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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Acknowledgements
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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Alili, L., Graczyk, P., Żak, T. (2015). On Inversions and Doob h-Transforms of Linear Diffusions. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) In Memoriam Marc Yor - Séminaire de Probabilités XLVII. Lecture Notes in Mathematics(), vol 2137. Springer, Cham. https://doi.org/10.1007/978-3-319-18585-9_6
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