On Inversions and Doob h-Transforms of Linear Diffusions

Abstract

Let X be a regular linear diffusion whose state space is an open interval \(E \subseteq \mathbb{R}\). We consider the dual diffusion X whose probability law is obtained as a Doob h-transform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. We provide a construction of X as a deterministic inversion I(X) of X, time changed with some random clock. Such inversions generalize the Euclidean inversions that intervene when X is a Brownian motion. The important case where X is X conditioned to stay above some fixed level is included. The families of deterministic inversions are given explicitly for the Brownian motion with drift, Bessel processes and the three-dimensional hyperbolic Bessel process.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of StatisticsThe University of WarwickCoventryUK
  2. 2.Département de MathématiquesUniversité d’Angers, UFR SciencesAngers Cedex 01France
  3. 3.Institute of Mathematics and Computer ScienceWrocław University of TechnologyWrocławPoland

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