Unimodality for free Levy processes

We will prove that: (1) A symmetric free Levy process is unimodal if and only if its free Levy measure is unimodal; (2) Every free Levy process with boundedly supported Levy measure is unimodal in sufficiently large time. (2) is completely different property from classical Levy processes. On the other hand, we find a free Levy process such that its marginal distribution is not unimodal for any time $s>0$ and its free Levy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free Levy measure in (2) cannot be dropped. For the proof we will (almost) characterize the existence of atoms and the existence of continuous probability densities of marginal distributions of a free Levy process in terms of Levy–Khintchine representation.