Résumé en anglais

We consider the long time limit for the solutions of a discrete wave equation with weak stochastic forcing. The multiplicative noise conserves energy, and in the unpinned case also conserves momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck equation for the limit wave function that holds for both square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.