×

The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest. (English) Zbl 0954.60060

The paper describes the local time process of a Brownian excursion or reflecting Brownian bridge as a solution of stochastic differential equation (SDE). The result is a development of the deep connection between Brownian excursions and branching processes.

MSC:

60J65 Brownian motion
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J55 Local time and additive functionals
60J60 Diffusion processes
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] ALDOUS, D. and PITMAN, J. 1994. Brownian bridge asymptotics for random mappings. Random Structures and Algorithms 5 487 512. · Zbl 0811.60057 · doi:10.1002/rsa.3240050402
[2] ALDOUS, D. J. 1991. The continuum random tree I. Ann. Probab. 19 1 28. · Zbl 0722.60013 · doi:10.1214/aop/1176990534
[3] ALDOUS, D. J. 1991. The continuum random tree II: an overview. In Stochastic AnalysisM. T. Barlow and N. H. Bingham, eds. 23 70. Cambridge Univ. Press. · Zbl 0791.60008 · doi:10.1017/CBO9780511662980.003
[4] ALDOUS, D. J. 1993. The continuum random tree III. Ann. Probab. 21 248 289. · Zbl 0791.60009 · doi:10.1214/aop/1176989404
[5] BERTOIN, J. and PITMAN, J. 1994. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math. 118 147 166. · Zbl 0805.60076
[6] BIANE, P. 1986. Relations entre pont et excursion du mouvement brownien reel. Ann. Inst. \' H. Poincare 22 1 7. \' · Zbl 0596.60079
[7] BIANE, P. and YOR, M. 1987. Valeurs principales associees aux temps locaux browniens. \' Bull. Sci. Math. 111 23 101. · Zbl 0619.60072
[8] BIANE, P. and YOR, M. 1988. Sur la loi des temps locaux browniens pris en un temps exponentiel. Seminaire de Probabilites XXII. Lecture Notes in Math. 1321 454 466. \' Śpringer, Berlin. · Zbl 0652.60081
[9] BLUMENTHAL, R. M. 1983. Weak convergence to Brownian excursion. Ann. Probab. 11 798 800. · Zbl 0515.60081 · doi:10.1214/aop/1176993525
[10] BORODIN, A. N. 1986. On the character of convergence to Brownian local time I. Probab. Theory Related Fields 72 231 250. · Zbl 0572.60078 · doi:10.1007/BF00699105
[11] BORODIN, A. N. 1986. On the character of convergence to Brownian local time II. Probab. Theory Related Fields 72 251 277. · Zbl 0572.60078 · doi:10.1007/BF00699105
[12] BORODIN, A. N. 1989. Brownian local time. Uspekhi Mat. Nauk. 44 7 48. · Zbl 0734.60081 · doi:10.1137/1134075
[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. · Zbl 0808.60066 · doi:10.1007/BF01204951
[14] CHAUMONT, L. 1997. An extension of Vervaat’s transformation and its consequences. Prepublication 402, Laboratoire de Probabilites, Univ. Paris VI. \' \' · Zbl 0952.60079
[15] CHUNG, K. L. 1996. Excursions in Brownian motion. Arkiv fur Matematik 14 155 177. \"
[16] DRMOTA, M. and GITTENBERGER, B. 1997. On the profile of random trees. Random Structures Algorithms 10 421 451. · Zbl 0882.60084 · doi:10.1002/(SICI)1098-2418(199707)10:4<421::AID-RSA2>3.0.CO;2-W
[17] DRMOTA, M. and GITTENBERGER, B. 1997. On the strata of random mappings a combinatorial approach. Preprint. Stochastic Process. Appl. · Zbl 0993.05009 · doi:10.1016/S0304-4149(99)00021-6
[18] DWASS, M. 1969. The total progeny in a branching process. J. Appl. Probab. 6 682 686. JSTOR: · Zbl 0192.54401 · doi:10.2307/3212112
[19] FELLER, W. 1951. The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Statist. 22 427 432. · Zbl 0043.34201 · doi:10.1214/aoms/1177729589
[20] FELLER, W. 1951. Diffusion processes in genetics. Proc. Second Berkeley Symp. Math. Statist. Probab. 227 246. Univ. California Press, Berkeley. · Zbl 0045.09302
[21] HARRIS, T. E. 1952. First passage and recurrence distributions. Trans. Amer. Math. Soc. 73 471 486. JSTOR: · Zbl 0048.36301 · doi:10.2307/1990803
[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. JSTOR: · Zbl 0547.60081 · doi:10.2307/3213612
[23] IMHOF, J. P. and KUMMERLING, P. 1986. Operational derivation of some Brownian motion \" results. Internat. Statist. Rev. 54 327 341. JSTOR: · Zbl 0624.60092 · doi:10.2307/1403062
[24] ITO, K. 1971. Poisson point processes attached to Markov processes. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 3 225 240. Univ. California Press, Berkeley.
