Skip to main content

White noise methods for stochastic partial differential equations

  • Conference paper
  • First Online:
Stochastic Partial Differential Equations and Their Applications

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 176))

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asch, J. and Potthoff, J.: Itô's lemma without non-anticipatory conditions; Probab. Th. Rel. Fields 88 (1991) 17–46

    Google Scholar 

  2. Chow, P.L.: Generalized solution of some parabolic equations with a random drift; J. Appl. Math. Optimization 20 (1989) 81–96

    Google Scholar 

  3. Chow, P.L., Cochran, G. and Potthoff, J.: On parabolic stochastic partial differential equations; in preparation

    Google Scholar 

  4. de Faria, M., Potthoff, J. and Streit, L.: The Feynman integrand as a Hida distribution; J. Math. Phys. 32 (1991) 2123–2127

    Google Scholar 

  5. Gaveau, B. and Trauber, P.: L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel; J. Funct. Anal. 46 (1982) 230–238

    Google Scholar 

  6. Hida, T.: Brownian Motion. Berlin, Heidelberg, New York: Springer (1980)

    Google Scholar 

  7. Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L.: White Noise: An Infinite Dimensional Calculus. Monograph in preparation

    Google Scholar 

  8. Hida, T. and Potthoff, J.: White noise analysis — an overview; in White Noise Analysis — Mathematics and Applications, T. Hida, H.-H. Kuo, J. Potthoff and L. Streit (eds.). Singapore: World Scientific (1990)

    Google Scholar 

  9. Hille, E. and Phillips, R.S.: Functional Analysis and Semigroups. Providence: American Mathematical Society Colloq. Publ. (1957)

    Google Scholar 

  10. Hitsuda, M.: Formula for Brownian partial derivatives; in Proc. 2nd Japan-USSR Symp. Probab. Th. 2, Academy of Science USSR (1972) 111–114

    Google Scholar 

  11. King, R.: Stochastic integrals and metadistributions. Applications to stochastic partial differential equations and quantum field theory. Ph. D. dissertation (1975), Cornell Univ.

    Google Scholar 

  12. Kondrat'ev, Ju.G.: Nuclear spaces of entire functions in problems of infinite-dimensional analysis; Soviet Math. Dokl. 22 (1980) 588–592

    Google Scholar 

  13. Krée, P.: Calcul d'intégrales et de dérivées en dimension infinie; J. Funct. Anal. 31 (1979) 150–186

    Google Scholar 

  14. Krée, M. and Krée, P.: Continuité de la divergence dans les espaces de Sobolev relatifs à l'espace de Wiener; C. R. Acad. Sci. Paris 296 (1983) 833–836

    Google Scholar 

  15. Kubo, I.: Itô formula for generalized Brownian functionals; in Theory and Application of Random Fields; G. Kallianpur (ed.). Berlin, Heidelberg, New York: Springer (1983)

    Google Scholar 

  16. Kubo, I. and Takenaka, S.: Calculus on Gaussian white noise III; Proc. Japan Acad. 57A (1981) 433–437

    Google Scholar 

  17. Kubo, I. and Yokoi, Y.: A remark on the space of testing random variables in the white noise calculus; Negoya Math. J. 115 (1989) 139–149

    Google Scholar 

  18. Kuo, H.-H.: Lectures on white noise analysis; Preprint (1990)

    Google Scholar 

  19. Kuo, H.-H. and Potthoff, J.: Anticipating stochastic integrals and stochastic differential equations; in White Noise Analysis — Mathematics and Applications, T. Hida, H.-H. Kuo, J. Potthoff and L. Streit (eds.). Singapore: World Scientific (1990)

    Google Scholar 

  20. Kuo, H.-H., Potthoff, J. and Streit, L.: A characterization of white noise test functionals; Nagoya Math. J. 121 (1991) 185–194

    Google Scholar 

  21. Kuo, H.-H., Potthoff, J. and Yan, J.-A.: Continuity of affine transformations of white noise test functionals and applications; Preprint (1990), to appear in Stoch. Proc. Appl.

