  The statistics of the condensed polaritons is described in terms of the
Wigner function. In the framework of the truncated Wigner method, the Wigner
function obeys a Fokker- Planck equation, which is solved analytically. The
second order correlations in the stationary state are in excellent agreement
with those obtained from the numerical solution of the master equation and show
a qualitative and, well above threshold, also quantitative agreement with
recent experiments. Furthermore, the contributions of the different noise
effects that influence the polariton ground state statistics are explicitly
defined. Exploiting the equivalence between Fokker-Planck and Langevin
descriptions of stochastic processes, the time dependent correlations of the
polaritons close to the stationary state are derived. Explicit expressions for
the linewidth and for the relaxation rate of the polariton intensity are
obtained, whose values are of the same order of magnitude of the experimental
data. Finally, the limit of validity of the truncated Wigner method in the
present model is discussed.