[25] ITO, K. and MCKEAN, H. P. 1965. Diffusion Processes and Their Sample Paths. Springer, New York. · Zbl 0127.09503
[26] JEANBLANC, M., PITMAN, J. and YOR, M. 1997. The Feynman Kac formula and decomposition of Brownian paths. Comput. Appl. Math. 16 27 52. · Zbl 0877.60027
[27] JEULIN, T. 1985. Temps local et theorie du grossissement: application de la theorie du \' ǵrossissement a l’etude des temps locaux browniens. Grossissements de filtrations: éxemples et applications. Lecture Notes in Math. 1118 197 304. Springer, Berlin. · Zbl 0562.60080
[28] KAWAZU, K. and WATANABE, S. 1971. Branching processes with immigration and related limit theorems. Theory Probab. Appl. 16 36 54. · Zbl 0242.60034
[29] KENNEDY, D. P. 1976. The distribution of the maximum Brownian excursion. J. Appl. Probab. 13 371 376. JSTOR: · Zbl 0338.60048 · doi:10.2307/3212843
[30] KERSTING, G. 1996. On the profile of a conditioned 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 Cramer von Mises ťests. · Zbl 0134.36707
[33] KNIGHT, F. 1997. Approximation of stopped brownian local time by diadic upcrossing chains. Stochastic Process. Appl. 66 253 270. · Zbl 0889.60083 · doi:10.1016/S0304-4149(96)00119-6
[34] KNIGHT, F. B. 1963. Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 197 36 56. JSTOR: · Zbl 0119.14604 · doi:10.2307/1993647
[35] KNIGHT, F. B. 1998. The moments of the area under reflected Brownian bridge conditional on its local time at zero. Preprint. Z · Zbl 0965.60081 · doi:10.1155/S1048953300000137
[36] KOLCHIN, V. F.. Random Mappings. Optimization Software, New York. Trans. of Russian. original. · Zbl 0930.12004
[37] KURTZ, T. G. and PROTTER, P. 1991. Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035 1070. · Zbl 0742.60053 · doi:10.1214/aop/1176990334
[38] KUSHNER, H. J. 1974. On the weak convergence of interpolated Markov chains to a diffusion. Ann. Probab. 2 40 50. · Zbl 0285.60064 · doi:10.1214/aop/1176996750
[39] LAMPERTI, J. 1967. Limiting distributions for branching processes. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 225 241. Univ. California Press, Berkeley. · Zbl 0238.60066
[40] LAMPERTI, J. 1967. The limit of a sequence of branching processes.Wahrsch. Verw. Gebiete 7 271 288. · Zbl 0154.42603 · doi:10.1007/BF01844446
[41] LEURIDAN, C. 1998. Le theoreme de Ray-Knight a temps fixe. Seminaire de Probabilites \' \' \' XXXII. Lecture Notes in Math. 1686 376 406. Springer, Berlin. · Zbl 0919.60073
[42] LEVY, P. 1939. Sur certains processus stochastiques homogenes. Compositio Math. 7 \' 283 339. · Zbl 0022.05903
[43] LINDVAL, T. 1972. Convergence of critical Galton Watson branching processes. J. Appl. Probab. 9 445 450. JSTOR: · Zbl 0238.60063 · doi:10.2307/3212815
[44] MCGILL, P. 1986. Integral representation of martingales in the Brownian excursion filtration. Seminaire de Probabilites XX. Lecture Notes in Math. 1204 465 502. Springer, \' \' Berlin. · Zbl 0635.60057
[45] NEVEU, J. and PITMAN, J. 1989. The branching process in a Brownian excursion. Seminaire \' de Probabilites XXIII. Lecture Notes in Math. 1372 248 257. Springer, Berlin. \' · Zbl 0741.60081
[46] NEVEU, J. and PITMAN, J. 1989. Renewal property of the extrema and tree property of a one-dimensional Brownian motion. Seminaire de Probabilites XXIII. Lecture Notes in \' Ḿath. 1372 239 247. Springer, Berlin. · Zbl 0741.60080
[47] 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. · Zbl 0611.60052 · doi:10.1007/BF00569993
[48] PAVLOV, YU. L. 1988. Distributions of the number of vertices in strata of a random forest. Theory Probab. Appl. 33 96 104. · Zbl 0668.60015 · doi:10.1137/1133009
[49] PAVLOV, YU. L. 1994. Limit distributions of the height of a random forest of plane rooted trees. Discrete Math. Appl. 4 73 88. · Zbl 0810.60083 · doi:10.1515/dma.1994.4.1.73
