Decomposition at the maximum for excursions and bridges of one-dimensional diffusions

  • Jim Pitman
  • Marc Yor


In his fundamental paper [25], Itô showed how to construct a Poisson point process of excursions of a strong Markov process X over time intervals when X is away from a recurrent point a of its statespace. The point process is parameterized by the local time process of X at a. Each point of the excursion process is a path in a suitable space of possible excursions of X,starting at a at time 0, and returning to a for the first time at some strictly positive time ζ, called the lifetime of the excursion. The intensity measure of the Poisson process of excursions is a σ-finite measure Λ on the space of excursions, known as Itô’s excursion law. Accounts of Itô’s theory of excursions can now be found in several textbooks [48, 46, 10]. His theory has also been generalized to excursions of Markov processes away from a set of states [34, 19, 10] and to excursions of stationary, not necessarily Markovian processes [38].


Brownian Motion Path Space Brownian Bridge Bessel Process Path Decomposition 


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Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • Jim Pitman
    • 1
  • Marc Yor
    • 2
  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA
  2. 2.Laboratoire de ProbabilitésUniversité Pierre et Marie CurieParis Cedex 05France

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