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On the Equivalence of Probability Spaces

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Abstract

For a general class of Gaussian processes W, indexed by a sigma-algebra \({\mathscr {F}}\) of a general measure space \((M,{\mathscr {F}}, \sigma )\), we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering \(\sigma (A)\), for \(A\in {\mathscr {F}}\), 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.

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It is a pleasure to thank the referee for his/her comments on the first version of the paper.

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Correspondence to Daniel Alpay.

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D. Alpay and P. Jorgensen thank the Binational Science Foundation Grant Number 2010117. One of the authors (P.J.) thanks colleagues at Ben-Gurion University for kind hospitality and for many very fruitful discussions. Part of this work was done while P.J. visited BGU in May and June 2014. D. Alpay thanks the Earl Katz family for endowing the chair which supported his research.

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Alpay, D., Jorgensen, P. & Levanony, D. On the Equivalence of Probability Spaces. J Theor Probab 30, 813–841 (2017). https://doi.org/10.1007/s10959-016-0667-7

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