Operator theory of multiple Ito-integrals

We study systems of Gaussian fields indexed by families F of positive sigma-finite measures μ. For a given μ, the corresponding Gaussian field W(μ) is centered and has quadratic variation equal to μ. Our focus is the induced multi-variable case of stochastic analysis and discrete time Gaussian random walk processes. The approach is operator-theoretic with three aims: (i) explicit formulas for the operators and Hilbert spaces involved; (ii) implications for Krein-Feller diffusion processes; and (iii) a study of operator systems and algebras generated by the W(μ)-induced Ito-isometries Vμ, for μ in F.