Abstract.
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.
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Received: 3 November 1998 / Published online: 30 March 2000
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Lindsay, J., Wills, S. Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise. Probab Theory Relat Fields 116, 505–543 (2000). https://doi.org/10.1007/s004400050261
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DOI: https://doi.org/10.1007/s004400050261
Keywords
- Weak Solution
- Stochastic Differential Equation
- Strong Solution
- Bounded Linear Operator
- Stochastic Integrator