  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.
