Evans, M. P. Existence of quantum diffusions. (English) Zbl 0667.60060 Probab. Theory Relat. Fields 81, No. 4, 473-483 (1989). 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 Cited in 2 ReviewsCited in 23 Documents 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) Keywords:quantum stochastic differential equations; quantum diffusions PDFBibTeX XMLCite \textit{M. P. Evans}, Probab. Theory Relat. Fields 81, No. 4, 473--483 (1989; Zbl 0667.60060) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.