We analyze the existence and properties of right inverses $K$ for
nonsymmetric Lévy processes $X$, extending recent work of Evans in the
symmetric setting. First, both $X$ and $-X$ have right inverses if and only if
$X$ is recurrent and has a nontrivial 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.'' Second, when only asking for right inverses of $X$, a certain
``strength of asymmetry'' is needed. Millar's notion of creeping turns out
necessary but not sufficient for the existence of right inverses. We analyze
this both in the bounded and unbounded variation case with a particular
emphasis on results in terms of the Lévy–Khintchine
characteristics.
Publié le : 2002-01-14
Classification:
Lévy processes,
subordinators,
excursions,
potential theory,
right inverses,
creeping,
60G51,
60J25,
60J45
@article{1020107772,
author = {Winkel, Matthias},
title = {Right Inverses of Nonsymmetric L\'evy Processes},
journal = {Ann. Probab.},
volume = {30},
number = {1},
year = {2002},
pages = { 382-415},
language = {en},
url = {http://dml.mathdoc.fr/item/1020107772}
}
Winkel, Matthias. Right Inverses of Nonsymmetric Lévy Processes. Ann. Probab., Tome 30 (2002) no. 1, pp. 382-415. http://gdmltest.u-ga.fr/item/1020107772/