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Zbl 1074.60082
Fourati, Sonia
Vervaat et Lévy. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 3, 461-478 (2005). ISSN 0246-0203

This paper presents the existing results on the relationship between the bridge and the excursion of a Brownian motion and extends these results for the special framework of Kuznetsov measures. Section 1 briefly surveys the problems, while Section 2 exposes necessary notions and technical background: Kuznetsov measures, indexed trajectories, Markov measures, the ACC hypothesis (absolute continuity of the resolvent) of Silverstein (1980). Sections 2 and 3 contain classical results and an appropriate definition of the processes represented by the Brownian bridge and the Brownian positive excursion in the framework of potential theory. Section 4 presents Vervaat's (1979) path transformation that gives the correspondence between the bridge and the excursion of a Brownian motion. Using the general theory of processes, the author shows the Vervaat correspondence for Lévy processes under the absolute continuity of the resolvent (Silverstein's ACC hypothesis). The final Section 5 proves by the potential theory framework, proposed for the Brownian bridge and excursion, that these two Markov processes become elementary transformations of two Kuznetsov measures.
[Neculai Curteanu (Iaşi)]

MSC 2000:: *60J45 Probabilistic potential theory
60G51 Processes with independent increments
60J65 Brownian motion
60A10 Probabilistic measure theory

Keywords: Lévy process; bridge of Brownian motion; excursion of Brownian motion; indexed trajectory; Kuznetsov measure; Markov measure; potential theory; general theory of processes

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