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

Let mu denote a Borel probability measure and let {mu(t)}(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. (C) 2016 Published by Elsevier Inc.