Skip to main content

Quantum stochastic calculus

  • Conference paper
  • First Online:
Stochastic Processes and Their Applications

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1203))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Accardi, L., Parthasarathy, K. R.: Stochastic calculus on local algebras. In: Quantum Probability and Applications, Accardi and von Waldenfels (ed.) (to appear).

    Google Scholar 

  2. Applebaum, D.B., Hudson, R.L.: Fermion Ito's formula and stochastic evolutions, Commun. Math. Phys. 96, 473–496 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  3. Applebaum, D.B., Quasifree stochastic evolutions, In: Quantum Probability and Applications, Accardi and von Waldenfels (ed.) Lecture Notes in Mathematics, Berlin, Heidelberg, New York, Tokyo, Springer (to appear).

    Google Scholar 

  4. Araki, H.: Factorisable representations of current algebra, Publications of RIMS, Kyoto University, Ser. A 5, 361–422 (1970).

    Article  MathSciNet  MATH  Google Scholar 

  5. Barchielli, A., Lupieri, G: Quantum stochastic calculus, Operation valued stochastic processes and continual measurements in quantum theory, preprint, University of Milan.

    Google Scholar 

  6. Barnett, C., Streater, R.F., Wilde, I.F.: The Ito-Clifford integral I-IV, J. Functional Analysis, 48, 172–212 (1982), J. London Math. Soc., 27, 373–384 (1983), Commun. Math. Phys., 89, 13–17 (1983), J. Operator Theory, 11, 255–271 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  7. Barnett, C., Streater, R.F., Wilde, I.F.: Stochastic integrals in an arbitrary probability gage space, Math. Proc. Cambridge Phil. Soc., 94, 541–551 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  8. Frigerio, A.: Diffusion processes, quantum stochastic differential equations and the classical KMS condition, J.Math. Phys. 25, 1050–65 (1984).

    Article  MathSciNet  Google Scholar 

  9. Frigerio, A., Gorini, V.: Markov dilations and quantum detailed balance, Commun. Math. Phys. 93, 517–32 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  10. Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of n-level system, J.Math. Phys. 17, 821–5 (1976).

    Article  MathSciNet  Google Scholar 

  11. Hudson, R.L., Lindsay, J.M.: Stochastic integration and a martingale representation theorem for non-Fock quantum Brownian motion, J. Functional Anal. 61, 202–221 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  12. Hudson, R.L., Karandikar, R.L., Parthasarathy, K.R.: Towards a theory of noncommutative semimartingales adapted to Brownian motion and a quantum Ito's formula; and Hudson, R.L., Parthasarathy, K.R.: Quantum diffusions. In: Theory and Applications of Random Fields, Kallianpur, (ed.) Lecture Notes in Control Theory and Information Sciences 49, Berlin, Heidelberg, New York, Tokyo: Springer 1983.

    Google Scholar 

  13. Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolutions, Commun. Math. Phys. 93, 301–323 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  14. Hudson, R.L., Parthasarathy, K.R.: Stochastic dilations of uniformly continuous completely positive semigroups, Acta Appl. Math. 2, 353–398 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  15. Hudson, R.L., Parthasarathy, K.R.: Generalised Weyl operators, In: Stochastic Analysis and Applications, Truman and Williams (ed.) Lecture Notes in Mathematics 1095, Berlin, Heidelberg, New York, Tokyo: Springer (1984).

    Chapter  Google Scholar 

  16. Hudson, R.L., Parthasarathy, K.R.: Construction of quantum diffusions, In: Quantum Probability and Applications, Accardi (ed.) Lecture Notes in Mathematics 1055, Berlin, Heidelberg, New York, Tokyo: Springer (1984).

    Google Scholar 

  17. Hudson, R.L., Parthasarathy, K.R.: Unification of Fermion and Boson stochastic calculus, Submitted to Commun. Math. Phys.

    Google Scholar 

  18. Hudson, R.L., Lindsay, J.M., Parthasarathy, K.R.: Stochastic integral representations of some martingales in Fock space. Preprint, Nottingham University.

    Google Scholar 

  19. Lindblad, G.: On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119–30 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  20. Parthasarathy, K.R., Schmidt, K.: Positive Definite Kernels, Continuous Tensor Products and Central Limit Theorems of Probability Theory, Lecture Notes in Mathematics 272, Berlin, Heidelberg, New York: Springer (1972).

    MATH  Google Scholar 

  21. Parthasarathy, K.R.: A remark on the integration of Schrödinger equation using quantum Ito's formula, Lett. Math. Phys. 8, 227–232 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  22. Parthasarathy, K.R.: Some remarks on the integration of Schrödinger equation using the quantum stochastic calculus, In: Quantum Probability and Applications, Accardi and von Waldenfels (ed.) (to appear).

    Google Scholar 

  23. Parthasarathy, K.R.: Boson stochastic calculus, Pramana (to appear).

    Google Scholar 

  24. Parthasarathy, K.R.: One parameter semigroups of completely positive maps on groups arising from quantum stochastic differential equations, Bolletino Mat. Ital. (to appear).

    Google Scholar 

  25. Segal, I.E.: Tensor algebras over Hilbert spaces, Ann. Math. 63, 160–175 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  26. Streater, R.F.: Current commutation relations, continuous tensor products and infinitely divisible group representations, Rendiconti Sc. Int. Fisica. Fermi, Vol. XI, 247–269 (1969).

    Google Scholar 

  27. von Waldenfels, W.: Ito solution of the linear quantum stochastic differential equation describing light emission and absorption, In: Quantum Probability and Applications, Accardi (ed.), Lecture Notes in Mathematics 1055, Berlin, Heidelberg, New York, Tokyo: Springer (1984).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Kiyosi Itô Takeyuki Hida

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Parthasarathy, K.R. (1986). Quantum stochastic calculus. In: Itô, K., Hida, T. (eds) Stochastic Processes and Their Applications. Lecture Notes in Mathematics, vol 1203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076881

Download citation

  • DOI: https://doi.org/10.1007/BFb0076881

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16773-0

  • Online ISBN: 978-3-540-39852-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics