A white-noise approach to stochastic calculus

During the past 15 years a new technique, called the stochastic limit of qu antum theory, has been applied to deduce new, unexpected results in a varie ty of traditional problems of quantum physics, such as quantum electrodynam ics, bosonization in higher dimensions, the emergence of the noncrossing di agrams 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 classica l and quantum stochastic calculus based on white noise which has suggested a natural nonlinear extension of this calculus. The natural theoretical fra mework of this new approach is the white-noise calculus initiated by T. Hid a as a theory of infinite-dimensional generalized functions. In this paper, we describe the main ideas of the white-noise approach to stochastic calcu lus 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 no ise and renormalization), some nontrivial extensions of known results in cl assical and quantum stochastic calculus can be obtained.