Continuous version of Kingman's subadditive ergodic theorem. Let $T$ be a measure-preserving transformation on the probability space $(\Omega, \mathcal{A},\mathbb{P})$.
Kingsman's subadditive ergodic theorem states the following :
Let $(g_n)_n$ a sequence of random variables in $L^1(\Omega)$ such that
$$g_{n+m}(\omega) \leq g_n(\omega) + g_m(T^n \omega), \quad \forall n,m>0$$
for almost every $w \in \Omega$. Then there exists $g : \Omega \rightarrow \mathbb{R}$ a $T$-invariant function such that
$$\underset{n \rightarrow + \infty}{\lim} \frac{g_n(\omega) }{n} = g(\omega)$$
for almost every $w \in \Omega$. Moreover, if $T$ verify the ergodic property (every $T$-invariant function is constant almost everywhere on $\Omega$), then $g$ is a constant function.
I am looking for a continuous version of this theorem.
I would like a theorem that works on the following framework :
Let $(\Omega, \mathcal{A}, \mathbb{P})$ a probability space equipped with a group of transformation $(\tau_x)_{x \in \mathbb{R}^d}$ such that $(\Omega, \mathcal{A}, \mathbb{P},(\tau_x)_{x \in \mathbb{R}^d} )$ forms a $d$-dimension dynamic system. Let $(w \mapsto g(x,\omega))_{x \in \mathbb{R}^d}$ a family of random variables in $L^1(\Omega)$ indexed by $x \in \mathbb{R}^d$ and assume that
$$g(x+z,\omega) \leq g(x,\omega) + g(x,\tau_z \omega), \quad \forall x,z \in \mathbb{R}^d$$
and for a.e $w \in \Omega$.
Is it true that we can find $h$ a $\tau$-invariant function (i.e $h(\tau_x \omega)=\omega, \ \forall x \in \mathbb{R}^d, \ a.e \ \omega \in \Omega$) such that
$$\underset{R \rightarrow + \infty}{\lim} \frac{1}{|B_R|} \int_{B_R} g(x,\omega) \ \mathrm{d}x = h(\omega)$$
for almost $w \in \Omega$, where $B_R$ stands for the ball in $\mathbb{R}^d$ of radius $R$ and center $0$.
Moreover, if I assume the following :

*

*the field $g : \mathbb{R}^d \times \Omega$ is stationary (i.e there exists a random variable $\tilde{g} \in L^1(\Omega)$ such that $g(x,\omega)=\tilde{g}(\tau_x \omega)$ for all $x \in \mathbb{R}^d$ and almost all $w \in \Omega$)


