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
Stochastic boundary value problems: a white noise functional approach
Download PDF
Download PDF
  • Published: September 1993

Stochastic boundary value problems: a white noise functional approach

  • Helge Holden1,
  • Tom Lindstrøm2,
  • Bernt Øksendal2,
  • Jan Ubøe3 &
  • …
  • Tu-Sheng Zhang2 

Probability Theory and Related Fields volume 95, pages 391–419 (1993)Cite this article

  • 186 Accesses

  • 16 Citations

  • Metrics details

Summary

We give a program for solving stochastic boundary value problems involving functionals of (multiparameter) white noise. As an example we solve the stochastic Schrödinger equation {ie391-1} whereV is a positive, noisy potential. We represent the potentialV by a white noise functional and interpret the product of the two distribution valued processesV andu as a Wick productV ◊u. Such an interpretation is in accordance with the usual interpretation of a white noise product in ordinary stochastic differential equations. The solutionu will not be a generalized white noise functional but can be represented as anL 1 functional process.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  • [C] Colombeau, J.F.: Generalized functions. Bull. Am. Math. Soc.23, 251–268 (1990)

    Google Scholar 

  • [D] Durrett, R.: Brownian motion and martingales in analysis. Belmont, C4: Wadsworth 1984

    Google Scholar 

  • [GHLØUZ] Gjessing, H., Holden, H., Lindstrøm, T., Øksendal, B., Ubøe, J., Zhang, T.-S.: The Wick product. In: Melnikov, A. (ed.) New trends in probability and statistics, vol. IV. Moscow: TVP Publishers (to appear)

  • [HKPS] Hida, T., Kuo, H.-H., Potthoff, J., Streit, L.: White noise analysis (Forth-coming book)

  • [HLØUZ] Holden, H., Lindstrøm T., Øksendal, B., Ubøe, J., Zhang, T.-S.: The Burgers equation with a noisy force. Manuscript, 1992 (to appear)

  • [I] Ito, K.: Multiple Wiener integral. J. Math. Soc. Japan3 157–169, (1951)

    Google Scholar 

  • [KP] Kuo, H.-H., Potthoff, J.: Anticipating stochastic integrals and stochastic differential equations. In: Hida, T., Kuo, H.-H., Potthoff, J., Streit, L. (eds.) White noise analysis, pp. 256–273. Singapore: World Scientific 1990.

    Google Scholar 

  • [LØU1] Lindstrøm, T., Øksendal, B., Ubøe, J.: Stochastic differential equations involving positive noise. In: Barlow, M., Bingham, N. (eds.) Stochastic analysis, pp. 261–303. Cambridge: Cambridge University Press 1991.

    Google Scholar 

  • [LØU2] Lindstrøm, T., Øksendal, B., Ubøe, J.: Wick multiplication and Ito-Skorohod stochastic differential equations. In: Albeverio, S., Fenstand, J.E., Holden, H., Lindstrøm T., (eds.) Ideas and methods in mathematical analysis, stochastics and applications, pp. 183–206. Cambridge: Cambridge University Press 1992

    Google Scholar 

  • [LØU3] Lindstrøm, T., Øksendal, B., Ubøe, J.: Stochastic modelling of fluid flow in porous media. In: Chen, S., Yong, J. (eds.). Control theory, stochastic analysis and applications, pp. 156–172. Singapore: World Scientific 1991

    Google Scholar 

  • [GV] Gelfand, I.M., Vilenkin, N. Ya.: Generalized functions, vol. 4: Applications of harmonic analysis, (english translation). New York: Academic Press 1964

    Google Scholar 

  • [M] Miranda, C.: Partial differential equations of elliptic type, 2nd edn. Berlin Heidelberg New York: Springer 1970

    Google Scholar 

  • [NZ] Nualart, D., Zakai, M.: Generalized Brownian functionals and the solution to a stochastic partial differential equation. J. Funct. Anal.84, 279–296 (1989)

    Google Scholar 

  • [Ø] Øksendal, B.: Stochastic differential equations, third edn. Berlin Heidelberg New York: Springer 1992

    Google Scholar 

  • [W] Walsh, J. B.: An introduction to stochastic partial differential equations. In: Carmona, R., Kesten, H., Walsh, J. B., (eds.) École d'été de probabilités de Saint-Flour XIV-1984. (Lect. Notes Math., vol. 1180, pp. 265–437) Berlin Heidelberg New York: Springer 1986

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, The Norwegian Institute of Technology, University of Trondheim, N-7034, Trondheim, Norway

    Helge Holden

  2. Department of Mathematics, University of Oslo, Blindern, P. O. Box 1053, N-0316, Oslo 3, Norway

    Tom Lindstrøm, Bernt Øksendal & Tu-Sheng Zhang

  3. Haugesund Maritime College, Skåregaten 103, N-5500, Haugesund, Norway

    Jan Ubøe

Authors
  1. Helge Holden
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tom Lindstrøm
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bernt Øksendal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jan Ubøe
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Tu-Sheng Zhang
    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

Holden, H., Lindstrøm, T., Øksendal, B. et al. Stochastic boundary value problems: a white noise functional approach. Probab. Th. Rel. Fields 95, 391–419 (1993). https://doi.org/10.1007/BF01192171

Download citation

  • Received: 17 December 1991

  • Revised: 22 September 1992

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01192171

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

  • 60 H 15
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