Unitary cocycles and processes on the full Fock space

TL;DR

This paper explores how free Lévy processes on non-commutative unitary groups can be constructed via infinitesimal convolution of additive free white noise, advancing understanding of non-commutative stochastic processes.

Contribution

It introduces a method to derive free Lévy processes on non-commutative unitary groups using infinitesimal convolution of free white noise, connecting cocycles, Schürmann triples, and Lévy processes.

Findings

Free Lévy processes can be constructed from additive free white noise.

A Schürmann triple characterizes the Lévy process on the non-commutative unitary group.

The approach links infinitesimal convolution with free probability structures.

Abstract

We consider a unitary cocycle or Sch\"urmann triple on the non-commutative unitary group fixed by a complex matrix which induces an additive free white noise or an additive free L\'evy process on the tensor algebra over the full Fock space. A L\'evy process on a Voiculescu dual semi-group is given by a generator or Sch\"urmann triple. We will show how a free L\'evy process on the non-commutative unitary group fixed by a complex matrix can be obtained by infinitesimally convolving the additive free white noise.