Journal of Operator Theory

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

Nonlinear free Lévy-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-Vallée cedex 2, France

Summary: There are two natural notions of Lévy 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 Lévy 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 Lévy processes. This gives a nonlinear free Lévy-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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