  Quantum stochastic cocycles provide a basic model for time-homogeneous
Markovian evolutions in a quantum setting, and a direct counterpart in
continuous time to quantum random walks, in both the Schrodinger and Heisenberg
pictures. This paper is a sequel to one in which correspondences were
established between classes of quantum stochastic cocycle on an operator space
or C*-algebra, and classes of Schur-action `global' semigroup on associated
matrix spaces over the operator space. In this paper we investigate the
stochastic generation of cocycles via the generation of their corresponding
global semigroups, with the primary purpose of strengthening the scope of
applicability of semigroup theory to the analysis and construction of quantum
stochastic cocycles. An explicit description is given of the affine
relationship between the stochastic generator of a completely bounded cocycle
and the generator of any one of its associated global semigroups. Using this,
the structure of the stochastic generator of a completely positive
quasicontractive quantum stochastic cocycle on a C*-algebra whose expectation
semigroup is norm continuous is derived, giving a comprehensive stochastic
generalisation of the Christensen--Evans extension of the GKS&L theorem of
Gorini, Kossakowski and Sudarshan, and Lindblad. The transformation also
provides a new existence theorem for cocycles with unbounded structure map as
stochastic generator. The latter is applied to a model of interacting particles
known as the quantum exclusion Markov process, in particular on integer
lattices in dimensions one and two.
