On the Hausdorff Continuity of Free L\`evy Processes and Free Convolution Semigroups

TL;DR

This paper proves that the support of free additive convolution semigroups varies continuously in the Hausdorff metric for all t > 1, using complex analysis and measure characterization techniques.

Contribution

It establishes Hausdorff continuity of free convolution semigroup supports, extending understanding of free probability dynamics.

Findings

Support varies continuously in Hausdorff metric for t > 1

Utilizes complex analytic methods and Huang's measure characterization

Provides a rigorous mathematical framework for free convolution semigroup support behavior

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.