Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\sigma$-algebra $\mathcal B(X)$ and an appropriate $\sigma$-ideal of null sets $\mathcal N(X)$. A measure on $(X,\mathcal B, \mathcal N)$ is called a random field on $M$ (this is also called a stochastic process, but I prefer to reserve that language for one-dimensional parameter spaces).
Since $M$ is homogeneous, there are natural actions arising from its symmetry group $G$. In our case $M = \mathbb R^d$, this means rotations and translations. Measures push-forward under actions, and we call a measure stationary (and isotropic) when it is invariant under these symmetries. A measure is ergodic if there is only one invariant set (up to null sets).
Let $\mathcal P^G$ denote the space of stationary, ergodic measures on $(X,\mathcal B, \mathcal N)$.
Side Question: Is there a nice characterization of this space $\mathcal P^G$?
More pertinently, I would like a numerical method to rapidly generate a stationary, ergodic random field. If the space $M$ were discrete, one plausible mechanism would be to use IID random variables. The independence is a strong form of ergodicity, and identical dependence is stationarity. On the other hand, this is no longer natural from the point of view of continuum random geometry.
One can take a dynamical approach, by starting with an arbitrary distribution on $X$, then transforming it by random transformations and taking averages. This is lengthy, though, and doesn't seem numerically efficient.
Main Question: Is there a nice class of stationary, ergodic distributions which one can easily sample from numerically?