Let $(\Omega,\mathcal{A}, \mu, (\tau_x)_{x \in \mathbb{R}^3}$) be a probability space endowed with an ergodic dynamic system $(\tau_x)_{x \in \mathbb{R}^3}$.
We say that a function $\phi :\Omega \times \mathbb{R}^3$ is stationary if there exists a random variable $a \in L^2(\Omega)$ such as
$$\phi(\omega,x)=a(\tau_x \omega)$$ for almost all $x$ and $\omega$. Usually, we will denote $\tilde{\phi}:=a$.
Now let $f_n$ be sequence of stationary random fields in $L^2(\Omega,L^2_{loc}(\mathbb{R}^3))$ such that
$$\mathbb{E} [ |\tilde{f_n}|^2 ] \leq C$$ with an absolute constant $C$ independent of $N$. Therefore, there exists $a_0 \in L^2(\Omega)$ such that $\tilde{f_n}$ weakly converges toward $a_0$. Using Ergodic theorem, we can show that there exists $f_0(\omega, \cdot) \in L^2_{loc}(\mathbb{R}^3)$ such that $f_n(\omega, \cdot)$ weakly converges toward $f_0(\omega,\cdot)$ in $L^2_{\text{loc}}(\mathbb{R}^3)$ for almost all $\omega$.
I would like to prove that the limiting point $f_0 : \Omega \times \mathbb{R}^3$ is a stationary random fields and that $\tilde{f_0}=a_0$. Concretely, i'm trying to prove that
$$f_0(\omega,x)=a_0(\tau_x \omega)$$ for almost all $x$ and $\omega$. I tried the following : Let $\omega$ a fixed realization in $\Omega$ (up to a set of zero measure) and $A$ a compact in $\mathbb{R}^3$. I know that :
$$\int_{A} |f_n(\omega,x) - \tilde{f_n}(\tau_x \omega)|^2 \ dx=0$$ and I want to pass to the limit $n \rightarrow + \infty$ in this equality. However, since all I have is weak convergences for $f_n(\omega,\cdot)$ and $\tilde{f_n}$ I can't pass to the limit.
Any help or thoughts are welcomed, feel free to ask question if you want details on the framework.
Since $f_n$ is stationary, it means that
$$f_n(\omega,x)=f_n(\tau_{-x} \omega,0)$$
almost everywhere. Therefore, the weak limit $f_0(\omega,\cdot)$ of $f_n(\omega,\cdot)$ (in $L^2_{loc}(\mathbb{R}^3)$) verifies
$$f_0(\omega,x)=f_0(\tau_{-x} \omega,0).$$
Then I pose $\tilde{f_0}(\omega)=f_0(\omega,0)$ for almost all $\omega$ and we can easily check that $f_0$ is a stationary random field. Remains to prove that $a_0=\tilde{f_0}$. By weak lower semi-continuity of the norm on $L^2(\Omega)$, we have :
$$\mathbb{E}\left[ |a_0(\cdot)- f_0(\cdot,0)|^2 \right] \leq \liminf_{n \rightarrow + \infty} \ \mathbb{E} \left[|\tilde{f_n}(\cdot) - f_n(\cdot,0)|^2 \right]$$ since $\tilde{f_n}$ weakly converges toward $a_0$ in $L^2(\Omega)$ and since we can show that $f_n(\cdot,0)$ weakly converges toward $f_0(\cdot,0)$ in $L^2(\Omega)$.