GITTENBERGER, B. (1997). On the strata of random mappings - a combinatorial approach. Preprint. Stochastic Process. Appl. To appear.
18. DWASS, M. (1969). The total progeny in a branching process. J. Appl. Probab. 6 682-686.
19. FELLER, W. (1951). The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statist. 22 427-432.
20. FELLER, W. (1951). Diffusion processes in genetics. Proc. Second Berkeley Symp. Math. Statist. Probab. 227-246. Univ. California Press, Berkeley.
21. HARRIS, T. E. (1952). First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 471-486.
22. IMHOF, J. P. (1984). Density factorization for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21 500-510.
23. IMHOF, J. P. and KÜMMERLING, P. (1986). Operational derivation of some Brownian motion results. Internat. Statist. Rev. 54 327-341.
24. ITÔ, K. (1971). Poisson point processes attached to Markov processes. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 3 225-240. Univ. California Press, Berkeley.
25. ITÔ, K. and MCKEAN, H. P. (1965). Diffusion Processes and Their Sample Paths. Springer, New York.
26. JEANBLANC, M., PITMAN, J. and YOR, M. (1997). The Feynman-Kac formula and decomposition of Brownian paths. Comput. Appl. Math. 16 27-52.
27. JEULIN, T. (1985). Temps local et théorie du grossissement: 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.
28. KAWAZU, K. and WATANABE, S. (1.971). Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 36-54.
29. KENNEDY, D. P. (1976). The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 371-376
30. KERSTING, G. (1996). On the profile of a conditioner Galton-Watson process. Unpublished manuscript.
31. KERSTING, G. (1998). On the height profile of a conditioned Galton-Watson tree. Unpublished manuscript.
32. KIEFER, J. (1959). K-sample analogues of the Kolmogorov-Smirnov and Cramér-von Mises tests.
33. KNIGHT, F. (1997). Approximation of stopped brownian local time by diadic upcrossing chains. Stochastic Process. Appl. 66 253-270.
34. KNIGHT, F. B. (1963). Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 197 36-56.
35. KNIGHT, F. B. (1998). The moments of the area under reflected Brownian bridge conditional on its local time at zero. Preprint.
36. KOLCHIN, V. F., Random Mappings. Optimization Software, New York. (Trans. of Russian original.)
37. KURTZ, T. G. and PROTTER, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070.
38. KUSHNER, H. J. (1974). On the weak convergence of interpolated Markov chains to a diffusion. Ann. Probab. 2 40-50.
39. LAMPERTI, J. (1967). Limiting distributions for branching processes. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 225-241. Univ California Press, Berkeley.
40. LAMPERTI, J. (1967). The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 271-288.
41. LE GALL, J. F. (1986). Une approche élémentaire des théorèmes de décomposition de Williams. Séminaire de Probabilités XX. Lecture Notes in Math. 1204 447-464. Springer, Berlin.
42. LE GALL, J. F. (1989). Marches aléatoires, mouvement Brownien et processus de branchement. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 258-274. Springer, Berlin.
43. LE GALL, J. F. (1991). Brownian excursions, trees and measure-valued branching processes. Ann. Probab. 19 1399-1439.
44. LE GALL, J. F. (1993). The uniform random tree in a Brownian excursion. Probab. Theory Related Fields 96 369-383.
45. LE GALL, J. F. and LE JAN, Y. (1998). Branching processes in Lévy processes: the exploration process. Ann. Probab. 26 213-252.
46. LE GALL, J. F. and YOR, J. (1986). Excursion browniennes et carrés de processus de Bessel. C. R. Acad. Sci. Paris Sér. 1 303 73-76.
47. LEURIDAN, C. (1998). Le théorème de Ray-Knight à temps fixe. Séminaire de Probabilités XXXII. Lecture Notes in Math. 1686 376-406. Springer, Berlin.
48. LÉVY, P. (1939). Sur certains processus stochastiques homogènes. Compositio Math. 7 283-339.
49. LINDVAL, T. (1972). Convergence of critical Galton-Watson branching processes. J. Appl. Probab. 9 445-450.
50. MCGILL, P. (1986). Integral representation of martingales in the Brownian excursion filtration. Séminaire de Probabilités XX. Lecture Notes in Math. 1204 465-502. Springer, Berlin.
