×

Un théorème de Ray-Knight lié au supremum des temps locaux browniens. (A Ray-Knight theorem related to suprema of Brownian local times). (French) Zbl 0688.60060

Let \(\alpha\) denote the first hitting time of 1 by the supremum over nonnegative values of the space variable of the Brownian local times. We give a complete description in terms of Bessel processes for the law of the process (in the space variable) of the Brownian local times at time \(\alpha\). The proofs rely on stochastic calculus, excursion theory and especially the Ray-Knight theorems on Brownian local times. This result is applied to give a different proof of Borodin’s result concerning the law of the supremum of Brownian local times taken at an independent exponential time.
Reviewer: N.Eisenbaum

MSC:

60J55 Local time and additive functionals
60J65 Brownian motion
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Abramovitz, M., Stegun, I.: Handbook of mathematical functions. Dover 1965
[2] Biane, P., Yor, M.: Sur la loi des temps locaux browniens pris en un temps exponentiel Séminaire de. Probabilités XXII. (Lect. Notes Math., vol. 1321, pp. 454-466) Berlin Heidelberg New York: Springer 1988 · Zbl 0652.60081
[3] Biane, P., Yor, M.: Valeurs principales associées aux temps locaux browniens. Bull. Sci. Math., II. Ser.111, 23-101 (1987) · Zbl 0619.60072
[4] Borodin, A.N.: Distribution of integral functionals of Brownian motion. Zap. Nauckn. Semin. Zeningr. Otd. Mat. Inst. Steklova119, 19-38 (1982). In english: J. Sov. Math.27, 3005-3021 (1984) · Zbl 0491.60082
[5] Csaki, E., Földes, A., Salminen, P.: On the joint distribution of the maximum and its location for a linear diffusion. Ann. Inst. Henri Poincaré23, 179-194 (1987) · Zbl 0621.60081
[6] Jeulin, T.: Application de la théorie du grossissement à l’étude des temps locaux browniens. (Lect. Notes Math. vol. 1118). Berlin Heidelberg New York: Springer 1985 · Zbl 0562.60080
[7] Karatzas, I., Shreve, S.: Brownian motion and stochastic calculus. Graduate texts in Math. Berlin Heidelberg New York: Springer 1987 · Zbl 0615.60075
[8] Knight, F.B.: Random walks and a sojourn density process of Brownian motion. Trans. Am. Math. Soc.109, 56-86 (1963) · Zbl 0119.14604
[9] Pitman, J., Yor, M.: A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheor. Verw. Geb.59, 425-457 (1982) · Zbl 0484.60062
[10] Pitman, J., Yor, M.: Asymptotic laws of planar Brownian motion. Ann. Probab.14, 733-779 (1986) · Zbl 0607.60070
[11] Ray, D.B.: Sojourn times of a diffusion process. Ill. J. Math.7, 615-630 (1963) · Zbl 0118.13403
[12] Walsh, J.B.: Excursions and local time. ?Temps locaux?. Astérisque52-53, 159-192 (1978)
[13] Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusion. Proc. Lond. Math. Soc., III. Ser.3, 738-768 (1974) · Zbl 0326.60093
[14] Williams, D.: Markov properties of Brownian local times. Bull. Am. Math. Soc.75, 1035-1036 (1969) · Zbl 0266.60060
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.