Regularity results for free L\'{e}vy processes

Abstract

Given a free additive convolution semigroup $\left(\mu_t\right)_{t\geq 0}$ and a probability measure $\nu$ on $\mathbb{R}$, we find the necessary and sufficient conditions for the process $\mu_t \boxplus \nu$ to be Lebesgue absolutely continuous with a positive and analytic density throughout $\mathbb{R}$ at all time $t>0$. For semigroups without this property, we find the necessary and sufficient conditions for the density of $\mu_t \boxplus \nu$ to be analytic at its zeros. These results are quantified by the L\'{e}vy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that $\mu_t \boxplus \nu$ has a finite number of connected components in its support if both the L\'{e}vy measure of $\left(\mu_t\right)_{t \geq 0}$ and the initial law $\nu$ do.