# On the Equivalence of Probability Spaces

On the Equivalence of Probability Spaces For a general class of Gaussian processes W, indexed by a sigma-algebra $${\mathscr {F}}$$ F of a general measure space $$(M,{\mathscr {F}}, \sigma )$$ ( M , F , σ ) , we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $$\sigma (A)$$ σ ( A ) , for $$A\in {\mathscr {F}}$$ A ∈ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

# On the Equivalence of Probability Spaces

, Volume 30 (3) – Jan 20, 2016
29 pages

/lp/springer_journal/on-the-equivalence-of-probability-spaces-FvSkhckQJO
Publisher
Springer US
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-016-0667-7
Publisher site
See Article on Publisher Site

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