    Google Scholar 

  22. Kuo, H.-H. and Russek, A.: White noise approach to stochastic integration; J. Multivariate Analysis 24 (1988) 218–236

    Google Scholar 

  23. Lindstrøm, T., Øksendal, B. and Ubøe, J.: Stochastic differential equations involving positive noise; Preprint (1990), to appear in Stochastic Analysis, M. Barlow and N. Bingham (eds.)

    Google Scholar 

  24. Lindstrøm, T., Øksendal, B. and Ubøe, J.: Wick multiplication and Itô-Skorokhod stochastic differential equations; Preprint (1991), to appear in Ideas and Methods in Mathematical Physics, S. Albeverio et al. (eds.)

    Google Scholar 

  25. Meyer, P.A. and Yan, J.-A.: Distributions sur l'espace de Wiener (suite); Séminaire de Probabilités XXIII; J. Azéma, P.A. Meyer and M. Yor, eds. Berlin, Heidelberg, New York: Springer (1989)

    Google Scholar 

  26. Meyer, P.A. and Yan, J.-A.: Les “fonctions caractéristiques” des distributions sur l'espace de Wiener; Preprint (1990)

    Google Scholar 

  27. Nualart, D. and Pardoux, E.: Stochastic calculus with anticipating integrands; Probab. Th. Rel. Fields 78 (1988) 535–581

    Google Scholar 

  28. Nualart, D. and Zakai, M.: Generalized stochastic integrals and the Malliavin calculus; Probab. Th. Rel. Fields 73 (1986) 255–280

    Google Scholar 

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

    Google Scholar 

  30. Ocone, D.: Malliavin's calculus and stochastic integral representation of functionals of diffusion processes; Stochastics 12 (1984) 161–185

    Google Scholar 

  31. Potthoff, J.: On Meyer's equivalence; Nagoya Math. J. 111 (1988) 99–109

    Google Scholar 

  32. Potthoff, J. and Streit, L.: A characterization of Hida distributions; Preprint (1989), to appear in J. Funct. Anal.

    Google Scholar 

  33. Potthoff, J. and Streit, L.: Invariant states on random and quantum fields: Ø-bounds and white noise analysis; BiBoS Preprint (1990)

    Google Scholar 

  34. Potthoff, J. and Streit, L.: Generalized Radon — Nikodym derivatives and Cameron — Martin theory; Preprint (1990), to appear in Proc. Int. Conf. Gaussian Random Fields, Nagoya, 1990

    Google Scholar 

  35. Potthoff, J. and Yan J.A.: Some results about test and generalized functionals of white noise; to appear in Proc. Singapore Probab. Conf. (1989), L.Y. Chen (ed.)

    Google Scholar 

  36. Simon, B.: Distributions and their Hermite expansions; J. Math. Physics 12 (1971) 140–148

    Google Scholar 

  37. Skorokhod, A. V.: On a generalization of a stochastic integral; Theory Probab. Appl. 20 (1975) 219–233

    Google Scholar 

  38. Streit, L. and Westerkamp, W.: A generalization of the characterization theorem for generalized functionals of white noise; Preprint (1991)

    Google Scholar 

  39. Watanabe, H.: The local time of self-intersection of Brownian motions as “generalized Brownian functionals”; Preprint (1991)

    Google Scholar 

  40. Watanabe, S.: Malliavin's calculus in terms of generalized Wiener functionals; in Theory and Application of Random Fields; G. Kallianpur (ed.). Berlin, Heidelberg, New York: Springer (1983)

    Google Scholar 

  41. Yan, J.-A.: A characterization of white noise functionals; Preprint (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Boris L. Rozovskii Richard B. Sowers

Rights and permissions

Reprints and permissions

Copyright information

© 1992 International Federation for Information Processing

About this paper

Cite this paper

Potthoff, J. (1992). White noise methods for stochastic partial differential equations. In: Rozovskii, B.L., Sowers, R.B. (eds) Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control and Information Sciences, vol 176. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0007337

Download citation

  • DOI: https://doi.org/10.1007/BFb0007337

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55292-5

  • Online ISBN: 978-3-540-47015-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics