Right Inverses of Nonsymmetric Lévy Processes

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.