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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.

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