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

Philippe Biane

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arXiv · 1908.00755

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

Philippe Biane

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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. 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.

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Philippe Biane. 2019-08-02. Nonlinear free L\'evy-Khinchine formula and conformal mapping. https://arxiv.org/abs/1908.00755

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