Th\'eor\`eme ergodique pour cocycle harmonique, applications au milieu al\'eatoire. Ergodic theorem for harmonic cocycle, applications in random environment

TL;DR

This paper proves a pointwise ergodic theorem for harmonic cocycles of degree 1 in stationary Z^d actions, extending previous results and applying them to random media, with implications for elliptic and non-elliptic cases.

Contribution

It establishes a new pointwise ergodic theorem for harmonic cocycles under optimal integrability conditions, broadening the scope of previous work and including applications in random environments.

Findings

Proves ergodic theorem for harmonic cocycles in Z^d actions.

Identifies optimal L^2 integrability condition in the elliptic case.

Demonstrates applications to random media environments.

Abstract

In this work we prove the pointwise ergodic theorem for harmonic degree 1 cocycle of a measurable stationary action of Z^d on a probability space. In a precedent paper Boivin and Derriennic (1991) studied this theorem for not necessarily harmonic cocycles. The harmonic hypothesis allows, in the elliptic case, to change the integrability condition to L^2, while Boivin and Derriennic showed that the optimum condition in the non-harmonic case is the finiteness of Lorentz's norm L_{d,1}. They showed in particular that L^d is not enough. Berger and Biskup published in 2007 a paper on the harmonic not elliptic case, but only in dimension d=2. Finally, applications of this theorem in random media are presented.