Unitary cocycles and processes on the full Fock space

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.