Homogenization of Elliptic Operators

Abstract

We start with an analytic description of random homogeneous fields on R ⁿ. In the case n = 1 they are called, usually, random stationary processes. Let us consider a probability space Ω, i.e. a set equipped with a σ-algebra F of measurable subsets and a countably additive non-negative measure μ on F normalized by μ(Ω) = 1. We always assume the measure μ to be complete. An n-dimensional dynamical system is defined as a family of selfmaps

$$T\left( x \right):\Omega \to \Omega , x \in {R^n},$$

with the following properties:

• (1) T(x + y) = T (x)T (y), x, y ∈ R ⁿ, and T (0) = I;

• (2) the map T(x) is measure preserving, i.e. for any x ∈ R ⁿ and for any μ-measurable subset U ⊂ Ω, the set T(x)U is μ-measurable and

  $$\mu \left( {T\left( x \right)u} \right) = \mu \left( u \right);$$

• (3) the map

  $$T:{R^n} \times \Omega \to \Omega , T:\left( {x,\omega } \right) \mapsto T\left( x \right)\omega ,$$

  is measurable, where R ⁿ × Ω is endowed with the measure dx ⨂μ,dx stands for the Lebesgue measure.