Continuity and growth of free multiplicative convolution semigroups

TL;DR

This paper studies how the support of free multiplicative convolution semigroups of probability measures on the positive half-line changes continuously with the parameter, and determines the asymptotic length of their support.

Contribution

It establishes the continuous variation of the support of free multiplicative convolution semigroups and calculates the asymptotic support length.

Findings

Support varies continuously with t

Asymptotic length of the support is determined

Provides insights into the structure of free convolution semigroups

Abstract

Let $\mu$ be a compactly supported probability measure on the positive half-line and let $\mu^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $\mu^{\boxtimes t}$ varies continuously as $t$ changes. We also obtain the asymptotic length of the support of these measures.