Unimodality for free L\'evy processes

We will prove that: (1) A symmetric free L\'evy process is unimodal if and only if its free L\'evy measure is unimodal; (2) Every free L\'evy process with boundedly supported L\'evy measure is unimodal in sufficiently large time. (2) is completely different property from classical L\'evy processes. On the other hand, we find a free L\'evy process such that its marginal distribution is not unimodal for any time $s>0$ and its free L\'evy measure does not have a bounded support. Therefore, we conclude that the boundedness of the support of free L\'evy 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 L\'evy process in terms of L\'evy--Khintchine representation.