On the Hausdorff Continuity of Free Lèvy Processes and Free Convolution Semigroups

Abstract: Let $\mu$ denote a Borel probability measure and let ${ \mu_{t} }_{t\geq 1}$ denote the free additive convolution semigroup of Nica and Speicher. We show that the support of these measures varies continuously in the Hausdorff metric for $t >1$. We utilize complex analytic methods and, in particular, a characterization of the absolutely continuous portion of these supports due to Huang.