Right inverses of nonsymmetric Lévy processes.
Let \(X=(X_t)_{t\geq 0}\) be a Lévy process. An increasing process \(K\) is called a right inverse of \(X\), if it satisfies \(X(K_x)=x\) for all \(x\geq 0\). \textit{S. N. Evans} [Probab. Theory Relat. Fields 118, 37--48 (2000; Zbl 0967.60060)] has characterised the symmetric Lévy processes that have right inverses. The present paper addresses the question what happens when symmetry fails. It is shown that both \(X\) and \(-X\) have right inverses if and only if \(X\) is recurrent and has a non-trivial Gaussian component. The paper also addresses the question when only \(X\) has a (partial) right inverse and gives a description of the excursion measure \(n^Z\) of the strong Markov process \(Z=X-L\) (reflected process) where \(L_t=\inf\{x>0\); \(K_x>t\}\).
scientific article; zbMATH DE number 1906056
scientific article