# Generating stationary, ergodic random fields on a homogeneous space

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?

• By the Riesz representation theorem, measures are exactly functionals on $X$, by the correspondance $\mu(f) = \int_M f(q) \, \mu(dq)$. Going down this route, one sees ergodic measures as the extreme points of an appropriate simplex, probably the Poulsen simplex as in Vaughn Climenhaga's question below. When we toss stationarity and especially isotropy into the mix, I really don't have a good intuition on what this space looks like. mathoverflow.net/questions/83981/… – Tom LaGatta Jun 20 '12 at 17:36
• You'll want to look at Yaglom's and Robert Adler's books if you haven't already. – Steve Huntsman Jun 21 '12 at 1:38