Quantum stochastic convolution cocycles III

Every Markov-regular quantum Lévy 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 Lévy 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.