Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
On dissipative stochastic equations in a Hilbert space
Download PDF
Download PDF
  • Published: December 1996

On dissipative stochastic equations in a Hilbert space

  • A. S. Holevo1 

Probability Theory and Related Fields volume 104, pages 483–500 (1996)Cite this article

  • 214 Accesses

  • 43 Citations

  • Metrics details

Summary

A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Barchielli, A., Holevo, A.S.: Constructing quantum processes via classical stochastic calculus. Stoch. Proc. Appl.58, 293–318 (1995)

    Google Scholar 

  2. Belavkin, V.P.: A new wave equation for a continuous nondemalition measurement. Phys. Lett. A140, 355–358 (1989)

    Google Scholar 

  3. Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics I. Berlin: Springer 1981

    Google Scholar 

  4. Chebotarev, A.M., Fagnola, F.: Sufficient conditions for conservativity of dynamical semigroup. J. Funct. Anal.118, 131–153 (1993)

    Google Scholar 

  5. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Cambridge: Cambridge University Press 1992

    Google Scholar 

  6. Davies, E.B.: Quantum dynamical semigroups and the neutron diffusion equation. Rep. Math. Phys.11, 169–188 (1977)

    Google Scholar 

  7. Diosi, D.: Continuous quantum measurement and Ito formalism. Phys. Lett. A129, 419–423 (1988)

    Google Scholar 

  8. Emery, M.: Stabilité des solutions des é quations différentielles stochastiques: application aux intégrales multiplicatives stochastiques. Z. Wahr. verw. Geb.47, 241–334 (1987)

    Google Scholar 

  9. Fagnola, F.: Characterization of isometric and unitary weakly differentiable cocylces in Fock space. Quant. Probab. Rel. TopicsVIII, 143–164 (1993)

    Google Scholar 

  10. Feller, W.: An introduction to probability theory, vol. II, Chap. XIV. New York: Wiley 1966

    Google Scholar 

  11. Gatarek, D., Gisin, N.: Continuous quantum jumps and infinite-dimensional stochastic equations. J. Math. Phys.32, 2152–2157 (1991)

    Google Scholar 

  12. Holevo, A.S.: A remark on classical stochastic representations of quantum dynamical semigroups and on nonlinear wave equations. Proc. Russian-Finnish Conf. on Probability Theory, Steklov Mathematical Institute, 1993

  13. Holevo, A.S.: Time-ordered exponentials in quantum stochastic calculus. Quant. Probab. Rel. TopicsVII, 175–202 (1992)

    Google Scholar 

  14. Holevo, A.S.: On the structure of covariant dynamical semigroups. J. Funct. Anal.131, 255–278 (1995)

    Google Scholar 

  15. Journé, J.-L.: Structure des cocycles markoviennes sur l'espace de Fock. Probab. Theory Related Fields75, 291–316 (1987)

    Google Scholar 

  16. Kato, T.: Perturbation theory for linear operators, 2nd edition. Berlin: Springer 1976

    Google Scholar 

  17. Liptser, R.Sh., Shiryayev, A.N.: Theory of martingales. Dodrecht: Kluwer 1986

    Google Scholar 

  18. Meyer, P.-A.: Quantum probability for probabilists (Lect. Notes Math., vol. 1538) Berlin Heidelberg New York: Springer 1993

    Google Scholar 

  19. Rozovskii, B.L.: Stochastic evolution systems. Linear theory and applications to nonlinear filtering. Dodrecht Boston London: Kluwer 1990

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Vavilova 42, 117966, Moscow, Russia

    A. S. Holevo

Authors
  1. A. S. Holevo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Holevo, A.S. On dissipative stochastic equations in a Hilbert space. Probab. Th. Rel. Fields 104, 483–500 (1996). https://doi.org/10.1007/BF01198163

Download citation

  • Received: 10 May 1994

  • Revised: 12 June 1995

  • Issue Date: December 1996

  • DOI: https://doi.org/10.1007/BF01198163

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (1991)

  • 60H15
  • 81S25
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature