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Existence of quantum diffusions. (English) Zbl 0667.60060

A quantum diffusion \((A,A^ t,j)\) comprises of unital *-algebras A and \(A'\) and a family of identity preserving *-homomorphisms \(j=(j_ t:t\geq 0)\) from A into \(A'\). Also j satisfies a system of quantum stochastic differential equations \[ dj_ t(x_ 0)=j_ t(\mu^ i_ j(x_ 0))dM^ j_ i,\quad j_ 0(x_ 0)=x_ 0\otimes I \] for all \(x_ 0\in A\) where \(\mu^ i_ j\), \(1\leq i,j\leq N\) are maps from A to itself and are known as the structure maps.
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.
Reviewer: M.P.Evans

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
81P20 Stochastic mechanics (including stochastic electrodynamics)
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[1] Evans, M. P.; Hudson, R. L.; Accardi, L.; von Waldenfels, W., Multidimension diffusions, Quantum probability III. Proceedings. Oberwolfach 1987, 69-88 (1987), Berlin Heidelberg New York Tokyo: Springer, Berlin Heidelberg New York Tokyo
[2] Hudson, R. L.; Prohorov, Y.; Sazonov, V. V., Quantum diffusions and cohomology of algebras, Proceedings of 1st World Congress of Bernoulli Society, vol. 1, 479-485 (1987), Utrecht: VNU Science Press, Utrecht · Zbl 0692.46068
[3] Hudson, R. L.; Parthasarathy, K. R., Quantum Ito’s formula and stochastic evolutions, Comm. Math. Phys., 93, 301-323 (1984) · Zbl 0546.60058
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