Journal of Operator Theory
Volume 93, Issue 2, Spring 2025 pp. 477-509.
Operator theory of multiple Ito-integralsAuthors: Palle E.T. Jorgensen (1), James Tian (2)
Author institution:(1) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A.
(2) Mathematical Reviews, 416 4th Str. Ann Arbor, MI 48103-4816, U.S.A.
Summary: We study systems of Gaussian fields indexed by families $\mathscr{F}$ of positive sigma-finite measures $\mu$. For a given $\mu$, the corresponding Gaussian field $W^{(\mu)}$ is centered and has quadratic variation equal to $\mu$. 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 $W^{(\mu)}$-induced Ito-isometries $V_{\mu}$, for $\mu$ in $\mathscr{F}$.
DOI: http://dx.doi.org/10.7900/jot.2023jun14.2428
Keywords: Hilbert space, reproducing kernel Hilbert space, harmonic analysis, Gaussian free fields, transforms, covariance, generalized It\^o-integration, generalized Brownian motion
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