Skip to main content
SpringerLink
Log in

Convolution of Probability Measures on Lie Groups and Homogenous Spaces

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden’s result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Applebaum, D.: Compound Poisson processes and Lévy processes in groups and symmetric spaces. J. Theo. Probab. 13, 383–425 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Applebaum, D.: Some L 2 properties of semigroups of measures on Lie groups. Semigroup Forum 79, 217–228 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dani, S.G., McCrudden, M.: Embeddability of infinitely divisible distributions on linear Lie group. Invent. Math. 110, 237–261 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dani, S.G., McCrudden, M.: Convolution roots and embedding of probability measures on Lie groups. Adv. Math. 209, 198–211 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dani, S.G., Guivarc’h, Y., Shah, R.: On the embeddability of certain infinitely divisible probability measures on Lie groups. (English summary). Math. Z. 272, 361–379 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gangolli, R.: Isotropic infinitely divisible measures on symmetric space. Acta Math. 111, 213–246 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  7. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press (1978)

  8. Heyer, H.: Probability Measures on Locally Compact Groups. Springer-Verlag (1977)

  9. Hunt, G.A.: Semigroups of measures on Lie groups. Trans. Am. Math. Soc. 81, 264–293 (1956)

    Article  MATH  Google Scholar 

  10. Liao, M.: Lévy processes in Lie groups. Cambridge University Press (2004)

  11. Liao, M., Wang, L.: Lévy-Khinchin formula and existence of densities for convolution semigroups on symmetric spaces. Potential Anal. 27, 133–150 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Valery, V., Volchkov, V., Volchkov, V.: Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group. Springer-Verlag (2009)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Ming Liao

Authors
  1. Ming Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ming Liao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liao, M. Convolution of Probability Measures on Lie Groups and Homogenous Spaces. Potential Anal 43, 707–715 (2015). https://doi.org/10.1007/s11118-015-9493-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-015-9493-2

Keywords

Mathematics Subject Classification (2010)

Search

Navigation