Markovian Cocycles on Operator Algebras Adapted to a Fock Filtration

The introduction of a Feller-type condition allows the study of Markovian, cocycles adapted to a Fock filtration to be extended from von Neumann algebras to C*-algebras. It is shown that every such cocycle which, along with its adjoint cocycle, is pointwise strongly continuous and whose associated semigroups are norm continuous, weakly satisfies a quantum stochastic differential equation (QSDE). The matrix of coefficients of this equation may thereby be considered as the generator of the cocycle. The QSDE is satisfied strongly in any of the following cases: when the cocycle is completely positive and contractive, or the driving quantum noise is finite dimensional, or the C*-algebra is finite dimensional and the cocycle generator is bounded. Applying the algebra results to Fock-adapted Markovian cocycles on a Hilbert space we obtain similar characterisations. In particular a contraction cocycle whose Markov semigroup is norm continuous strongly satisfies a QSDE. A representation of cocycles in terms of a family of associated semigroups is central to the present analysis, providing the connection with QSDEs through a parallel work (2000, J. M. Lindsay and S. J. Wills, Probab. Theory Related Fields116, 505–543).