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arXiv · 1408.5983

Free Subordination and Belinschi-Nica Semigroup

Octavio Arizmendi·Takahiro Hasebe
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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.

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Octavio Arizmendi, Takahiro Hasebe. 2015-10-19. Free Subordination and Belinschi-Nica Semigroup. https://doi.org/10.1007/s11785-015-0500-9

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