Schurmann'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
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.