×

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

Citations:

Zbl 0967.60060
PDF BibTeX XML Cite
Full Text: DOI

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.