The Annals of Probability

Right Inverses of Nonsymmetric Lévy Processes

Matthias Winkel

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Abstract

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.

Article information

Source
Ann. Probab., Volume 30, Number 1 (2002), 382-415.

Dates
First available in Project Euclid: 29 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1020107772

Digital Object Identifier
doi:10.1214/aop/1020107772

Mathematical Reviews number (MathSciNet)
MR1894112

Zentralblatt MATH identifier
1040.60040

Subjects
Primary: 60G51: Processes with independent increments; Lévy processes
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Keywords
Lévy processes subordinators excursions potential theory right inverses creeping

Citation

Winkel, Matthias. Right Inverses of Nonsymmetric Lévy Processes. Ann. Probab. 30 (2002), no. 1, 382--415. doi:10.1214/aop/1020107772. https://projecteuclid.org/euclid.aop/1020107772


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