Free Subordination and Belinschi-Nica Semigroup

TL;DR

This paper introduces a new realization of the Belinschi-Nica semigroup as free multiplicative subordination, extending the framework to more general semigroups and establishing related differential equations and connections to various stable laws.

Contribution

It provides a novel realization of the Belinschi-Nica semigroup as free multiplicative subordination and extends the concept to broader semigroups with associated differential equations.

Findings

Realization of the Belinschi-Nica semigroup as free multiplicative subordination

Derivation of a generalized complex Burgers equation for these semigroups

Identification of relations to Markov-Krein transform, Boolean, and monotone stable laws

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.