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

Abstract

Let $\mu$ denote a Borel probability measure and let $\{{\mu }_t{\}}_{t\ge 1}$ 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.

Small revisions at the referees behest. To be published in the Journal of Mathematical Analysis and its Applications