The ergodic theorem for additive cocycles of ℤ^d or ℝ^d

Abstract

Let us consider (Ω, , μ, G) a measure-preserving dynamical system, (Ω, , μ) is a probability space. The group G, which is supposed to be either ℤd or ℝd (d ≥ 1), acts on Ω by measure-preserving transformations. This action is denned by a mapwhich is jointly measurable, such that Tx+y = TxTy and Txμ = μ

Degree two ergodic theorem for divergence-free stationary random fields

“…The family of integrals of such a field, on triangular surfaces, forms a degree 2 cocycle for the action of translations. Thus this theorem constitutes an ergodic theorem for degree 2 cocycles, analogous to the ergodic theorem for degree 1 cocycles of actions of Ê d , proved in [1]. Similarly to this last reference, the required condition of integrability for the pointwise convergence is finiteness of a Lorentz norm.…”

“…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].…”

“…We specify the connection between these concepts and those of algebraic cocycles and coboundaries of degree 2, and we prove that weak functional coboundaries of degree 2 in Ä p,q (Ω) are dense in the space of weak functional cocycles of degree 2 in Ä p,q (Ω). We start by exposing a phenomenon which exhibits the difference between the ergodic theorem for cocycle of degree 1 proved in [1] and our theorem for the degree 2.…”

“…The ergodic theorem for cocycle of degree 1 of [1] has a converse statement, with the following weak version: if a pseudo-cocycle F (x)(ω) of degree 1 satisfies the local mean ergodic theorem in Ä 1 (Ω), and if we denote by f (ω) = (f i (ω)) 1≤i≤3 the limit field, defined by…”

Tail estimates for homogenization theorems in random media

“…Then by [3], Lemma 2, p. 26, there is a constant C < ∞, which depends only on the dimension d, and there is a set of paths C from z 0 to z 1 such that…”