Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
| ||

``Probabilités et Modèles Aléatoires''
| ||

**Auteur(s): **

**Code(s) de Classification MSC:**

- 60J25 Markov processes with continuous parameter
- 60G18 Self-similar processes

**Résumé:** Let $\xi$ be a real valued Lévy process that drifts to $-\infty$ and satisfies Cramer's condition, and $X$ a self--similar Markov process
associated to $\xi$ via Lamperti's [22] transformation. In this case, $X$ has $0$ as a trap and fulfills the assumptions of Vuolle-Apiala [34]. We deduce from [34] that there exists a unique excursion measure $\exc,$ compatible with the semigroup of $X$ and such that $\exc(X_{0+}>0)=0.$ Here, we give a precise description of $\exc$ via its associated entrance law. To that end, we construct a self--similar process $X^{\natural},$ which can be viewed as $X$ conditioned to never hit $0,$ and then we construct $\exc$ in a similar way like the Brownian excursion measure is constructed via the law of a Bessel(3) process. An alternative description of $\exc$ is given by specifying the law of the excursion process conditioned to have a given length. We establish some duality relations from which we determine the image under time reversal of $\exc$.

**Mots Clés:** *Self--similar Markov process ; description of excursion measures ; weak duality ;
Lévy processes
*

**Date:** 2003-07-07

**Prépublication numéro:** *PMA-838*