Continuity and growth of free multiplicative convolution semigroups

Abstract

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

11 pages