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Absolute continuity of distributions of one-dimensional Lévy processes

Part of: Miscellaneous topics of analysis in the complex domain Integral operators

Published online by Cambridge University Press:  15 September 2017

Tongkeun Chang
Tongkeun Chang*
Affiliation:
Yonsei University
*
* Postal address: Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu, Seoul, 03722, Korea. Email address: [email protected]
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Abstract

In this paper we study the existence of Lebesgue densities of one-dimensional Lévy processes. Equivalently, we show the absolute continuity of the distributions of one-dimensional Lévy processes. Compared with the previous literature, we consider Lévy processes with Lévy symbols of a logarithmic behavior at ∞.

Keywords

Lévy processLebesgue densitysubordinator

MSC classification

Primary: 45P05: Integral operators
Secondary: 30E25: Boundary value problems
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 852 - 872
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge Unversity Press. Google Scholar
[2] Blumenthal, R. M. and Getoor, R. K. (1960). Some theorems on stable processes. Trans. Amer. Math. Soc. 95, 263–273. Google Scholar
[3] Dziubański, J. (1991). Asymptotic behaviour of densities of stable semigroups of measures. Prob. Theory. Relat. Fields 87, 459–467. Google Scholar
[4] Folland, G. B. (1999). Real Analysis, Modern Techniques and Their Applications, 2nd edn. John Wiley, New York. Google Scholar
[5] Głowacki, P. (1993). Lipschitz continuity of densities of stable semigroups of measures. Colloq. Math. 66, 29–47. CrossRefGoogle Scholar
[6] Głowacki, P. and Hebisch, W. (1993). Pointwise estimates for densities of stable semigroups of measures. Studia Math. 104, 243–258. Google Scholar
[7] Hartman, P. and Wintner, A. (1942). On the infinitesimal generators of integral convolutions. Amer. J. Math. 64, 273–298. Google Scholar
[8] Kallenberg, O. (1981). Splitting at backward times in regenerative sets. Ann. Prob. 9, 781–799. CrossRefGoogle Scholar
[9] Knopova, V. and Schilling, R. L. (2013). A note on the existence of transition probability densities of Lévy processes. Forum Math. 25, 125–149. Google Scholar
[10] Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, 2nd edn. Springer, Heidelberg. CrossRefGoogle Scholar
[11] Pruitt, W. E. and Taylor, S. J. (1969). The potential kernel and hitting probabilities for the general stable process in ℝ N . Trans. Amer. Math. Soc. 146, 299–321. Google Scholar
[12] Sato, K.-I. (1999). Lévy processes and Infinitely Divisible Distributions (Camb. Studies Adv. Math. 68). Cambridge University Press. Google Scholar
[13] Tucker, H. G. (1965). On a necessary and sufficient condition that an infinitely divisible distribution be absolutely continuous. Trans. Amer. Math. Soc. 118, 316–330. Google Scholar
[14] Zaigraev, A. (2006). On asymptotic properties of multidimensional α-stable densities. Math. Nachr. 279, 1835–1854. Google Scholar