Excursions of the integral of the Brownian motion
Emmanuel Jacob
Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 46, Number 3
(2010), 869-887.
Abstract
The integrated Brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Itô’s excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been introduced recently by Bertoin.
Résumé
L’intégrale du mouvement Brownien est parfois appelée processus de Langevin. Lachal a étudié plusieurs lois d’excursions qui lui sont associées. Nous suivons ici un point de vue différent, développé par Pitman, pour les processus stationnaires. Nous construisons d’abord un processus de Langevin stationnaire avant d’en déterminer explicitement la mesure d’excursion stationnaire. Ce travail permet alors de fournir une nouvelle description de la mesure d’excursion d’Itô du processus de Langevin réfléchi sur une barrière inélastique, introduit récemment par Bertoin.
Secondary Subjects:
60G18
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aihp/1281100402
Digital Object Identifier: doi:10.1214/09-AIHP322
Zentralblatt MATH identifier: 05795087
Mathematical Reviews number (MathSciNet): MR2682270
References
[1] J. Azéma. Théorie générale des processus et retournement du temps. Ann. Sci. École Norm. Sup. (4) 6 (1973) 459–519.
Mathematical Reviews (MathSciNet):
MR365725
[2] J. Bertoin. Reflecting a Langevin process at an absorbing boundary. Ann. Probab. 35 (2007) 2021–2037.
[3] J. Bertoin. A second order SDE for the Langevin process reflected at a completely inelastic boundary. J. Eur. Math. Soc. (JEMS) 10 (2008) 625–639.
[4] R. K. Getoor. Excursions of a Markov process. Ann. Probab. 7 (1979) 244–266.
Mathematical Reviews (MathSciNet):
MR525052
[5] J. P. Gor’kov. A formula for the solution of a certain boundary value problem for the stationary equation of Brownian motion. Dokl. Akad. Nauk SSSR 223 (1975) 525–528.
Mathematical Reviews (MathSciNet):
MR386033
[6] A. Lachal. Sur le premier instant de passage de l’intégrale du mouvement brownien. Ann. Inst. H. Poincaré Probab. Statist. 27 (1991) 385–405.
[7] A. Lachal. Application de la théorie des excursions à l’intégrale du mouvement brownien. In Séminaire de Probabilités XXXVII 109–195. Lecture Notes in Math. 1832. Springer, Berlin, 2003.
[8] M. Lefebvre. First-passage densities of a two-dimensional process. SIAM J. Appl. Math. 49 (1989) 1514–1523.
[9] B. Maury. Direct simulation of aggregation phenomena. Commun. Math. Sci. 2 (2004) 1–11.
[10] H. P. McKean Jr. A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2 (1963) 227–235.
Mathematical Reviews (MathSciNet):
MR156389
[11] J. Pitman. Stationary excursions. In Séminaire de Probabilités, XXI 289–302. Lecture Notes in Math. 1247. Springer, Berlin, 1987.
Mathematical Reviews (MathSciNet):
MR941992
