  Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is
shown to be equivalent to one governed by a quantum stochastic differential
equation, and the generating functionals of norm-continuous convolution
semigroups on a multiplier C*-bialgebra are then completely characterised.
These results are achieved by extending the theory of quantum Levy processes on
a compact quantum group, and more generally quantum stochastic convolution
cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier
C*-bialgebras. Strict extension results obtained by Kustermans, together with
automatic strictness properties developed here, are exploited to obtain
existence and uniqueness for coalgebraic quantum stochastic differential
equations in this setting. Then, working in the universal enveloping von
Neumann bialgebra, we characterise the stochastic generators of Markov-regular,
*-homomorphic (respectively completely positive and contractive), quantum
stochastic convolution cocycles.
