×
Carmona, Ph.; Petit, F.; Yor, M.

Some extensions of the arc sine law as partial consequences of the scaling property of Brownian motion. (English) Zbl 0808.60066

Probab. Theory Relat. Fields 100, No. 1, 1-29 (1994).
Summary: The scaling property of Brownian motion is exploited systematically in order to extend Paul Lévy’s arc sine law to Brownian motion perturbed by its local time at 0. Other important ingredients of the proofs are some Ray-Knight theorems for local times.

Cited in 1 Review
Cited in 19 Documents

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion

Keywords:

Lévy’s arc sine law; Brownian motion; Ray-Knight theorems; local times
PDFBibTeX XMLCite
\textit{Ph. Carmona} et al., Probab. Theory Relat. Fields 100, No. 1, 1--29 (1994; Zbl 0808.60066)
Full Text: DOI

References:

[1] Barlow, M.T., Pitman, J.W., Yor, M.: On Walsh’s Brownian motions. Sém. Probas. XXIII. (Lect. Notes Math., vol. 1372, pp. 275-293) Berlin Heidelberg New York: Springer 1989 · Zbl 0747.60072
[2] Barlow, M.T., Pitman, J.W., Yor, M.: Une extension multidimensionnelle de la loi de l’arc sinus. Sém. Probas. XXIII. (Lect. Notes Math., vol. 1372, pp. 294-314) Berlin Heidelberg New York: Springer 1989 · Zbl 0738.60072
[3] Biane, Ph., Yor, M.: Sur la loi des temps locaux browniens pris en un temps expenentiel. Sém. Probas. XXII. (Lect. Notes Math., vol. 1321, pp. 454-466) Berlin Heidelberg New York: Springer 1988 · Zbl 0652.60081
[4] Biane, Ph., Le Gall, J.F., Yor, M.: Un processus qui ressemble au pont Brownien. Sém Probas. XXI. (Lect. Notes Math., vol. 1247, pp. 270-275) Berlin Heidelberg New York: Springer 1987
[5] Carmona, P., Petit, F., Yor, M.: Sur les fonctionnelles exponentielles de certains processus de Lévy,. Stochastics Stochastic Rep. (to appear 1994). · Zbl 0830.60072
[6] Fitzsimmons, P., Pitman, J.W., Yor, M.: Markovian Bridges: Construction, Palm interpretations, and Splicing. In: Seminar on Stochastic Processes, pp. 102-133 (1992). Basel Boston Berlin: Birkhäuser 1993 · Zbl 0844.60054
[7] Getoor, R.K., Sharpe M.J.: Excursions of Brownian motion and Bessel processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.47 83-106 (1979) · Zbl 0399.60074 · doi:10.1007/BF00533253
[8] Ikeda, N., Watanabe, S.: Stochastic differential equations and Diffusion processes. (Second edition) North Holland: Kodansha 1989 · Zbl 0684.60040
[9] Jeulin, T., Yor, M.: Sur les distributions de certaines fonctionnelles du mouvement brownien. Sém. Probas. XV. (Lect. Notes Math., vol. 850, pp. 210-226) Berlin Heidelberg New York: Springer 1981 · Zbl 0462.60077
[10] Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Berlin Heidelberg New York: Springer 1987 · Zbl 0615.60075
[11] Kent, J.: Some probabilistic properties of Bessel functions. Ann. Probab.6, 760-770 (1978) · Zbl 0402.60080 · doi:10.1214/aop/1176995427
[12] Le Gall, J.F., Yor, M.: Excursions browniennes et carrés de processus de Bessel. Comptes Rendus Acad. Sci. Paris. Sér. I,303, 73-76 (1986). · Zbl 0589.60070
[13] Le Gall, J.F., Yor, M.: Enlacements du mouvement brownien autour des courbes de l’espace. Trans. Am. Math. Soc.317, 687-722 (1990) · Zbl 0696.60072 · doi:10.2307/2001484
[14] Lebedev, N.N.: Special functions and their applications. New York: Dover Publications, Inc. 1972 · Zbl 0271.33001
[15] Lévy, P.: Sur certains processus stochastiques homogènes. Compos. Math.7, 283-339 (1939) · JFM 65.1346.02
[16] Mc Kean, H.P.: Brownian local time. Adv. Math.16, 91-111 (1975) · Zbl 0309.60054 · doi:10.1016/0001-8708(75)90102-4
[17] Meyer, P.A.: Intégrales stochastiques IV. Sém. Probas. I. (Lect. Notes Math., vol. 39, pp. 142-162) Berlin Heidelberg New York: Springer 1967
[18] Perman, M., Pitman, J., Yor, M.: Size-biased sampling of Poisson point processes, and excursions. Probab. Theory Relat. Fields92, (1) 21-40 (1992) · Zbl 0741.60037 · doi:10.1007/BF01205234
[19] Petit, F.: Sur le temps passé par le mouvement brownien au-dessus d’un multiple de son supremum, et quelques extensions de la loi de l’arc sinus. Part of a Thèse de Doctorat, Université Paris VII, February 1992.
[20] Petit, F.: Quelques extensions de la loi de l’arc sinus. Comptes rendus Acad. Sci. Paris, Sér. I315, 855-858 (1992) · Zbl 0756.60077
[21] Pitman, J.W., Yor, M.: A decomposition of Bessel Bridges. Z. Wahrscheinlichkeitstheor. Verw. Geb.59, 425-457 (1982) · Zbl 0484.60062 · doi:10.1007/BF00532802
[22] Pitman, J.W., Yor, M.: Are sine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc.65, (3) 326-356 (1992) · Zbl 0769.60014 · doi:10.1112/plms/s3-65.2.326
[23] Pitman, J.W., Yor, M.: Random scaling of Brownian and Bessel Bridges. University of California 1992 (Preprint)
[24] Pitman, J.W., Yor, M.: Dilatations d’espace-temps, réarrangements des trajectoires browniennes, et quelques extensions d’une identité de Knight. Comptes Rendus Acad. Sci. Paris. Série I316, 723-726 (1993) · Zbl 0789.60059
[25] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Berlin Heidelberg New York: Springer 1991 · Zbl 0731.60002
[26] Walsh, J.B.: A diffusion with discontinuous local time. In: ?Temps locaux?. Astérisque 52-53 (Société Mathématique de France) pp. 37-45 (1978)
[27] Watanabe, S.: Generalized arc-sine laws for one-dimensional diffusion processes and random walks. (Preprint 1993)
[28] Williams, D.: Markov properties of Brownian local time. Bull. Am. Math. Soc.76, 1035-1036 (1969) · Zbl 0266.60060 · doi:10.1090/S0002-9904-1969-12350-5
[29] Yor, M.: Some Aspects of Brownian Motion, Part I: Some Special Functionals. (Lect. Math.) ETH Zürich: Birkhäuser 1992 · Zbl 0779.60070
[30] Yor, M.: Random Brownian scaling and some absolute continuity relationships. (Preprint 1993). Laboratoire de Probabilités, Paris VI. In the Proceedings of the Ascona Meeting (1993) Basel Boston Berlin: Birkhäuser (1994) (to appear)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.