Gaussian Hilbert space from white noise

Question

There are two common ways of obtaining Gaussian processes on Hilbert space: either from a random element of a larger space, or directly as a collection of random variables indexed by the Hilbert space. My question involves the passage from the former to the latter.

Let $\Phi\to H$ be a rigged Hilbert space (a nuclear space densely embedded in Hilbert space) and let $\Phi'$ denote the continuous dual of $\Phi$. Embed $H$ in $\Phi'$ such that the duality pairing on $\Phi'\times \Phi$ and the inner product of $H$ are consistent. By the Minlos theorem there is a unique Borel measure $\mu$ (which we call the white noise measure) on $\Phi'$ satisfying $$ \mu(\exp i\langle \cdot,f\rangle)=\exp \Bigl(-\tfrac12\langle f,f\rangle\Bigr),\qquad f\in \Phi. $$ A standard Gaussian process on $H$ is a collection of (centered) Gaussian random variables $\{X_f\colon f\in H\}$ defined on a common probability space $(\Omega,\nu)$ satisfying $\text{Cov}(X_f,X_g)=\langle f,g\rangle$.

At the formal level, one obtains a standard Gaussian process by setting $(\Omega,\nu)=(\Phi',\mu)$ and $X_f=\langle \cdot,f\rangle$. However, it is not clear that this is well-defined since $\langle \cdot,\cdot\rangle$ was defined on $\Phi'\times \Phi$ and $H\times H$, but not on $\Phi'\times H$.

Question. What is an explicit choice of $\{X_f\colon f\in H\}$ and $(\Omega,\nu)$ given $(\Phi',\mu)$ that gives a standard Gaussian process on $H$?

Answer 1

Here is a potential alternative, not necessarily a full answer: in Hida's Brownian Motion, section 3.1, the setup is exactly the same as in this question (e.g. $\Phi \subset H \subset \Phi'$, $\Phi$ a nuclear space, $H$ a Hilbert space, $\Phi'$ the dual space), but the sequence of random variables is indexed in $\Phi$, not $H$.

And usually, this is good enough for what we want: we can get the white noise measure just from this by setting $\Phi$ to be the Schwartz space, and the white noise emerges from the process you've described.