The Annals of Probability

The SDE Solved By Local Times of a Brownian Excursion or Bridge Derived From the Height Profile of a Random Tree or Forest

Jim Pitman

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Abstract

Let $B$ be a standard one-dimensional Brownian motion started at 0. Let $L_{t,v}(|B|)$ be the occupation density of $|B|$ at level $v$ up to time $t$. The distribution of the process of local times $(L_{t,v}(|B|)v\geq0)$ conditionally given $B_{t}= 0$ and $L_{t,0}(|B|)=l$ is shown to be that of the unique strong solution $X$ of the Itô SDE,

\[dX_v = \Big\{4 - X^2_v\Big(t - \textstyle\int_0^v X_u\,du\Big)^{-1}\Big\}dv + 2\sqrt{X_v}\,dB_v\]

on the interval $[0,V_{t}(X))$, where $V_{t}(X):= \inf{v:\int_0^v X_u,du=t}$, and $X_v= 0$ for all $v\geqV_t(X)$. This conditioned from study of the Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as $n\rightarrow\infty$ and $2k\sqrt{n\rightarrowl}$ of the height profile of a uniform rooted random forest of $k$ trees labeled by a set of $n$ elements, as obtained by conditioning a uniform random mapping of the set to itself to have $k$ cyclic points. The SDE is the continuous analog of a simple description of a Galton-Watson branching process conditioned on its total progeny. For $l = 0$, corresponding to asymptotics of a uniform random tree, the SDE gives a description of the process of local times of a Brownian excursion which is equivalent to Jeulin’s description of these local times as a time change of twice a Brownian excursion. Another corollary is the Biane-Yor description of the local times of a reflecting Brownian ridge as a time-changed reversal of twice a Brownian meander of the same length.

Article information

Source
Ann. Probab. Volume 27, Number 1 (1999), 261-283.

Dates
First available: 29 May 2002

Permanent link to this document
http://projecteuclid.org/euclid.aop/1022677262

Mathematical Reviews number (MathSciNet)
MR1681110

Digital Object Identifier
doi:10.1214/aop/1022677262

Zentralblatt MATH identifier
0954.60060

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C05: Trees

Keywords
Local time Bessel process Galton-Watson branching process Brownian meander Ray-Knight theorems random mapping

Citation

Pitman, Jim. The SDE Solved By Local Times of a Brownian Excursion or Bridge Derived From the Height Profile of a Random Tree or Forest. The Annals of Probability 27 (1999), no. 1, 261--283. doi:10.1214/aop/1022677262. http://projecteuclid.org/euclid.aop/1022677262.


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References

  • 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 AnalysisM. 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 reel. Ann. Inst. ´ H. Poincare 22 1 7. ´
  • 7 BIANE, P. and YOR, M. 1987. Valeurs principales associees 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. Seminaire de Probabilites 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. Prepublication 402, Laboratoire de Probabilites, Univ. Paris VI. ´ ´
  • 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.
  • 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 KUMMERLING, P. 1986. Operational derivation of some Brownian motion ¨ results. Internat. Statist. Rev. 54 327 341.
  • 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.
  • 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 theorie du grossissement: application de la theorie du ´ ´ grossissement a l'etude 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. 1971. 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 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 ´ 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. Z
  • 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.Wahrsch. Verw. Gebiete 7 271 288.
  • 47 LEURIDAN, C. 1998. Le theoreme de Ray-Knight a temps fixe. Seminaire de Probabilites ´ ´ ´ XXXII. Lecture Notes in Math. 1686 376 406. Springer, Berlin.
  • 48 LEVY, P. 1939. Sur certains processus stochastiques homogenes. 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. Seminaire de Probabilites XX. Lecture Notes in Math. 1204 465 502. Springer, ´ ´ Berlin.
  • 51 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. ´
  • 52 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 ´ ´ 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.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 andrandom 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.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 Ito'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. Ill. J. Math. 7 615 630.
  • 68 REVUZ, D. and YOR, M. 1994. 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.Wahrsch. Verw. Gebiete 27 37 46.
  • 72 TAKACS, L. 1995. Brownian local times. J. Appl. Math. Stochastic Anal. 3 209 232. ´
  • 73 TAKACS, L. 1995. On the local time of the Brownian motion. Ann. Appl. Probab. 5 ´ 741 756.
  • 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. Birkhauser, ¨ Boston.
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