Journal of Theoretical Probability

, Volume 18, Issue 3, pp 567–593 | Cite as

On First Range Times of Linear Diffusions

  • Paavo SalminenEmail author
  • Pierre Vallois


In this paper we consider first range times (with randomised range level) of a linear diffusion on R. Inspired by the observation that the exponentially randomised range time has the same law as a similarly randomised first exit time from an interval, we study a large family of non-negative 2-dimensional random variables (X,X′) with this property. The defining feature of the family is F c (x,y)=F c (x+y,0), ∀ x ≥ 0, y ≥ 0, where F c (x,y):=P (X > x, X′ > y) We also explain the Markovian structure of the Brownian local time process when stopped at an exponentially randomised first range time. It is seen that squared Bessel processes with drift are serving hereby as a Markovian element.


Bessel bridges Bessel functions Brownian motion h-transforms infimum Ray–Knight theorem supremum 


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© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Mathematical DepartmentÅbo Akademi UniversityÅboFinland
  2. 2.Département de MathématiqueUniversité Henri PoincaréVandoeuvre-les-NancyFrance

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