Regularity results for free Lévy processes

Given a free additive convolution semigroup (?t)t?0 and probability measure ? on R, we find the necessary sufficient conditions for process ?t?? to be Lebesgue absolutely continuous with positive analytic density throughout R at all time t>0. For semigroups without this property, of its zeros. These results are quantified by Lévy semigroup, making it fairly easy construct many concrete examples. Finally, show that has finite number connected components in support if both initial law do.