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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
Article

Abstract

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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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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