Regularity results for free Lévy processes

Abstract

Given a free additive convolution semigroup ${\left({\mu }_t\right)}_{t\ge 0}$ and a probability measure $\nu$ on $\mathbb{R}$, we find the necessary and sufficient conditions for the process ${\mu }_t⊞\nu$ 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 ${\mu }_t⊞\nu$ to be analytic at its zeros. These results are quantified by the Lévy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that ${\mu }_t⊞\nu$ has a finite number of connected components in its support if both the Lévy measure of ${\left({\mu }_t\right)}_{t\ge 0}$ and the initial law $\nu$ do.

Accepted by Advances in Mathematics