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

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Lindsay, J. Martin
Wills, Stephen J.
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Quantum stochastic differential equations of the form govern stochastic flows on a C *-algebra ?. 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 ?. 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 ? 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.
Quantum stochastic , Completely positive , Completely bounded , Stochastic flows , Quantum Markov semigroup , Quantum diffusion
Lindsay, J.M. and Wills, S.J. (2000) ‘Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise’, Probability Theory and Related Fields, 116(4), pp. 505–543. https://doi.org/10.1007/s004400050261
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© Springer-Verlag 2000. This version of the article has been accepted for publication, after peer review and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1007/s004400050261