Probability Theory and Related Fields

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

Existence of quantum diffusions

  • M. P. Evans


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.


Differential Equation Stochastic Process Probability Theory Operator Norm Statistical Theory 
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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

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

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