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.Comment: 21 pages, minor corrections, Complex Analysis and Operator Theory,
First online: 16 October 201
