# Journal of Operator Theory

Volume 85, Issue 1, Winter 2021 pp. 79-99.

Nonlinear free L\'evy--Khinchine formula and conformal mapping**Authors**: Philippe Biane

**Author institution:**CNRS, Institut Gaspard-Monge, Univ. Gustave Eiffel, 5 Boulevard Descartes, Champs-sur-Marne, 77454, Marne-la-Vall\'ee cedex 2, France

**Summary:**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.

**DOI:**http://dx.doi.org/10.7900/jot.2019aug02.2267

**Keywords:**free probability, Nevanlinna functions, L\'evy--Khinchine formula

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