Nonlinear free L\'evy-Khinchine formula and conformal mapping

There are two natural notions of L\'evy processes in free probability: the first one has free increments with homogeneous distributions and the other has homogeneous transition probabilities. In the two cases one can associate a Nevanlinna function to a free L\'evy process. The Nevanlinna functions appearing in the first notion were characterised by Bercovici and Voiculescu. I give an explicit parametrisation for the Nevanlinna functions associated with the second kind of free L\'evy processes. This gives a nonlinear free L\'evy-Khinchine formula.