Right inverses of nonsymmetric Lévy processes. (English) Zbl 1040.60040

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 \(x\geq 0\). 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 \(L_t=\inf\{x>0\); \(K_x>t\}\).


60G51 Processes with independent increments; Lévy processes
60J25 Continuous-time Markov processes on general state spaces
60J45 Probabilistic potential theory


Zbl 0967.60060
Full Text: DOI


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