On the homogenization of random stationary elliptic operators in divergence form

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.