Japanese Journal of Mathematics

, Volume 2, Issue 1, pp 97–127 | Cite as

On the excursion theory for linear diffusions

Special Feature: Award of the 1st Gauss Prize to K. Ito


We present a number of important identities related to the excursion theory of linear diffusions. In particular, excursions straddling an independent exponential time are studied in detail. Letting the parameter of the exponential time tend to zero it is seen that these results connect to the corresponding results for excursions of stationary diffusions (in stationary state). We characterize also the laws of the diffusion prior and posterior to the last zero before the exponential time. It is proved using Krein’s representations that, e.g. the law of the length of the excursion straddling an exponential time is infinitely divisible. As an illustration of the results we discuss the Ornstein–Uhlenbeck processes.

Keywords and phrases:

Brownian motion last exit decomposition local time infinite divisibility spectral representation Ornstein–Uhlenbeck process 

Mathematics Subject Classification (2000):

60J65 60J60 


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Copyright information

© The Mathematical Society of Japan and Springer 2007

Authors and Affiliations

  1. 1.Mathematical DepartmentÅbo Akademi UniversityÅboFinland
  2. 2.Département de MathématiqueUniversité Henri PoincaréVandoeuvre les NancyFrance
  3. 3.Laboratoire de Probabilités et Modèles aléatoiresUniversité Pierre et Marie CurieParis Cedex 05France

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