# Homogenization of Elliptic Operators

• Alexander Pankov
Chapter
Part of the Mathematics and Its Applications book series (MAIA, volume 422)

## 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. (1)

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

2. (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. (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.

## Keywords

Vector Field Periodic Function Elliptic Operator Ergodic Theorem Maximal Monotone
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