# 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 a diffusion $X^*$ which 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 $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details.
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https://hal.archives-ouvertes.fr/hal-00735182
Contributor : Piotr Graczyk <>
Submitted on : Wednesday, October 10, 2012 - 9:25:38 AM
Last modification on : Monday, March 9, 2020 - 6:15:58 PM
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• HAL Id : hal-00735182, version 2

### Citation

Larbi Alili, Piotr Graczyk, Tomasz Zak. On inversions and Doob $h$-transforms of linear diffusions. 2012. ⟨hal-00735182v2⟩

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