On the homogenization of random stationary elliptic operators in divergence form

TL;DR

This paper extends the homogenization theory for random elliptic operators to cases where the coefficient field is stationary but not necessarily ergodic, showing almost sure homogenization to a constant-coefficient operator.

Contribution

It generalizes classical homogenization results to non-ergodic stationary measures and provides explicit formulas for the homogenized coefficients in Gaussian cases.

Findings

Almost sure homogenization to a constant-coefficient operator.

Disintegration formula relates law of homogenized operator to ergodic components.

Explicit formula for homogenized coefficients in Gaussian coefficient fields.

Abstract

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.