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For a general class of Gaussian processes W, indexed by a sigma-algebra F of a

general measure space (M,F, _), we give necessary and sufficient conditions for the validity

of a quadratic variation representation for such Gaussian processes, thus recovering _(A),

for A 2 F, as a quadratic variation of W over A. We further provide a harmonic analysis

representation for this general class of processes. We apply these two results to: (i) a computation

of generalized Ito-integrals; and (ii) a proof of an explicit, and measure-theoretic

equivalence formula, realizing an equivalence between the two approaches to Gaussian processes,

one where the choice of sample space is the traditional path-space, and the other

where it is Schwartz’ space of tempered distributions.


This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Journal of Theoretical Probability in 2016 following peer review. The final publication is available at Springer via

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