  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.
