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Lithuanian Mathematical Journal

, Volume 48, Issue 1, pp 61–69 | Cite as

On stochastic processes associated with relativistic stable distributions

  • B. Grigelionis
Article

Abstract

Lévy processes with marginal relativistic α-stable distributions are described. Strictly stationary Ornstein-Uhlenbeck type processes with one-dimentional relativistic α-stable distributions are constructed. The exponential family as Esscher transforms of distributions on D [0,∞)(R d ) of relativistic α-stable Lévy processes is obtained and the corresponding mixed exponential processes are characterized.

Keywords

Lévy process mixed exponential process Ornstein-Uhlenbeck type process relativistic α-stable distribution relativistic Schrödinger operator subordinated Gaussian Lévy process Thorin subordinator 

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References

  1. 1.
    D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, Cambridge, 2004.MATHGoogle Scholar
  2. 2.
    R. Carmona, W.C. Masters, and B. Simon, Relativistic Schrödinger operators: Asymptotic behaviour of the eigenfunctions, J. Funct. Anal., 91:117–142, 1990.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    B. Grigelionis, Mixed exponential processes, in V. Korolyuk, N. Portenko, and H. Syta(Eds.), Skorokhod’s Ideas in Probability Theory, Kyiv, 2000.Google Scholar
  4. 4.
    B. Grigelionis, Thorin classes of Lévy processes and their transforms, Preprint, 37, Institute of Mathematics and Informatics, Vilnius, 2007.Google Scholar
  5. 5.
    B. Grigelionis, On subordinated multivariate Gaussian Lévy processes, Acta Appl. Math., 96:233–246, 2007.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. Jacod and A.N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin Heidelberg, 1987.MATHGoogle Scholar
  7. 7.
    P. Kim and Y.-R. Lee, Generalized 3G theorem and application to relativistic stable process on non-smooth open sets, J. Funct. Anal., 246:113–143, 2007.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J.V. Linnik and I.V. Ostrovski, Decomposition of Random Variables and Vectors, Am. Math. Soc., Providence, RI, 1977.MATHGoogle Scholar
  9. 9.
    M. Ryznar, Estimates of Green function for relativistic α-stable process, Potential Anal., 17:1–23, 2002.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.MATHGoogle Scholar
  11. 11.
    G.N. Watson, Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1958.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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