Regularity Results for Multiplicative Free Lévy Processes

Abstract We find the necessary and sufficient conditions for a free convolution $\mu \boxtimes \nu $ to have an analytic and positive density, where $\mu $ is $\boxtimes $-infinitely divisible. If the density of $\mu \boxtimes \nu $ vanishes at some points, we also study the analyticity of this density at such zeros. Finally, we show that $\mu \boxtimes \nu $ has a finite number of connected components in its support if both the free Lévy measure of $\mu $ and the measure $\nu $ do. These results are applied to multiplicative free Lévy processes on the positive line and the unit circle.