Théorème ergodique pour cocycle harmonique, applications au milieu aléatoire. Ergodic theorem for harmonic cocycle, applications in random environment

Jérôme Depauw

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arXiv · 1309.1566

Théorème ergodique pour cocycle harmonique, applications au milieu aléatoire. Ergodic theorem for harmonic cocycle, applications in random environment

Jérôme Depauw

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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.

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Jérôme Depauw. 2013-09-06. Théorème ergodique pour cocycle harmonique, applications au milieu aléatoire. Ergodic theorem for harmonic cocycle, applications in random environment. https://arxiv.org/abs/1309.1566

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