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

## TL;DR

This paper introduces a nonlinear free Le9vy-Khinchine formula by explicitly parametrizing Nevanlinna functions linked to a second type of free Le9vy processes, expanding the theoretical framework of free probability.

## Contribution

It provides an explicit parametrization of Nevanlinna functions for the second kind of free Le9vy processes, leading to a nonlinear free Le9vy-Khinchine formula.

## Key findings

- Explicit parametrization of Nevanlinna functions for second kind free Le9vy processes
- Derivation of a nonlinear free Le9vy-Khinchine formula
- Enhanced understanding of free probability processes

## 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.