  In this note we comment on the homogenization of a random elliptic operator
in divergence form $-\nabla \cdot a\nabla$, where the coefficient field $a$ is
distributed according to a stationary, but not necessarily ergodic, probability
measure $P$. We generalize the well-known case for $P$ stationary and ergodic
by showing that the operator $-\nabla \cdot a(\frac{\cdot}{\varepsilon})\nabla$
almost surely homogenizes to a constant-coefficient, random operator $-\nabla
\cdot A_h\nabla$. Furthermore, we use a disintegration formula for $P$ with
respect to a family of ergodic and stationary probability measures to show that
the law of $A_h$ may be obtained by using the standard homogenization results
on each probability measure of the previous family. We finally provide a more
explicit formula for $A_h$ in the case of coefficient fields which are a
function of a stationary Gaussian field.
