  We have developed a mean-field model to describe the dynamics of a
non-equilibrium Bose-Einstein condensate of exciton-polaritons in a
semiconductor microcavity. The spectrum of elementary excitations around the
stationary state is analytically studied in different geometries. A diffusive
behaviour of the Goldstone mode is found in the spatially homogeneous case and
new features are predicted for the Josephson effect in a two-well geometry.
