SpringerLink
Log in

Free Subordination and Belinschi–Nica Semigroup

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We realize the Belinschi–Nica semigroup of homomorphisms as a free multiplicative subordination. This realization allows to define more general semigroups of homomorphisms with respect to free multiplicative convolution. For these semigroups we show that a differential equation holds, generalizing the complex Burgers equation. We give examples of free multiplicative subordination and find a relation to the Markov–Krein transform, Boolean stable laws and monotone stable laws. A similar idea works for additive subordination, and in particular we study the free additive subordination associated to the Cauchy distribution and show that it is a homomorphism with respect to monotone, Boolean and free additive convolutions.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Notes

  1. The parametrization is different from that in the original paper [1], and the dilation is omitted here.

References

  1. Arizmendi, O., Hasebe, T.: On a class of explicit Cauchy-Stieltjes transforms related to monotone stable and free Poisson laws. Bernoulli 19(5B), 2750–2767 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arizmendi, O., Hasebe, T.: Semigroups related to additive and multiplicative, free and Boolean convolutions. Stud. Math. 215(2), 157–185 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arizmendi, O., Hasebe, T.: Classical scale mixtures of Boolean stable laws. Trans. Am. Math. Soc. ar**v:1405.2162 (to appear)

  4. Arizmendi, O., Pérez-Abreu, V.: The S-transform for symmetric probability measures with unbounded supports. Proc. Am. Math. Soc. 137, 3057–3066 (2009)

    Article  MATH  Google Scholar 

  5. Belinschi, S.T., Bercovici, H.: Partially defined semigroups relative to multiplicative free convolution. Int. Math. Res. Not. 2, 65–101 (2005)

    Article  MathSciNet  Google Scholar 

  6. Belinschi, S., Bercovici, H.: A new approach to subordination results in free probability. J. d’Anal. Math. 101, 357–365 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Belinschi, S.T., Nica, A.: On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution. Indiana Univ. Math. J. 57(4), 1679–1713 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bercovici, H.: Multiplicative monotonic convolution. Illinois J. Math. 49(3), 929–951 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Bercovici, H., Voiculescu, D.: Free convolution of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bertoin, J., Fujita, T., Roynette, B., Yor, M.: On a particular class of self-decomposable random variables: the duration of a Bessel excursion straddling an independent exponential time. Prob. Math. Stat. 26, 315–366 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Biane, P.: Processes with free increments. Math. Z. 227(1), 143–174 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Franz, U.: Monotone and boolean convolutions for non-compactly supported probability measures. Indiana Univ. Math. J. 58(3), 1151–1186 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Franz, U.: Boolean convolution of probability measures on the unit circle. Anal. Prob. Sémin. Congr. 16, 83–93 (2009)

    MathSciNet  Google Scholar 

  14. Hasebe, T.: Free infinite divisibility for beta distributions and related ones. Electron. J. Probab. 19(81), 1–33 (2014)

    MathSciNet  Google Scholar 

  15. Hasebe, T.: Monotone convolution and monotone infinite divisibility from complex analytic viewpoint. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13(1), 111–131 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kerov, S.: Interlacing measures. Am. Math. Soc. Transl. Ser. 2, 181, Am. Math. Soc. Providence, RI, pp. 35–83 (1998)

  17. Lenczewski, R.: Decompositions of the free additive convolution. J. Funct. Anal. 246, 330–365 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lenczewski, R.: Operators related to subordination for free multiplicative convolutions. Indiana Univ. Math. J. 57, 1055–1103 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106, 409–438 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Muraki, N.: Monotonic convolution and monotonic Lévy–Hinčin formula. (2000) (preprint)

  21. Raj Rao, N., Speicher, R.: Multiplication of free random variables and the S-transform: The case of vanishing mean. Elect. Commun. Probab. 12, 248–258 (2007)

    MATH  Google Scholar 

  22. Speicher, R., Woroudi, R.: Boolean convolution. In: Voiculescu, D. (ed.) Free Probability Theory. Fields Inst. Comm. 12, pp. 267–280. Amer. Math. Soc (1997)

  23. Voiculescu, D.: Multiplication of certain noncommuting random variables. J. Oper. Theory 18, 223–235 (1987)

    MathSciNet  MATH  Google Scholar 

  24. Voiculescu, D.: The analogues of entropy and of Fisher’s information measure in free probability theory I. Commun. Math. Phys. 155, 71–92 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Octavio Arizmendi was supported by CONACYT Grant 222668. Takahiro Hasebe was supported by European Commission, Marie Curie Actions—International Incoming Fellowships (Project 328112 ICNCP) at University of Franche-Comté; supported also by JSPS, Global COE program “Fostering top leaders in mathematics—broadening the core and exploring new ground” at Kyoto university

Author information

Authors and Affiliations

  1. Department of Probability and Statistics, CIMAT, Guanajuato, Mexico

    Octavio Arizmendi

  2. Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kitaku, Sapporo, 060-0810, Japan

    Takahiro Hasebe

Authors
  1. Octavio Arizmendi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takahiro Hasebe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takahiro Hasebe.

Additional information

Communicated by Yuri Kondratiev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arizmendi, O., Hasebe, T. Free Subordination and Belinschi–Nica Semigroup. Complex Anal. Oper. Theory 10, 581–603 (2016). https://doi.org/10.1007/s11785-015-0500-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-015-0500-9

Keywords

Mathematics Subject Classification

Search

Navigation