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In this work we prove the pointwise ergodic theorem for harmonic degree 1cocycle of a measurable stationary action of Z^d on a probability space. In aprecedent paper Boivin and Derriennic (1991) studied this theorem for notnecessarily harmonic cocycles. The harmonic hypothesis allows, in the ellipticcase, to change the integrability condition to L^2, while Boivin and Derriennicshowed that the optimum condition in the non-harmonic case is the finiteness ofLorentz'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 randommedia are presented.

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