[50] PERKINS, E. 1982. Local time is a semimartingale.Wahrsch. Verw. Begiete 60 79 117. · Zbl 0468.60070 · doi:10.1007/BF01957098
[51] PERMAN, M. 1996. An excursion approach to Ray Knight theorems for perturbed Brownian motion. Stochastic Process. Appl. 63 67 74. · Zbl 0909.60067 · doi:10.1016/0304-4149(96)00066-X
[52] PERMAN, M. and WERNER, W. 1998. Perturbed Brownian motions. Probab. Theory Related Fields 108 357 383. · Zbl 0884.60082 · doi:10.1007/s004400050113
[53] PITMAN, J. 1975. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. in Appl. Probab. 7 511 526. JSTOR: · Zbl 0332.60055 · doi:10.2307/1426125
[54] PITMAN, J. 1996. Cyclically stationary Brownian local time processes. Probab. Theory Related Fields 106 299 329. · Zbl 0857.60074 · doi:10.1007/s004400050066
[55] 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. URL:
[56] PITMAN, J. 1998. Enumerations of trees and forests related to branching processes andrandom walks. In Microsurveys in Discrete Probability D. Aldous and J. Propp, eds. 163 180. Amer. Math. Soc., Providence, RI. · Zbl 0908.05027
[57] PITMAN, J. and YOR, M. 1982. A decomposition of Bessel bridges.Wahrsch. Verw. Gebiete 49 425 457. · Zbl 0484.60062 · doi:10.1007/BF00532802
[58] PITMAN, J. and YOR, M. 1996. Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In Ito’s Stochastic Calculus and Probability Theory 293 310. Springer, New York. · Zbl 0877.60053
[59] PITMAN, J. and YOR, M. 1997. The two-parameter Poisson Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855 900. · Zbl 0880.60076 · doi:10.1214/aop/1024404422
[60] PROSKURIN, G. V. 1973. On the distribution of the number of vertices in strata of a random mapping. Theory Probab. Appl. 18 803 808. · Zbl 0324.60009 · doi:10.1137/1118106
[61] RAY, D. B. 1963. Sojourn times of a diffusion process. Ill. J. Math. 7 615 630. · Zbl 0118.13403
[62] REVUZ, D. and YOR, M. 1994. Continuous Martingales and Brownian motion, 2nd ed. Springer, Berlin. · Zbl 0804.60001
[63] ROGERS, L. C. G. 1987. Continuity of martingales in the Brownian excursion filtration. Probab. Theory Related Fields 76 291 298. · Zbl 0611.60075 · doi:10.1007/BF01297486
[64] ROGERS, L. C. G. and WALSH, J. B. 1991. The intrinsic local time sheet of Brownian motion. Probab. Theory Related Fields 88 363 379. · Zbl 0722.60079 · doi:10.1007/BF01418866
[65] SHIGA, T. and WATANABE, S. 1973. Bessel diffusions as a one-parameter family of diffusion processes.Wahrsch. Verw. Gebiete 27 37 46. · Zbl 0327.60047 · doi:10.1007/BF00736006
[66] TAKACS, L. 1995. Brownian local times. J. Appl. Math. Stochastic Anal. 3 209 232. \' · Zbl 0845.60078 · doi:10.1155/S1048953395000207
[67] TAKACS, L. 1995. On the local time of the Brownian motion. Ann. Appl. Probab. 5 \' 741 756. · Zbl 0845.60082 · doi:10.1214/aoap/1177004703
[68] VERVAAT, W. 1979. A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 143 149. · Zbl 0392.60058 · doi:10.1214/aop/1176995155
[69] WILLIAMS, D. 1969. Markov properties of Brownian local time. Bull. Amer. Math. Soc. 75 1035 1036. · Zbl 0266.60060 · doi:10.1090/S0002-9904-1969-12350-5
[70] WILLIAMS, D. 1970. Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 871 873. · Zbl 0233.60066 · doi:10.1090/S0002-9904-1970-12591-5
[71] WILLIAMS, D. 1974. Path decomposition and continuity of local time for one-dimensional () diffusions I. Proc. London Math. Soc. 3 28 738 768. · Zbl 0326.60093 · doi:10.1112/plms/s3-28.4.738
[72] YOR, M. 1992. Some Aspects of Brownian Motion I: Some Special Functionals. Birkhauser, \" Boston. · Zbl 0779.60070
[73] BERKELEY, CALIFORNIA 94720-3860 E-MAIL: pitman@stat.berkeley.edu
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.