On the equivalence of probability spaces

Abstract

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

To appear in Journal of Theoretical Probability

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• Perturbative test of single parameter scaling for 1D random media
• Resolution of unity for fermionic Gaussian operators
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