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

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Right inverses of non-symmetric LÚvy processes


Code(s) de Classification MSC:

RÚsumÚ: 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 ClÚs: LÚvy processes ; subordinators ; excursions ; potential theory ; creeping

Date: 2000-10-20

Prépublication numéro: PMA-618

Fichier postscript : PMA-618.ps