Complex Analysis and Operator Theory

, Volume 10, Issue 3, pp 581–603 | Cite as

Free Subordination and Belinschi–Nica Semigroup



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.


Free subordination Boolean stable law Monotone stable law Markov–Krein transform 

Mathematics Subject Classification



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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Probability and StatisticsCIMATGuanajuatoMexico
  2. 2.Department of MathematicsHokkaido UniversitySapporoJapan

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