51. NEVEU, J. and PITMAN, J. (1989). The branching process in a Brownian excursion. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 248-257. Springer, Berlin.
52. NEVEU, J. and PITMAN, J. (1989). Renewal property of the extrema and tree property of a one-dimensional Brownian motion. Séminaire de Probabilités XXIII. Lecture Notes in Math. 1372 239-247. Springer, Berlin.
53. NORRIS, J. R., ROGERS, L. C. G. and WILLIAMS, D. (1987). Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Related Fields 74 271-287.
54. PAVLOV, YU. L. (1988). Distributions of the number of vertices in strata of a random forest. Theory Probab. Appl. 33 96-104.
55. PAVLOV, YU. L. (1994). Limit distributions of the height of a random forest of plane rooted trees. Discrete Math. Appl. 4 73-88.
56. PERKINS, E. (1982). Local time is a semimartingale. Z. Wahrsch. Verw. Begiete 60 79-117.
57. PERMAN, M. (1996). An excursion approach to Ray-Knight theorems for perturbed Brownian motion. Stochastic Process. Appl. 63 67-74.
58. PERMAN, M. and WERNER, W. (1998). Perturbed Brownian motions. Probab. Theory Related Fields 108 357-383.
59. PITMAN, J. (1975). One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 511-526.
60. PITMAN, J. (1996). Cyclically stationary Brownian local time processes. Probab. Theory Related Fields 106 299-329.
61. PITMAN, J. (1997). Abel-Cayley-Hurwitz multinomial expansion associated with random mappings, forests and subsets. Technical Report 498, Dept. Statistics, Univ. California Berkeley. Available via http://www.stat .berkeley.edu/users/pitman.
62. PITMAN, J. (1998). Enumerations of trees and forests related to branching processes and random walks. In Microsurveys in Discrete Probability (D. Aldous and J. Propp, eds.) 163-180. Amer. Math. Soc., Providence, RI.
63. PITMAN, J. and YOR, M. (1982). A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 49 425-457.
64. PITMAN, J. and YOR, M. (1996). Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In Itô's Stochastic Calculus and Probability Theory 293-310. Springer, New York.
65. PITMAN, J. and YOR, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900.
66. PROSKURIN, G. V. (1973). On the distribution of the number of vertices in strata of a random mapping. Theory Probab. Appl. 18 803-808.
67. RAY, D. B. (1963). Sojourn times of a diffusion process. III. J. Math. 7 615-630.
68. REVUZ, D. and YOR, M. 11994). Continuous Martingales and Brownian motion, 2nd ed. Springer, Berlin.
69. ROGERS, L. C. G. (1987). Continuity of martingales in the Brownian excursion filtration. Probab. Theory Related Fields 76 291-298.
70. ROGERS, L. C. G. and WALSH, J. B. (1991). The intrinsic local time sheet of Brownian motion. Probab. Theory Related Fields 88 363-379.
71. SHIGA, T. and WATANABE, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrsch. Verw. Gebiete 27 37-46.
72. TAKÁCS, L. (1995). Brownian local times. J. Appl. Math. Stochastic Anal. 3 209-232.
73. TAKÁCS, L. (1995). On the local time of the Brownian motion. Ann. Appl. Probab. 5 741-756.
74. VAN DER HOFSTAD, R., DEN HOLLANDER, F. and KÖNIO, W. (1997). Central limit theorem for the Edwards model. Ann. Probab. 25 573-597
75. VERVAAT, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 143-149.
76. WILLIAMS, D. (1969). Markov properties of Brownian local time. Bull. Amer. Math. Soc. 75 1035-1036.
77. WILLIAMS, D. (1970). Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 871-873.
78. WILLIAMS, D. (1974). Path decomposition and continuity of local time for one-dimensional diffusions I. Proc. London Math. Soc. (3) 28 738-768.
79. YOR, M. (1992). Some Aspects of Brownian Motion I: Some Special Functionals. Birkhaüser, Boston.