Abstract
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].
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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.
J. Depauw, Théorème ergodique ponctuel pour cocycle de degré deux, Comptes Rendus de l’Académie des Sciences, Paris 325, Séries I (1997), 87–90.
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.
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.
R. A. Hunt, On L(p, q) spaces, Expositiones Mathematicae 12 (1966), 249–275.
V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1991.
A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms, Ergodic Theory and Dynamical Systems 15 (1995), 569–592.
G. W. Mackey, Ergodic Theory And Virtual Groups, Mathematische Annalen 166 (1966), 187–207.
S. Mac Lane, Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Band 114, Springer-Verlag, Berlin, 1963.
E. Stein, The differentiability of functions in ℝ n, Archiv für Mathematik 113 (1981), 383–385.
N. Wiener, The ergodic theorem, Duke Mathematical Journal 5 (1939), 1–18.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Depauw, J. Degree two ergodic theorem for divergence-free stationary random fields. Isr. J. Math. 157, 283–308 (2007). https://doi.org/10.1007/s11856-006-0012-4
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11856-006-0012-4