## 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 $$L_t=\inf\{x>0$$; $$K_x>t\}$$.

### MSC:

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

Zbl 0967.60060
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### References:

 [1] BERG, C. and FORST, G. (1975). Potential Theory on Locally Compact Abelian Groups. Springer, Berlin. · Zbl 0308.31001 [2] BERTOIN, J. (1996). Lévy Processes. Cambridge Univ. Press. · Zbl 0861.60003 [3] BERTOIN, J. (1999). Subordinators: Examples and Applications. Ecole d’été de Probabilités de St. Flour XXVII. Lecture Notes in Math. 1717. Springer, Berlin. · Zbl 0955.60046 [4] BLUMENTHAL, R. M. and GETOOR, R. K. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 493-516. · Zbl 0097.33703 [5] BRETAGNOLLE, J. (1971). Résultats de Kesten sur les processus à accroissements indépendants. Séminaire de Probabilités V. Lecture Notes in Math. 21-36. Springer, Berlin. [6] DELLACHERIE, C., MAISONNEUVE, B. and MEYER, P. A. (1992). Probabilités et potentiel 5, Chapters XVII-XXIV. Hermann, Paris. [7] EVANS, S. (2000). Right inverses of Lévy processes and stationary stopped local times. Probab. Theory Related Fields 118 37-48. · Zbl 0967.60060 [8] GETOOR, R. K. and SHARPE, M. J. (1981). Two results on dual excursions. In Seminar on Stochastic Processes (E. Çinlar, K. L. Chung and M. J. Sharpe, eds.) 31-52. Birkhäuser, Boston. · Zbl 0531.60069 [9] MILLAR, P. W. (1973). Exit properties of stochastic processes with stationary independent increments. Trans. Amer. Math. Soc. 178 459-479. · Zbl 0268.60065 [10] ROGERS, L. C. G. (1983). Itô excursion theory via resolvents.Wahrsch. Verw. Gebiete 63 237-255. · Zbl 0528.60073 [11] ROGERS, L. C. G. and WILLIAMS, D. (1987). Diffusions, Markov Processes and Martingales 2: Itô Calculus. Wiley, New York. · Zbl 0977.60005 [12] SATO, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press. · Zbl 0973.60001 [13] SIMON, T. (1999). Subordination in the wide sense for Lévy processes. Probab. Theory Related Fields 115 445-477. · Zbl 0944.60049 [14] VIGON, V. (2001). Votre Lévy rampe-t-il? Un critère pratique pour le savoir. J. London Math. Soc. · Zbl 1016.60054
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