  Schuermann's theory of quantum Levy processes, and more generally the theory
of quantum stochastic convolution cocycles, is extended to the topological
context of compact quantum groups and operator space coalgebras. Quantum
stochastic convolution cocycles on a C*-hyperbialgebra, which are
Markov-regular, completely positive and contractive, are shown to satisfy
coalgebraic quantum stochastic differential equations with completely bounded
coefficients, and the structure of their stochastic generators is obtained.
Automatic complete boundedness of a class of derivations is established,
leading to a characterisation of the stochastic generators of *-homomorphic
convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum
Levy process on a compact quantum group are given and, with respect to both of
these, it is shown that an equivalent process on Fock space may be
reconstructed from the generator of the quantum Levy process. In the examples
presented, connection to the algebraic theory is emphasised by a focus on full
compact quantum groups.
