Israel Journal of Mathematics

, Volume 157, Issue 1, pp 283–308 | Cite as

Degree two ergodic theorem for divergence-free stationary random fields

  • Jérôme Depauw


We prove the ergodic theorem for surface integrals of divergence-free stationary random fields of ℝ3. Mean convergence in \( \mathbb{L}^p \) spaces takes place as soon as the field is \( \mathbb{L}^p \)-integrable. The condition of integrability for the pointwise convergence is expressed by a Lorentz norm. This theorem is an ergodic theorem for cocycles of degree 2, analogous to the ergodic theorem for cocycles of degree 1 proved in [1].


Ergodic Theorem Weak Sense Cyclic Permutation Lorentz Space Pointwise Convergence 
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  1. [1]
    D. Boivin and Y. Derriennic, The ergodic theorem for additive cocycles of ℤ d or ℝ d, Ergodic Theory and Dynamical Systems 11 (1991), 19–39.zbMATHMathSciNetGoogle Scholar
  2. [2]
    J. Depauw, Théorème ergodique ponctuel pour cocycle de degré deux, Comptes Rendus de l’Académie des Sciences, Paris 325, Séries I (1997), 87–90.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    J. Depauw, Génération des cocycles de degré ≥ 2 d’une action mesurable stationnaire de ℤ d, Ergodic Theory and Dynamical Systems 22 (2002), 153–169.zbMATHMathSciNetGoogle Scholar
  4. [4]
    J. Feldman and C. C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebra. I, Transactions of the American Mathematical Society 234 (1977), 289–324.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. A. Hunt, On L(p, q) spaces, Expositiones Mathematicae 12 (1966), 249–275.zbMATHGoogle Scholar
  6. [6]
    V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1991.Google Scholar
  7. [7]
    A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms, Ergodic Theory and Dynamical Systems 15 (1995), 569–592.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    G. W. Mackey, Ergodic Theory And Virtual Groups, Mathematische Annalen 166 (1966), 187–207.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    S. Mac Lane, Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 114, Springer-Verlag, Berlin, 1963.Google Scholar
  10. [10]
    E. Stein, The differentiability of functions in ℝ n, Archiv für Mathematik 113 (1981), 383–385.Google Scholar
  11. [11]
    N. Wiener, The ergodic theorem, Duke Mathematical Journal 5 (1939), 1–18.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Hebrew University of Jerusalem 2007

Authors and Affiliations

  • Jérôme Depauw
    • 1
  1. 1.Laboratoire de Mathématiques et Physique Théorique, Faculté des Sciences et TechniquesUniversité de ToursToursFrance

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