Advertisement

Probability Theory and Related Fields

, Volume 81, Issue 4, pp 473–483 | Cite as

Existence of quantum diffusions

  • M. P. Evans
Article

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 (x0=jt(μ j i (x0))dM i i , j0(x0)=x0I for all x0A where μ j i , 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.

Keywords

Differential Equation Stochastic Process Probability Theory Operator Norm Statistical Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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 1987Google Scholar
  2. 2.
    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 1987Google Scholar
  3. 3.
    Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolutions. Comm. Math. Phys. 93, 301–323 (1984)Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • M. P. Evans
    • 1
  1. 1.Mathematics DepartmentNottingham UniversityNottinghamUK

Personalised recommendations