On the Hausdorff continuity of free Lèvy processes and free convolution semigroups

Let μ denote a Borel probability measure and let {μt}t≥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.