Advertisement

On multi-dimensional markovian cocycles

  • L. Accardi
  • J. L. Journé
  • J. M. Lindsay
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1396)

Abstract

Under a weak differentiability condition quantum Markov cocycles on Fock space satisfy quantum stochasticdifferential equations of the form Open image in new window where {Fαβ} is a matrix of operators with common dense domain, Λαβ are the basic martingles of Hudson-Parthasarathy calculus in n-dimensions and Λoo is time.

Keywords

Springer Lecture Note Wiener Space Quantum Brownian Motion Quantum Stochastic Calculus Quantum Stochastic Differential Equation 
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. [Fri]
    A. Frigerid: Positive contraction semigroups on B(H) and quantum stochastic differential equations. Preprint Udine, Italy 1987.Google Scholar
  2. [Gui]
    A. Guichardet: Symmetric Hilbert spaces and related topics. Lecture Notes in Mathematics 261, Springer 1972.Google Scholar
  3. [HuL]
    R.L. Hudson, J.M. Lindsay: On characterising quantum stochastic evolutions. Math. Proc. Camb. Phil. Soc. 102 (1987), 363–369.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [HuP1]
    R.L. Hudson, K.R. Parthasarathy: Quantum Itô's formula and stochastic evolutions Commun. Math. Phys. 93, (1984) 301–323.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [HuP2]
    R.L. Hudson, K.R. Parthasarathy: Stochastic dilations of uniformly continuous completely positive semigroups. Acta. Appl. Math. 2 (1984), 353–398.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [Jou]
    J.L. Journé: Structure des cocycles markoviens sur l'espace de Fock. Prob. Th. Rel. Fields. 75 (1987), 291–316.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [LiP]
    J.M. Lindsay, K.R. Parthasarathy: Cohomology of power sets with applications. Preprint, New Delhi, 1988.Google Scholar
  8. [Maa]
    H. Maassen: Quantum markov processes on Fock space described by integral kernels in Quantum Probability and Applications II. ed. L. Accardi and W. von Waldenfels. Springer Lecture Notes in Mathematics 1136 (1985) 361–374.Google Scholar
  9. [Mey]
    P.A.Meyer: Eléments de probabilités quantiques. in Sem. Prob. XX 1984/85 Springer Lecture Notes in Mathematics 1204 (1986) 186–312. Also: Calcul stochastique non-commutatif. Sem.Bourbaki, No. 672, 39 ème année 1986–87.Google Scholar
  10. [Par]
    K.R. Parthasarathy: Quantum stochastic calculus. in Stochastic Processes and their Applications. Proceedings, Nagoya 1985. Ed. K.Itô and T.Hida.Springer Lecture Notes in Mathematics 1203 (1986) 177–196.Google Scholar
  11. [PaS]
    K.R. Parthasarathy, K.B. Sinha: Stochastic integral representation of bounded quantum martingales in Fock space. J.Funct. Anal. 67 (1986) 126–151.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • L. Accardi
    • 1
  • J. L. Journé
    • 2
  • J. M. Lindsay
    • 3
  1. 1.Dipartimento di MatematicaII Università degli Studi di Roma(La Romanina), RomaItaly
  2. 2.Department of MathematicsPrinceton UniversityPrincetonU.S.A.
  3. 3.Department of MathematicsKings CollegeLondonU.K.

Personalised recommendations