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Gaussian Hilbert spaces. (English) Zbl 0887.60009
Cambridge Tracts in Mathematics. 129. Cambridge: Cambridge University Press. x, 340 p. (1997).
Let \(\{X(t)\}\) be a centerd Gaussian process. Because the Gaussian law has finite moments of any order, the totality of polynomials of \(X(t)\)’s, say \(X(t_1),\;X(t_1)^3+2X(t_2)X(t_3)\), etc., make a pre-Hilbert space with respect to the covariance as the inner product. The author calls the completion \(L^2(\Omega,P)\) a Gaussian Hilbert space. This Gaussian Hilbert space appears in various fields in different contexts, as the Fock space in quantum theory, as homogeneous chaos in N. Wiener’s book “Nonlinear problems in random theory” (1958; Zbl 0121.12302), and the fundamental spaces in which the ItĂ´ calculus and the generalizations – white noise calculus and Malliavin calculus – have been developed. The above fields share the same fundamental structure, but are written in their own formalisms. The author intends to unify the formalisms and give us a bird view of this subject. The book has 16 chapters and some appendices. Each chapter treats a topic, for instance, Wiener chaos, Wick product, Stochastic integrals, \(U\)-statistics, Malliavin calculus, and so on. Readers interested in some of these topics can easily reach books and papers which contain deeper results by historical notes and references.

60B05 Probability measures on topological spaces
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H07 Stochastic calculus of variations and the Malliavin calculus
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