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Homogenization of Elliptic Operators

  • Alexander Pankov
Chapter
  • 232 Downloads
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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1997

Authors and Affiliations

  • Alexander Pankov
    • 1
  1. 1.Vinnitsa Polytechnical InstituteVinnitsaUkraine

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