Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-13T22:46:34.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Kac's formula, levy's local time and brownian excursion

Published online by Cambridge University Press:  14 July 2016

G. Louchard
G. Louchard*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Université Libre de Bruxelles, Faculté des Sciences, Laboratoire d'informatique Théorique, Campus Plaine CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

Kac's formula for Brownian functionals and Levy's local time decomposition are shown to be useful tools in analysing Brownian excursion properties. These tools are applied to maximum, local time and area distributions. Some curious connections between some of these distributions are explained by simple probabilistic arguments.

Keywords

BROWNIAN EXCURSION MAXIMUMLOCAL TIMEAREA
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 3 , September 1984 , pp. 479 - 499
Copyright
Copyright © Applied Probability Trust 1984 

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.)

References

[1] Abramowitz, M. and Stegun, I. A. (1965) Handbook of Mathematical Functions. Dover, New York.Google Scholar
[2] Chung, K. L. (1976) Excursions in Brownian motion. Ark. Mat. 14, 155–177.Google Scholar
[3] Chung, K. L. and Durrett, R. (1976) Downcrossings and local time. Z. Wahrscheinlichkeitsth. 35, 147–149.Google Scholar
[4] 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
[5] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G. (1954) Tables of Integral Transforms, Vol. 1. McGraw-Hill, New York.Google Scholar
[6] Flajolet, P. (1981) Analyse d'algorithmes de manipulation d'arbres et de fichiers. Cahiers du BURO, 34–35.Google Scholar
[7] Getoor, R. K. and Sharpe, M. J. (1979) Excursions of Brownian motion and Bessel processes. Z. Wahrscheinlichkeitsth. 47, 83–106.Google Scholar
[8] Hooghiemstra, G. (1982) On the explicit form of the density of Brownian excursion local time. Proc. Amer. Math. Soc. 84, 127–130.Google Scholar
[9] Imhof, J. P. (1984) Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Prob. 21, 500–510.Google Scholar
[10] Ito, K. and Mckean, H. P. Jr (1974) Diffusion Processes and their Sample Paths, 2nd edn. Springer-Verlag, Berlin.Google Scholar
[11] Kennedy, D. P. (1976) The distribution of the maximum Brownian excursion. J. Appl. Prob. 13, 371–376.Google Scholar
[12] Knight, F. B. (1969) Brownian local times and taboo processes. Trans. Amer. Math. Soc. 143, 173–185.Google Scholar
[13] Knight, F. B. (1980) On the excursion process of Brownian motion. Trans. Amer. Math. Soc. 588, 77–86.Google Scholar
[14] Knight, F. B. (1980) On the excursion process of Brownian motion. Zbl. Math. 426, abstract 60073.Google Scholar
[15] Levy, P. (1948) Processus stochastiques et mouvement brownien, Chapitre VI. Gauthier-Villars, Paris.Google Scholar
[16] Louchard, G. (1965) Hausdorff-dimensional measure of local time. Unpublished note, Rockefeller University.Google Scholar
[17] Louchard, G. (1968) Mouvement brownien et valeurs propres du Laplacien. Ann. Inst. H. Poincaré (IV) 4, 331–342.Google Scholar
[18] Rice, S. O. (1982) The integral of the absolute value of the pinned Wiener process — Calculation of its probability density by numerical integration. Ann. Prob. 10, 240–243.Google Scholar
[19] Shepp, L. A. (1982) On the integral of the absolute value of the pinned Wiener process. Ann. Prob. 10, 234–239.Google Scholar
[20] Vervaat, W. (1979) A relation between Brownian bridge and Brownian excursion. Ann. Prob. 7, 143–149.Google Scholar