Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Quantum stochastic differential equations of the form dk_t = k_t ○ θ_β^α dΛ_α^β(t) govern stochastic flows on a C*-algebra script A sign. 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 script A sign. 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 script A sign 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: dX_t = F_β^αX_t dΛ_α^β(t), dY_t = Y_tF_β^α dΛ_α^β(t) , 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.