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




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


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© Springer Basel 2015

Authors and Affiliations

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

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