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Quantum stochastic cocycles provide a basic model for time-homogeneousMarkovian evolutions in a quantum setting, and a direct counterpart incontinuous time to quantum random walks, in both the Schrodinger and Heisenbergpictures. This paper is a sequel to one in which correspondences wereestablished between classes of quantum stochastic cocycle on an operator spaceor C*-algebra, and classes of Schur-action `global' semigroup on associatedmatrix spaces over the operator space. In this paper we investigate thestochastic generation of cocycles via the generation of their correspondingglobal semigroups, with the primary purpose of strengthening the scope ofapplicability of semigroup theory to the analysis and construction of quantumstochastic cocycles. An explicit description is given of the affinerelationship between the stochastic generator of a completely bounded cocycleand the generator of any one of its associated global semigroups. Using this,the structure of the stochastic generator of a completely positivequasicontractive quantum stochastic cocycle on a C*-algebra whose expectationsemigroup is norm continuous is derived, giving a comprehensive stochasticgeneralisation of the Christensen--Evans extension of the GKS&L theorem ofGorini, Kossakowski and Sudarshan, and Lindblad. The transformation alsoprovides a new existence theorem for cocycles with unbounded structure map asstochastic generator. The latter is applied to a model of interacting particlesknown as the quantum exclusion Markov process, in particular on integerlattices in dimensions one and two.

Quantum stochastic cocycles and completely bounded semigroups on operator spaces II