# 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.
keyword :
Type de document :
Pré-publication, Document de travail
2012
Domaine :

Littérature citée [23 références]

https://hal.archives-ouvertes.fr/hal-00735182
Contributeur : Piotr Graczyk <>
Soumis le : mercredi 10 octobre 2012 - 09:25:38
Dernière modification le : mercredi 19 décembre 2018 - 14:08:04
Document(s) archivé(s) le : vendredi 16 décembre 2016 - 22:25:10

### Fichier

Involution_Submitted10_10.pdf
Fichiers produits par l'(les) auteur(s)

### Identifiants

• 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〉

### Métriques

Consultations de la notice

## 221

Téléchargements de fichiers