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2015 | OriginalPaper | Buchkapitel

On Inversions and Doob h-Transforms of Linear Diffusions

verfasst von : Larbi Alili, Piotr Graczyk, Tomasz Żak

Erschienen in: In Memoriam Marc Yor - Séminaire de Probabilités XLVII

Verlag: Springer International Publishing

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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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Titel: On Inversions and Doob h-Transforms of Linear Diffusions
verfasst von: Larbi Alili
Piotr Graczyk
Tomasz Żak
Verlag: Springer International Publishing
Buch: In Memoriam Marc Yor - Séminaire de Probabilités XLVII
Print ISBN: 978-3-319-18584-2

Electronic ISBN: 978-3-319-18585-9

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-319-18585-9_6