  A L\'evy process on a *-bialgebra is given by its generator, a conditionally
positive hermitian linear functional vanishing at the unit element. A *-algebra
homomorphism k from a *-bialgebra C to a *-bialgebra B with the property that k
respects the counits maps generators on B to generators on C. A tranformation
between the corrresponding two L\'evy processes is given by forming
infinitesimal convolution products.
  This general result is applied to various situations, e.g., to a *-bialgebra
and its associated primitive tensor *-bialgebra (called "generator process") as
well as its associated group-like *-bialgebra (called Weyl-*-bialgebra). It
follows that a L\'evy process on a *-bialgebra can be realized on Bose Fock
space as the infinitesimal convolution product of its generator process such
that the vacuum vector is cyclic for the L\e'vy process. Moreover, we obtain
convolution approximations of the Az\'ema martingale by the Wiener process and
vice versa.
