Journal of Theoretical Probability

, Volume 27, Issue 2, pp 315–357 | Cite as

Inhomogeneous Lévy Processes in Lie Groups and Homogeneous Spaces

  • Ming Liao


We obtain a representation of an inhomogeneous Lévy process in a Lie group or a homogeneous space in terms of a drift, a matrix function and a measure function. Since the stochastic continuity is not assumed, our result generalizes the well-known Lévy–Itô representation for stochastic continuous processes with independent increments in ℝ d and its extension to Lie groups.


Lévy processes Lie groups Homogeneous spaces 

Mathematics Subject Classification

60J25 58J65 



Helpful comments from David Applebaum and an anonymous referee have led to a considerable improvement of the exposition.


  1. 1.
    Applebaum, D., Kunita, H.: Lévy flows on manifolds and Lévy processes on Lie groups. J. Math. Kyoto Univ. 33, 1105–1125 (1993) MathSciNetGoogle Scholar
  2. 2.
    Ethier, S.N., Kurtz, T.G.: Markov processes, characterization and convergence. Wiley, New York (1986) CrossRefMATHGoogle Scholar
  3. 3.
    Feinsilver, P.: Processes with independent increments on a Lie group. Trans. Am. Math. Soc. 242, 73–121 (1978) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, San Diego (1978) MATHGoogle Scholar
  5. 5.
    Heyer, H., Pap, G.: Martingale characterizations of increment processes in a locally compact group. In: Infin. Dime. Anal. Quantum Probab. Rel. Top., vol. 6, pp. 563–595 (2003) Google Scholar
  6. 6.
    Hunt, G.A.: Semigroups of measures on Lie groups. Trans. Am. Math. Soc. 81, 264–293 (1956) CrossRefMATHGoogle Scholar
  7. 7.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003) CrossRefMATHGoogle Scholar
  8. 8.
    Kallenberg, O.: Foundations of Modern Probability. Springer, Berlin (1997) MATHGoogle Scholar
  9. 9.
    Liao, M.: Lévy Processes in Lie Groups. Cambridge Univ. Press, Cambridge (2004) CrossRefMATHGoogle Scholar
  10. 10.
    Liao, M.: A decomposition of Markov processes via group actions. J. Theor. Probab. 22, 164–185 (2009) CrossRefMATHGoogle Scholar
  11. 11.
    Pap, G.: General solution of the functional central limit problems on a Lie group. In: Infin. Dime. Anal. Quantum Probab. Rel. Top., vol. 7 pp. 43–87 (2004) Google Scholar
  12. 12.
    Siebert, E.: Continuous hemigroups of probability measures on a Lie group. In: Lect. Notes Math., vol. 928, 362-402. Springer, Belrin (1982) Google Scholar
  13. 13.
    Stroock, D.W., Varadhan, S.R.S.: Limit theorems for random walks on Lie groups, Sankhyā: Indian J. Stat. Ser. A 35, 277–294 (1973) MATHMathSciNetGoogle Scholar
  14. 14.
    Sato, K.: Lévy Processes and Infinitely Devisible Distributions. Cambridge Studies Adv. Math., vol. 68. Cambridge Univ. Press, Cambridge (1999). Translated from 1990 Japanese original and revised by the author Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsAuburn UniversityAuburnUSA

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