On the equivalence of probability spaces

TL;DR

This paper establishes conditions under which Gaussian processes indexed by measure spaces can be represented via quadratic variations and harmonic analysis, linking different conceptual frameworks for such processes.

Contribution

It provides necessary and sufficient conditions for quadratic variation representations and a harmonic analysis framework for Gaussian processes indexed by measure spaces.

Findings

Quadratic variation representation for Gaussian processes established

Harmonic analysis representation developed for these processes

Explicit measure-theoretic equivalence between different Gaussian process frameworks

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.