Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T05:10:16.428Z Has data issue: false hasContentIssue false
  • English
  • Français

The Brownian excursion multi-dimensional local time density

Part of: Markov processes Combinatorial probability

Published online by Cambridge University Press:  14 July 2016

Bernhard Gittenberger  and
Guy Louchard
Bernhard Gittenberger*
Affiliation:
Technische Universitat Wien
Guy Louchard*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Department of Geometry, TU Wien, Wiedner Hauptstraße 8–10/113, A-1040 Wien, Austria. Email address: [email protected]
∗∗Postal address: Université Libre de Bruxelles, Département d'Informatique C.P. 212, Campus de la Plaine, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: a direct method based on Kac's formula for Brownian functionals and an indirect one based on a limit theorem for Galton–Watson trees.

Keywords

Brownian excursionlocal timerandom trees

MSC classification

Primary: 60J55: Local time and additive functionals
Secondary: 60C05: Combinatorial probability
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 350 - 373
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This author's work was supported by the Austrian Science Foundation FWF, grant P10187-PHY.

References

Aldous, D. (1991). The continuum random tree II: an overview. In Stochastic Analysis, eds. Barlow, M. T. and Bingham, N. H. CUP, Cambridge.Google Scholar
Chung, K. L. (1976). Excursions in Brownian motion. Ark. Mat. 14, 155–177.Google Scholar
Cohen, J. W., and Hooghiemstra, G. (1981). Brownian excursion, the M/M/1 queue and their occupation times. Math. Operat. Res. 6, 608–629.Google Scholar
Drmota, M. (1997). On nodes of given degree in random trees. In Probabilistic Methods in Discrete Mathematics, eds. Kolchin, V. F. et al. VSP, Utrecht, pp. 31–44.Google Scholar
Drmota, M., and Gittenberger, B. (1997). On the profile of random trees. Rand. Str. Alg. 10, 421–451.3.0.CO;2-W>CrossRefGoogle Scholar
Drmota, M., and Soria, M. (1995). Marking in combinatorial constructions: generating functions and limiting distributions. Theoret. Comp. Sci. 144, 67–99.Google Scholar
Le Gall, J.-F. (1991). Brownian excursions, trees and measure-valued branching processes. Ann. Prob. 19, 1399–1439.Google Scholar
Getoor, R. K., and Sharpe, M. J. (1979). Excursions of Brownian motion and Bessel processes. Z. Wahrscheinlichkeitsth. 47, 83–106.Google Scholar
Gittenberger, B. (1999). On the contour of random trees. To appear in SIAM J. Discrete Math.Google Scholar
Gutjahr, W., and Pflug, G. C. (1992). The asymptotic contour process of a binary tree is a Brownian excursion. Stoch. Proc. Appl. 41, 69–89.Google Scholar
Hooghiemstra, G. (1982). On the explicit form of the density of Brownian excursion local time. Proc. Amer. Math. Soc. 84, 127–130.Google Scholar
Itô, K., and McKean, H. P. Jr. (1974). Diffusion Processes and their Sample Paths, 2nd edn. Springer, Berlin.Google Scholar
Kemp, R. (1982). On the average oscillation of a stack. Combinatorica 2, 157–176.Google Scholar
Kemp, R. (1984). Fundamentals of the Average Case Analysis of Particular Algorithms. Wiley–Teubner, New York.CrossRefGoogle Scholar
Kennedy, D. P. (1976). The distribution of the maximum Brownian excursion. J. Appl. Prob. 13, 371–376.Google Scholar
Knight, F. B. (1980). On the excursion process of Brownian motion. Zbl. Math. 426, Abstract 60073.Google Scholar
Knuth, D. E. (1973). The Art of Computer Programming, Vol. 1, 2nd edn. Addison-Wesley, Reading, MA.Google Scholar
Levy, P. (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris, Chapitre VI.Google Scholar
Louchard, G. (1984). Kac's formula, Levy's local time and Brownian Excursion. J. Appl. Prob. 21, 479–499.CrossRefGoogle Scholar
Louchard, G. (1987). Random walks, Gaussian processes and list structures. Theoret. Comp. Sci. 53, 99–134.Google Scholar
Louchard, G., Randrianarimanana, B., and Schott, R. (1992). Dynamic algorithms in D. E. Knuth's model: a probabilistic analysis. Theoret. Comp. Sci. 93, 201–225.Google Scholar
Meir, A., and Moon, J. W. (1978). On the altitude of nodes in random trees. Can. J. Math. 30, 997–1015.Google Scholar
Takács, L. (1991). On the distribution of the number of vertices in layers of random trees. J. Appl. Math. Stoch. Anal. 4, 175–186.Google Scholar
Whittaker, E. T., and Watson, G. N. (1996). A Course of Modern Analysis, 4th edn. CUP, Cambridge.Google Scholar