*the system $(\Omega, \mathcal{A}, \mathbb{P},(\tau_x)_{x \in \mathbb{R}^d} )$ is ergodic
can I say that $h$ is a constant function (yes thanks to ergodicity) and do we have $h=\mathbb{E}[\tilde{g}]$ ?
 A: $d$-dimensional, continuous ergodic theorem
If $g$ is stationary and integrable, then a continuous version of the ergodic theorem implies that
\begin{equation*}
\lim_{R \to \infty} \frac{1}{|B_{R}|} \int_{B_{R}} g(x,\omega) \, dx = \tilde{g}(\omega),
\end{equation*}
where $\tilde{g}$ is a $\{\tau_{x}\}_{x \in \mathbb{R}^{d}}$-invariant random variable (i.e. $\tilde{g}(\tau_{x}\omega) = \tilde{g}(\omega)$).  When the group action is ergodic, $\tilde{g} = \mathbb{E}(g)$.  Note that none of this requires the sub-additivity property you postulated concerning $g$.
On the contrary, the way to see the above is to define a function $A(\cdot,\omega)$ on measurable sets by $A(E,\omega) = \int_{E} g(x,\omega) \, dx$.  This has the following additivity property:
\begin{equation*}
A(E,\omega) = \sum_{i = 1}^{N} A(E_{i},\omega) \quad \text{if} \, \, E = \bigcup_{i = 1}^{N} E_{i}, \, \, E_{i} \cap E_{j} = \phi.
\end{equation*}
Further, it is stationary and integrable in the sense that
\begin{equation*}
A(E + x, \omega) = A(E,\tau_{x}\omega), \quad \mathbb{E}(|A(B_{R},\omega)|) \leq C |B_{R}|.
\end{equation*}
Using these three properties, a continuous version of the ergodic theorem implies that
\begin{equation*}
\lim_{R \to \infty} \frac{1}{|B_{R}|} A(B_{R},\omega) = \tilde{g}(\omega).
\end{equation*}
$d$-dimensional, continuous sub-additive ergodic theorem
As in the case of space averages, the $d$-dimensional version of sub-additivity I will mention uses sub-additivity over sets instead of points.
Let's go back a step and simply consider the $d$-dimensional sub-additive lemma.  Here is a typical example: pick a constant $F$ and define the electrostatistic energy $E_{F}(U)$ in a domain $U \subseteq \mathbb{R}^{d}$ by
\begin{equation*}
E_{F}(U) = \inf \left\{ \int_{U} \left(\frac{1}{2} \|Du(x)\|^{2} + F u(x) \right) \, dx \, \mid \, u \in C^{\infty}_{c}(U) \right\}.
\end{equation*}
If $U = \cup_{i = 1}^{N} U_{i}$ and $\{U_{1},\dots,U_{N}\}$ is pairwise disjoint, then
\begin{equation*}
E_{F}(U) \leq \sum_{i = 1}^{N} E_{F}(U_{i}).
\end{equation*}
Working directly with cubes and mimicking the proof of Fekete's sub-additive lemma, one can show that there is a constant $\epsilon_{F} \geq 0$ such that
\begin{equation*}
\epsilon_{F} = \lim_{R \to \infty} \frac{1}{|Q_{R}|} E_{F}(Q_{R}) = \inf \left\{ \frac{E_{F}(Q_{R})}{|Q_{R}|} \, \mid \, R > 0 \right\}.
\end{equation*}
(Here $Q_{R}$ is the ball centered at $0$ with radius $R$ with respect to the $\ell^{\infty}$-norm, i.e. a cube centered at zero with sidelength $2R$.
Actually, the Euclidean norm would give the same result, which is another good exercise.)
[Actually, one can show that $E_{F}(Q_{R}) = \epsilon_{F} R^{d}$, but that's not the point!]
The same thing is true in a random setting.  Let $f : \mathbb{R}^{d} \times \Omega \to [0,\infty)$ be a stationary random field satisfying $\mathbb{E}\left(\int_{B(0,1)} |f(x)| \, dx\right) < \infty$.  Imagine $f$ captures the microscopic properties of some material, which, in this case, vary from one place to another.  This time, let's define the electrostatic energy $E(U,\omega)$ by
\begin{equation*}
E(U,\omega) = \inf \left\{ \int_{U} \left(\frac{1}{2} \|Du(x)\|^{2} - f(x) u(x) \right) \, dx \, \mid \, u \in C^{\infty}_{c}(U) \right\}.
\end{equation*}
Notice that $E(,\omega)$ has the following sub-additivity property: if $U = \sum_{i = 1}^{N} U_{i}$, then
\begin{equation*}
E(U,\omega) \leq \sum_{i = 1}^{N} E(U_{i},\omega).
\end{equation*}
Additionally, it is stationary in the sense that
\begin{equation*}
E(U + x, \omega) = E(U,\tau_{x} \omega).
\end{equation*}
One can show that there is a $\tau$-invariant random variable $\tilde{\varepsilon}$ such that
\begin{equation*}
\tilde{\varepsilon} = \lim_{R \to \infty} R^{-d} E(Q(0,R),\omega),
\end{equation*}
where $Q(0,R)$ is the ball centered at $0$ with radius $R$ with respect to the $L^{\infty}$-norm.  (One could replace $Q(0,R)$ by $B(0,R)$ with respect to any norm, in fact.)  If you want, $\tilde{\varepsilon}$ is the macroscopic electrostatic energy density of the material.
When everything is ergodic, the density is constant and is given by
\begin{equation*}
\tilde{\epsilon} = \inf \left\{ \frac{\mathbb{E}(E(Q_{R}))}{|Q_{R}|} \, \mid \, R > 0\right\}.
\end{equation*}
Reference:
The standard reference for this is Krengel's book Ergodic Theorems.
