Skip to main content
SpringerLink
Log in
Book cover

Stochastic Analysis and Applications in Physics pp 415–439Cite as

An Introduction to White Noise Analysis

  • Chapter

  • 238 Accesses

Part of the book series: NATO ASI Series ((ASIC,volume 449))

Abstract

The Gaussian white noise measure μ (on the Borel algebra over cylinder sets of real, tempered distributions ω ∈ S *(R d)) is conveniently described by its characteristic function:

$$C(f) = E({e^{i < \omega ,f > }}) = \int\limits_{S*} {d\mu [\omega ]{e^{i < \omega ,f > }}} = {e^{ - \tfrac{1}{2}\int {{f^{2(t)dt}}} }},f \in S({R^d})$$
((1.1))

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Reference

  1. Albeverio, S., Hida, T., Potthoff, J., Röckner, M. and Streit, L.: “Dirichlet forms in terms of White Noise Analysis: I - Construction and QFT examples”. Rev. Math. Phys. 1 (1990) 291–312. “II - Construction of infinite dimensional diffusions.” Rev. Math. Phys. 1 (1990) 313–312.

    Google Scholar 

  2. Albeverio, S., Høegh-Krohn, R.: “Quasi-invariant measures, symmetric diffusion processes and quantum fields.” In: Les Méthodes Mathématiques de la Théorie Quantique des Champs, colloques Internacionaux du C.N.R.S. 248

    Google Scholar 

  3. Albeverio, S., Kondratiev, Ju. G. and Streit, L.: “How to generalize White Noise Analysis to non-Gaussian measures.” Bielefeld preprint (1992), to appear in Proc. IX. Bielefeld Encounters in Mathematics and Physics.

    Google Scholar 

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

    Article  Google Scholar 

  5. Cochran, C. and Potthoff, J.: “Fixed point principles for stochastic partial differential equations.” In “Dynamics of Complex and Irregular Systems” (Ph. Blanchard et al., eds.). World Scientific, 1993.

    Google Scholar 

  6. Doss, H.: Sur une resolution stochastique de l’équation de Schrödinger à coéfficients anaalytiques. Comm. math. Phys. 73 (1980) 247–264.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Hida, T.: “Infinite-dimensional rotation group and unitary group.” Lecture Notes in Mathematics 1379. Springer. 1989. pp. 125–134.

    Google Scholar 

  9. Hida, T., Kuo, H. -H., Potthoff, J. and Streit, L.: White Noise - an Infinite Dimensional Calculus. Kluwer-Academic, 1993.

    MATH  Google Scholar 

  10. Hida, T., Potthoff, J. and Streit, L.: “Dirichlet forms and white noise analysis.” Comm. math. Phys. 116 (1988) 235–245.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hida, T. and Streit, L.: “Generalized Brownian functionals and the Feynman integral.” Stoch. Proc. Appl. 16 (1983) 55–69.

    MathSciNet  Google Scholar 

  12. Khandekar, D. C. and Streit, L.: “Constructing the Feynman integrand.” Ann. Physik 1 (1992) 49.

    Article  MathSciNet  Google Scholar 

  13. Kondratiev, Ju. G. and Streit, L.: “Spaces of white noise distributions. Constructions, descriptions, applications I.” Rep. Math. Phys. 33 (1993) 341–366.

    Article  MathSciNet  Google Scholar 

  14. Kondratiev, Ju. G., Leukert, P., Potthoff, J., Streit, L., Westerkamp, W.: “Generalized Functionals in Gaussian Spaces - the Characterization Theorem Revisited”. Univ. Mannheim preprint, 1994.

    Google Scholar 

  15. Kondratiev, Ju. G., Leukert, P., and Streit, L.: “Wick Calculus in Gaussian Analysis”. Preprint 1994.

    Google Scholar 

  16. Kuo, H. -H.: “The Fourier transform in white noise calculus.” J. Multivariate Anal. 31 (1989) 311–327.

    Article  MathSciNet  MATH  Google Scholar 

  17. Lascheck, A., Leukert, P., Streit, L. and Westerkamp, W.: “Quantum mechanical propagators in terms of Hida distributions.” Rep. Math. Phys. 33 (1993) 221–232.

    Article  MathSciNet  MATH  Google Scholar 

  18. Lascheck, A., Leukert, P., Streit, L. and Westerkamp, W.: “More about Donsker’s Delta Function”. Preprint 1993, to appear in Soo-chow Math. J. (1994).

    Google Scholar 

  19. Lindstrom, T., Oksendal, B. and Uboe: “Dynamical systems in random media: a white noise functional approach.” Preprint. Oslo. 1990.

    Google Scholar 

  20. Meyer, P. A. and YAN, J. -A.: “Les fonctions caracteristiques” des distributions sur l’espace de Wiener.“ Strasbourg preprint, 1991.

    Google Scholar 

  21. Potthoff, J. and Röckner, M.: “On the contraction property of energy forms on infinite-dimensional space.” J. Funct. Anal. 92 (1990)

    Google Scholar 

  22. Potthoff, J. and Streit, L.: “Invariant states on random and quantum fields. 0-bounds and White Noise Analysis.” J. Funct. Anal. 111 (1993) 259–277.

    Article  MathSciNet  Google Scholar 

  23. Streit, L. and Westerkamp, W.: “A generalization of the characterization theorem for generalized functionals of white noise.” In “Dynamics of Complex and Irregular Systems” (Ph. Blanchard et al., eds.). World Scientific, 1993.

    Google Scholar 

  24. Watanabe, H.: “The local time of self-interactions of Brownian motions as generalized Brownian functionals”. Lett. Math.Phys. 23 (1991) 1–9.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CCM, Universidade da Madeira BiBoS, Univ., Bielefeld, Germany

    Ludwig Streit

Authors
  1. Ludwig Streit
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. TELECOM Portugal, Funchal(Madeira), Portugal

    Ana Isabel Cardoso

  2. CCM, Universidade da Madeira, Funchal(Madeira), Portugal

    Margarida de Faria

  3. Fakultät für Mathematik und Informatik, Universität Mannheim, Mannheim, Germany

    Jürgen Potthoff

  4. Centre de Physique Théorique, Ecole Polytechnique, Palaiseau, France

    Roland Sénéor

  5. CCM, Universidade da Madeira, Funchal(Madeira), Portugal

    Ludwig Streit

  6. BiBoS, Universität Bielefeld, Bielefeld, Germany

    Ludwig Streit

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Streit, L. (1994). An Introduction to White Noise Analysis. In: Cardoso, A.I., de Faria, M., Potthoff, J., Sénéor, R., Streit, L. (eds) Stochastic Analysis and Applications in Physics. NATO ASI Series, vol 449. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0219-3_17

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-0219-3_17

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4098-3

  • Online ISBN: 978-94-011-0219-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation