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
Contents
Full-Text PDF