TY - JOUR

T1 - On the Equivalence of Probability Spaces

AU - Alpay, Daniel

AU - Jorgensen, Palle

AU - Levanony, David

N1 - Funding Information:
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.
Publisher Copyright:
© 2016, Springer Science+Business Media New York.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - For a general class of Gaussian processes W, indexed by a sigma-algebra F of a general measure space (M, F, σ) , we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering σ(A) , for 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.

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

KW - Equivalence of measures

KW - Gaussian processes

KW - Stochastic calculus

UR - http://www.scopus.com/inward/record.url?scp=84954484122&partnerID=8YFLogxK

U2 - 10.1007/s10959-016-0667-7

DO - 10.1007/s10959-016-0667-7

M3 - Article

AN - SCOPUS:84954484122

SN - 0894-9840

VL - 30

SP - 813

EP - 841

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 3

ER -