Abstract. Quantum stochastic differential equations of the form dkt = kt • θ α β dΛ β α (t) govern stochastic flows on a C * -algebra A. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite dimensional, and the coefficient matrix θ consists of bounded linear operators on A. Weak and strong forms of solution are distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when A is a von Neumann algebra and the components of θ are normal, these in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and * -homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy equations:, in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes. The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown to have bounded generators, and to be necessarily strong solutions. IntroductionThe quantum stochastic differential equation (QSDE)

The quantum stochastic differential equation dkt=kt^θrβαdΛαrβ(t) is considered on a unital C*‐algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non‐commutative topological space. 2000 Mathematics Subject Classification 81S25, 46L07, 46L53, 47D06.

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Abstract. When a Fock-adapted Feller cocycle on a C * -algebra is regular, completely positive and contractive it possesses a stochastic generator that is necessarily completely bounded. Here necessary and sufficient conditions are given, in the form of a sequence of identities, for a completely bounded map to generate a weakly multiplicative cocycle. These are derived from a product formula for iterated quantum stochastic integrals. Under two alternative assumptions, one of which covers all previously considered cases, the first identity in the sequence is shown to imply the rest.