zbMATH — the first resource for mathematics

Dual representations of Laplace transforms of Brownian excursion and generalized meanders. (English) Zbl 1391.60180
Summary: The Laplace transform of the \(d\)-dimensional distribution of Brownian excursion is expressed as the Laplace transform of the \((d + 1)\)-dimensional distribution of an auxiliary Markov process, started from a \(\sigma\)-finite measure and with the roles of arguments and times interchanged. A similar identity holds for the Laplace transform of a generalized Brownian meander, which is expressed as the Laplace transform of the same auxiliary Markov process, with a different initial law.

60J25 Continuous-time Markov processes on general state spaces
Full Text: DOI
[1] Bertoin, J.; Pitman, J., Path transformations connecting Brownian bridge, excursion and meander, Bull. Sci. Math., 118, 2, 147-166, (1994) · Zbl 0805.60076
[2] Bertoin, J.; Yor, M., On subordinators, self-similar Markov processes and some factorizations of the exponential variable, Electron. Commun. Probab., 6, 95-106, (2001) · Zbl 1024.60030
[3] Biane, P., Processes with free increments, Math. Z., 227, 1, 143-174, (1998) · Zbl 0902.60060
[4] Biane, P.; Le Gall, J.-F.; Yor, M., Un processus qui ressemble au pont brownien, (Séminaire de Probabilités, XXI, Lecture Notes in Math., vol. 1247, (1987), Springer, Berlin), 270-275 · Zbl 0621.60086
[5] Bryc, W.; Wang, Y., The local structure of \(q\)-Gaussian processes, Probab. Math. Statist., 36, 2, 335-352, (2016) · Zbl 1357.60081
[6] Bryc, W., Wang, Y., 2017. Limit fluctuations for density of asymmetric simple exclusion processes with open boundaries, arxiv http://arxiv.org/abs/1707.07350.
[7] Bryc, W.; Wesołowski, J., Asymmetric simple exclusion process with open boundaries and quadratic harnesses, J. Stat. Phys., 167, 2, 383-415, (2017) · Zbl 1376.82055
[8] Durrett, R. T.; Iglehart, D. L.; Miller, D. R., Weak convergence to Brownian meander and Brownian excursion, Ann. Probab., 5, 1, 117-129, (1977) · Zbl 0356.60034
[9] Hirsch, F.; Yor, M., On temporally completely monotone functions for Markov processes, Probab. Surv., 9, 253-286, (2012) · Zbl 1245.60071
[10] Imhof, J.-P., Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications, J. Appl. Probab., 21, 3, 500-510, (1984) · Zbl 0547.60081
[11] Itô, K.; McKean, H. P., Diffusion processes and their sample paths, Grundlehren Math. Wiss., 125, (1965) · Zbl 0127.09503
[12] Janson, S., Brownian excursion area, wright’s constants in graph enumeration, and other Brownian areas, Probab. Surv., 4, 80-145, (2007) · Zbl 1189.60147
[13] Lebedev, N. N., Special functions and their applications, xii+308, (1965), Prentice-Hall, Inc., Englewood Cliffs, N.J., Revised English edition. Translated and edited by Richard A. Silverman
[14] Mansuy, R.; Yor, M., Aspects of Brownian motion, (Universitext, (2008), Springer-Verlag, Berlin), xiv+195 · Zbl 1162.60022
[15] Pitman, J., Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times, Electron. J. Probab., 4, 11, 33, (1999) · Zbl 0935.60068
[16] Pitman, J., (Combinatorial Stochastic Processes, Lecture Notes in Mathematics, vol. 1875, (2006), Springer-Verlag, Berlin), x+256, Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002, With a foreword by Jean Picard
[17] Pitman, J.; Yor, M., A decomposition of Bessel bridges, Z. Wahrsch. Verw. Gebiete, 59, 4, 425-457, (1982) · Zbl 0484.60062
[18] Revuz, D.; Yor, M., (Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, (1999), Springer-Verlag, Berlin), xiv+602 · Zbl 0917.60006
[19] Wang, Y., Extremes of \(q\)-Ornstein-Uhlenbeck processes, Stochastic Process. Appl., (2017), (in press)
[20] Yen, J.-Y.; Yor, M., Local times and excursion theory for Brownian motion, (Lecture Notes in Mathematics, vol. 2088, (2013), Springer, Cham), x+135, A tale of Wiener and Itô measures
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.