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

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

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

- 60J25 Markov processes with continuous parameter
- 60J30 Processes with independent increments
- 60J45 Probabilistic potential theory, See also {31Cxx, 31D05}

**Résumé:** We analyse the existence and properties of right inverses $K$ for non-symmetric L\'evy processes $X$, extending recent
work of Evans \cite{Eva-99} in the symmetric setting. First, both $X$ and $-X$ have right inverses if and only if $X$ is
recurrent and has a non-trivial Gaussian component. Our main result is then a description of the excursion measure $n^Z$ of the strong
Markov process $Z=X-L$ (reflected process) where $L_t=\inf\{x>0:K_x>t\}$. Specifically, $n^Z$ is essentially the restriction of $n^X$
to the 'excursions starting negative'. When only asking for right inverses of $X$, a certain 'strength of asymmetry' is needed.
Millar's \cite{Mil-73} notion of creeping turns out necessary but not sufficient for the existence of right inverses. We analyse this
both in the bounded and unbounded variation case with a particular emphasis on results in terms of the L\'evy-Khintchine
characteristics.

**Mots Clés:** *Lévy processes ; subordinators ; excursions ; potential theory ; creeping
*

**Date:** 2000-10-20

**Prépublication numéro:** *PMA-618*

**Fichier postscript : **PMA-618.ps