Acta Applicandae Mathematica

, Volume 63, Issue 1–3, pp 3–25 | Cite as

A White-Noise Approach to Stochastic Calculus

  • Luigi Accardi
  • Yun Gang Lu
  • Igor V. Volovich


During the past 15 years a new technique, called the stochastic limit of quantum theory, has been applied to deduce new, unexpected results in a variety of traditional problems of quantum physics, such as quantum electrodynamics, bosonization in higher dimensions, the emergence of the noncrossing diagrams in the Anderson model, and in the large-N-limit in QCD, interacting commutation relations, new photon statistics in strong magnetic fields, etc. These achievements required the development of a new approach to classical and quantum stochastic calculus based on white noise which has suggested a natural nonlinear extension of this calculus. The natural theoretical framework of this new approach is the white-noise calculus initiated by T. Hida as a theory of infinite-dimensional generalized functions. In this paper, we describe the main ideas of the white-noise approach to stochastic calculus and we show that, even if we limit ourselves to the first-order case (i.e. neglecting the recent developments concerning higher powers of white noise and renormalization), some nontrivial extensions of known results in classical and quantum stochastic calculus can be obtained.

quantum probability white noise stochastic limit 


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  1. 1.
    Accardi, L., Frigerio, A. and Lu, Y. G.: On the weak coupling limit problem, In: Quantum Probability and Applications IV, Lecture Notes in Math. 1396, Springer, New York, 1987, pp. 20–58.Google Scholar
  2. 2.
    Accardi, L., Frigerio, A. and Lu, Y. G.: The weak coupling limit as a quantum functional central limit, Comm. Math. Phys. 131(1990), 537–570.Google Scholar
  3. 3.
    _Accardi, L., Lu, Y. G. and Volovich, I. V.: The stochastic sector of quantum field theory, Volterra Preprint 138, 1993; Mat. Zam. (1994).Google Scholar
  4. 4.
    Accardi, L., Lu, Y. G. and Volovich, I. V.: Non-commutative (quantum) probability, master fields and stochastic bosonization, Volterra Preprint CVV-198-94, hep-th/9412241.Google Scholar
  5. 5.
    Accardi, L. and Mohari, A.: Stochastic flows and imprimitivity systems, In: Quantum Probability and Related Topics, World Scientific, Singapore, 1994, pp. 43–65. Volterra Preprint, 1994.Google Scholar
  6. 6.
    Accardi, L., Lu, Y. G. and Volovich, I. V.: Nonlinear extensions of classical and quantum stochastic calculus and essentially infinite dimensional analysis, In: L. Accardi and Chris Heyde (eds), Probability Towards 2000, Lecture Notes in Statist. 128, Springer, New York, 1998, pp. 1–33.Google Scholar
  7. 7.
    Accardi, L. and Volovich, I. V.: Quantum white noise with singular non-linear interaction, Volterra Preprint 278, 1997.Google Scholar
  8. 8.
    Accardi, L., Lu, Y. G. and Volovich, I. V.: A White Noise Approach to Classical and Quantum Stochastic Calculus, Volterra Preprint 375, Rome, July 1999, World Scientific, Singapore, 2000.Google Scholar
  9. 9.
    Cochran, W. G, Kuo, H.-H. and Sengupta, A.: A new class of white noise generalized functions, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(1998), 43–67.Google Scholar
  10. 10.
    Hida, T.: Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes 13, 1975.Google Scholar
  11. 11.
    Hida, T.: The impact of classical functional analysis on white noise calculus, Volterra Preprints 90, 1992.Google Scholar
  12. 12.
    Hida, T., Kuo, H.-H., Potthoff, J. and Streit L.: White Noise. An Infinite Dimensional Calculus, Kluwer Acad. Publ., Dordrecht, 1993, pp. 185–231.Google Scholar
  13. 13.
    Hida, T., Potthoff, J. and Streit, L.:White Noise Analysis and Applications, Kluwer Acad. Publ., Dordrecht, 1994.Google Scholar
  14. 14.
    Hudson, R. L. and Parthasarathy, K. R.: Construction of quantum diffusions, In: L. Accardi, A. Frigerio and V. Gorini (eds), Quantum Probability and Applications to the Quantum Theory of Irreversible Processes, Lecture Notes in Math. 1055, Springer, New York, 1984.Google Scholar
  15. 15.
    Hudson, R. L. and Parthasarathy, K. R.: Quantum Itô's formula and stochastic evolutions, Comm. Math. Phys. 93(1984), 301–323.Google Scholar
  16. 16.
    Kuo, H.-H.: White Noise Distribution Theory, CRC Press, Boca Raton, 1996.Google Scholar
  17. 17.
    Parthasarathy, K. R.: Quantum stochastic calculus, Preprint, 1995.Google Scholar
  18. 18.
    Obata, N.: Coherent state representation and unitarity condition in white noise calculus, J. Korean Math. Soc. (2000), to appear; White noise operator theory: fundamental concepts and developing applications, to appear in Proc. Volterra Internat. School, World Scientific, Singapore, 2000.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Luigi Accardi
    • 1
  • Yun Gang Lu
    • 1
    • 2
  • Igor V. Volovich
    • 1
    • 3
  1. 1.Centro Matematico Vito VolterraUniversità di RomaRomeItaly. e-mail
  2. 2.Dipartimento di MatematicaUniversità di BariBariItaly
  3. 3.Steklov Mathematical Institute, Russian Academy of SciencesMoscowRussia

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