On Inversions and Doob h-Transforms of Linear Diffusions

Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)


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.


Brownian Motion Infinitesimal Generator Dual Process Linear Diffusion Bessel Process 
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We thank the referee for numerous comments that helped to improve the paper. We would like to thank Julien Berestycki who asked the first author a question which led to Corollary 2. We are greatly indebted to l’Agence Nationale de la Recherche for the research grant ANR-09-Blan-0084-01.


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© 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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