Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Recurrent extensions of self-similar Markov processes and Cramer's condition

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