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

Abstract

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 (P.~Biane, \textit{Math. Z.} {\bf 227}(1998), 143--174). 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 characterized by Bercovici and Voiculescu, \textit{Pacific J. Math.} {\bf 153}(1992), 217--248. I give an explicit parametrization for the Nevanlinna functions associated with the second kind of free L\'evy processes. This gives a nonlinear free L\'evy--Khinchine formula.