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White noise distribution theory. (English) Zbl 0853.60001
Probability and Stochastics Series. Boca Raton, FL: CRC Press. 378 p. (1996).
This is a monograph on “white noise calculus”, which has been developed in recent twenty years by T. Hida and his colleagues including the author. White noise means the centered normalized Gaussian measure \(\mu\) on the space of generalized functions, for example the Schwartz space \({\mathcal S}' (\mathbb{R}^d)\). As the function \(B(t)= \langle \chi_{[0, t]} (\cdot), \xi(\cdot) \rangle\), \(\xi \in{\mathcal S}' (\mathbb{R})\), can be identified with the standard Brownian motion with respect to the probability \(\mu\), an element \(F\in L^2 (S', \mu)\) is considered as function of the Brownian motion \(B(t)\) (or its derivatives \(dB(t)/ dt\), \(t\in \mathbb{R}\)), and called a Brownian functional. The motivation of white noise calculus is to develop an advanced calculus for Brownian functionals. We can define a Gel’fand triplet \((L^+) \supset L^2 (S', \mu) \supset (L^-)\), and the space \((L^+)\) is called the space of generalized functionals of white noise. Using this infinitely-many-variables analogue of generalized functions, some tools of advanced calculus are developed, for example differentiation, Fourier transforms, Laplacians, integrations etc. Finally, a formulation of Feynman’s path integral is obtained.

60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H99 Stochastic analysis