The Annals of Probability

Random Brownian scaling identities and splicing of Bessel processes

Jim Pitman and Marc Yor

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An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of Brownian excursions, due to Williams and Bismut, each involving back-to-back splicing of fragments of two independent three-dimensional Bessel processes. Generalizations of both splicing constructions are described, which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.

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Ann. Probab., Volume 26, Number 4 (1998), 1683-1702.

First available in Project Euclid: 31 May 2002

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G18: Self-similar processes 60J60: Diffusion processes [See also 58J65]

Brownian bridge Brownian excursion Brownian scaling path transformation Williams' decomposition local time Bessel process range process


Pitman, Jim; Yor, Marc. Random Brownian scaling identities and splicing of Bessel processes. Ann. Probab. 26 (1998), no. 4, 1683--1702. doi:10.1214/aop/1022855878.

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  • [1] Biane, P. (1986). Relations entre pont et excursion du mouvement Brownien r´eel. Ann. Inst. H. Poincar´e 22 1-7.
  • [2] Biane, P. (1988). Sur un calcul de F. Knight. S´eminaire de Probabilit´es XXII Lecture Notes in Math. 1321. 190-197. Springer, Berlin.
  • [3] Biane, P., Le Gall, J. F. and Yor, M. (1987). Un processus qui ressemble au pont brownien. S´eminaire de Probabilit´es XXI Lecture Notes in Math. 1247 270-275. Springer, Berlin.
  • [4] Biane, P. and Yor, M. (1987). Valeurs principales associ´ees aux temps locaux Browniens. Bull. Sci. Math. 111 23-101.
  • [5] Bismut, J.-M. (1985). Last exit decompositions and regularity at the boundary of transition probabilities.Wahrsch. Verw. Gebiete. 69 65-98.
  • [6] 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.
  • [7] Fitzsimmons, P., Pitman, J. and Yor, M. (1993). Markovian Bridges: Construction, Palm Interpretation, and Splicing. (E. Çinlar, K. Chung and M. Sharpe, eds.) 101-134. Birkh¨auser, Boston.
  • [8] Getoor, R. K. (1979). The Brownian escape process. Ann. Probab. 7 864-867.
  • [9] Hsu, P. and March, P. (1988). Brownian excursions from extremes. S´eminaire de Probabilit´es XXII. Lecture Notes in Math. 1321 190-197. Springer, Berlin.
  • [10] Imhof, J. P. (1992). A construction of the Brownian path from BES3 pieces. Stochastic Process. Appl. 43 345-353.
  • [11] It o, K. (1971). Poisson point processes attached to Markov processes. Proc. Sixth Berkley Symp. Math. Statist. Probab. 3 225-240. Univ. California Press, Berkeley.
  • [12] Jeanblanc, M., Pitman, J. and Yor, M. (1997). The Feynman-Kac formula and decomposition of Brownian paths. Comput. Appl. Math. 16 27-52.
  • [13] Kent, J. (1978). Some probabilistic properties of Bessel functions. Ann. Probab. 6 760-770.
  • [14] Knight, F. B. (1988). Inverse local times, positive sojourns, and maxima for Brownian motion. Colloque Paul L´evy sur les Processus Stochastiques. Asterisque 157-158 233-247.
  • [15] Molchanov, S. A. and Ostrovski, E. (1969). Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 14 128-131.
  • [16] Pitman, J. (1986). Stationary excursions. S´eminaire de Probabilit´es XXI. Lecture Notes in Math. 1247 289-302. Springer, Berlin.
  • [17] Pitman, J. (1996). Cyclically stationary Brownian local time processes. Probab. Theory Related Fields 106 299-329.
  • [18] Pitman, J. and Yor, M. (1982). A decomposition of Bessel bridges.Wahrsch. Verw. Gebiete 59 425-457.
  • [19] Pitman, J. and Yor, M. (1992). Arcsine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc. 3 65 326-356.
  • [20] Pitman, J. and Yor, M. (1993). Dilatations d'espace-temps, r´earrangements des trajectoires browniennes, et quelques extensions d'une identit´e de Knight. C.R. Acad. Sci. Paris Ser. I Math. 316 723-726.
  • [21] Pitman, J. and Yor, M. (1996). Decomposition at the maximum for excursions and bridges of one-dimensional diffusions. In It o's Stochastic Calculus and Probability Theory (N. Ikeda, S. Watanabe, M. Fukushima and H. Kunita, eds.) 293-310. Springer, New York.
  • [22] Pitman, J. and Yor, M. (1997). On the relative lengths of excursions derived from a stable subordinator. S´eminaire de Probabilit´es XXXI Lecture Notes in Math. 1655 287-305. Springer, Berlin.
  • [23] Pitman, J. and Yor, M. (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855-900.
  • [24] Pitman, J. and Yor, M. (1998). Ranked functionals of Brownian excursions. C.R. Acad. Sci. Paris Ser. I Math. 326 93-97.
  • [25] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion, 2nd ed. Springer, Berlin.
  • [26] Rogers, L. C. G. and Williams, D. (1987). Diffusions, Markov Processes and Martingales II: It o Calculus. Wiley, New York.
  • [27] Shiga, T. and Watanabe, S. (1973). Bessel diffusions as a one-parameter family of diffusion processes.Wahrsch. Verw. Gebiete 27 37-46.
  • [28] Vallois, P. (1991). Sur la loi conjointe du maximum et de l'inverse du temps local du mouvement brownien: applications a un th´eor eme de Knight. Stochastics Stochastics Rep. 35 175-186.
  • [29] Vallois, P. (1995). Decomposing the Brownian path via the range process. Stochastic Process. Appl. 55 211-226.
  • [30] Vervaat, W. (1979). A relation between Brownian bridge and Brownian excursion. Ann. Probab. 7 143-149.
  • [31] Wilks, S. (1962). Mathematical Statistics. Wiley, New York.
  • [32] Williams, D. (1970). Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 871-873.
  • [33] Williams, D. (1974). Path decomposition and continuity of local time for one dimensional diffusions I. Proc. London Math. Soc. 3 28 738-768.
  • [34] Williams, D. (1979). Diffusions, Markov Processes, and Martingales I: Foundations. Wiley, New York.
  • [35] Williams, D. (1990). Brownian motion and the Riemann zeta-function. In Disorder in Physical Systems (G. R. Grimmett and D. J. A. Welsh, eds.) 361-372. Clarendon Press, Oxford.
  • [36] Yor, M. (1992). Some Aspects of Brownian Motion, Part I: Some Special Functionals. Birkh¨auser, Basil.
  • [37] Yor, M. (1995). Random Brownian scaling and some absolute continuity relationships. In Seminar on Stochastic Analysis, Random Fields and Applications (E. Bolthausen, M. Dozzi, and F. Russo, eds.) 243-252. Birkh¨auser, Boston.
  • [38] Yor, M. (1997). Some Aspects of Brownian Motion, Part II: Some Recent Martingale Problems. Birkh¨auser, Basil.