Summary
A quantum diffusion (A, A′, j) comprises of unital *-algebras A and A′ and a family of identity preserving *-homomorphisms j=(j t : t≧0) from A into A′. Also j satisfies a system of quantum stochastic differential equations dj t (x 0=j t(μ i j (x 0))dM i i , j 0(x 0)=x 0⊗I for all x 0∈A where μ i j , 1≦i, j≦N are maps from A to itself and are known as the structure maps. In this paper an existence proof is given for a class of quantum diffusions, for which the structure maps are bounded in the operator norm sense. A solution to the system of quantum stochastic differential equations is first produced using a variation of the Picard iteration method. Another application of this method shows that the solution is a quantum diffusion.
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Evans, M.P., Hudson, R.L.: Multidimension diffusions. In: Accardi, L., Waldenfels, W. von (eds.) Quantum probability III. Proceedings. Oberwolfach 1987. (Lect. Notes Math., vol. 1303, pp. 69–88). Berlin Heidelberg New York Tokyo: Springer 1987
Hudson, R.L.: Quantum diffusions and cohomology of algebras. In: Prohorov, Y., Sazonov, V.V. (eds.) Proceedings of 1st World Congress of Bernoulli Society, vol. 1, pp. 479–485. Utrecht: VNU Science Press 1987
Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolutions. Comm. Math. Phys. 93, 301–323 (1984)
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Evans, M.P. Existence of quantum diffusions. Probab. Th. Rel. Fields 81, 473–483 (1989). https://doi.org/10.1007/BF00367298
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DOI: https://doi.org/10.1007/BF00367298