Gaussian random fields and Wiener integral Let $(\Omega, \mathcal{A}, P)$ be a probability space,
and let $W(x): \Omega \to \mathbb{R}$ be r.v. for each $x \in \mathbb{R}^d$.
We assume that $\{ W(x) \}_{x \in \mathbb{R}^d}$ be a Gaussian random field, that is, for arbitrary $N \in \mathbb{N}$ and $x^{(1)}, \dots, x^{(N)} \in \mathbb{R}^d$, the distribution of $(W(x^{(1)}), \dots, W(x^{(N)}))$ is $N$-dimensional Gaussian distribution.
Then, can we define Wiener integral
$$\int_{\mathbb{R}^d} f(x) dW(x) \, (\in L^2(\Omega))$$
for each $f \in L^2(\mathbb{R}^d)$ ?
I don't know a lot about probability theory, but I need it suddenly.
 A: Yes, this construction of the Wiener integral is given in many books on Stochastic Calculus, for example, Brownian Motion, Martingales, and Stochastic Calculus by JF Le Gall, from which I am pulling.
To simplify the notation, I am going to write $G(f) := \int_{\mathbb{R}^d} f dW$ for $f \in L^2\left(\mathbb{R}^d, \mathcal{B}\left(\mathbb{R}^d\right), \lambda \right)$. For any $A \in \mathcal{B}\left(\mathbb{R}^d\right)$, we define $G\left(\mathbf{1}_A\right)$ as $\mathcal{N}\left(0, \lambda(A)\right)$-random variable and the covariance be any two such random variables as $\mathbb{E}\left[G\left(\mathbf{1}_A) G\right(\mathbf{1}_B)\right] = \lambda\left(A\cap B\right)$. Let $\mathcal{H}^{pre}$ be the span of the random variables with these inner product define by
$\left<G\left(\mathbf{1}_A), G\right(\mathbf{1}_B)\right>_{\mathcal{H}} = \mathbb{E}\left[G\left(\mathbf{1}_A) G\right(\mathbf{1}_B)\right]$
with the obvious extension by linearity to any element of $\mathcal{H}^{pre}$. Then we can define the Hilbert space $\mathcal{H} = \text{cl}\left({\mathcal{H}^{pre}}\right)$ with the same inner product.
$\mathcal{H}$ is a Gaussian space that is a closed subspace of $L^2\left(\Omega, \mathscr{F}, \mathbb{P}\right)$ formed by centered Gaussian random variables and defining the Wiener integral amounts to showing that $G$ as a map from $ L^2\left(\mathbb{R}^d, \mathcal{B}\left(\mathbb{R}^d\right), \lambda \right)$ to $\mathcal{H}$
is well-defined. Observe that by our construction $G$ is an isometry from the simple functions into $\mathcal{H}$ thus there exists a unique continuous extension to all of $L^2$ by density. Thus for any $f\in L^2$, $G(f)$ is a mean zero Gaussian random variable with variance $\left\|f\right\|_{L^2}$ and for two $L^2$ functions, $g,f$, the covariance their Wiener integrals is $Cov\left(G(f), G(g)\right) = \mathbb{E}\left[G(f)G(g)\right] = \left<f, g\right>_{L^2}$.
To get some intuition as to why this is the correct way of constructing the Wiener integral consider how we would compute the integral numerically. Fix some $f$ in $L^2$ and let's assume it's uniformly continuous so it decays nicely and we don't have to deal with issues of it not being defined everywhere. For $n \in \mathbb{N}$ define
$$f_n = \sum_{z \in \mathbb{Z}^d \; s.t. \; |z|_\infty < 2^n} f\left(\frac{z}{n}\right) \mathbf{1}_{\prod_i^d \left[\frac{2z_i - 1}{2n}, \frac{2z + 1}{2n}\right)}$$
this an approximation of $f$ that is constant on cubes of side length $\frac{1}{n}$ and has support on $\|x\|_\infty \leq 2^n/n$. It is easy to show that $f_n \to f$ in $L^2$. The obvious definition of $G(f_n)$ would be
$$\int_{\mathbb{R}^d} f_n dW = \sum_{z \in \mathbb{Z}^d \; s.t. \; |z|_\infty < 2^n} f\left(\frac{z}{n}\right) \frac{1}{n^{d/2}} \xi_z$$
where $\left(\xi_z\right)_{z\in \mathbb{Z}^d}$ are independent standard normal random variables. I will leave it to you to check that the right hand side would go to zero if $\frac{1}{n^{d}}$ was used instead. One can check that 1) $\mathbb{E}\left[G\left(f_n\right)^2\right] = \left\|f_n\right\|_{L^2}^2$, 2) the sequence $\left(G\left(f_n\right)\right)_n$ is Cauchy in mean square i.e. $\mathbb{E}\left[\left(G\left(f_n\right) - G\left(f_m\right)\right)^2\right] \to 0$ as $n,m \to \infty$, and 3) for another $g$ and similar defined approximations $g_n$, $\mathbb{E}\left[G\left(f_n\right)G\left(g_n\right)\right] = \left<f_n, g_n\right>$.
Lastly, since $L^2$ is a separable Hilbert space $G$ can be characterized in terms of a given orthonormal basis of $L^2$: let $\left\{\varphi_n\right\}$ be an orthonormal basis of $L^2$ then
$$ G(f) = \sum \xi_n \left<f, \varphi_n\right>$$
where $\left(\xi_n\right)_n$ are independent standard normal random variables. This characterization is likely useful for many applications.
