  A Fokker-Planck equation for the Wigner function evolution in a noisy Kerr
medium ($\chi^{(3)}$ non-linearity) is presented. We numerically solved this
equation taking a coherent state as an initial condition. The dissipation
effects are discussed. We provide examples of quantum interference, sub-Planck
phase space structures, and Gaussian versus non-Gaussian dynamical evolution of
the state. The results also apply to the description of a nanomechanical
resonator with an intrinsic Duffing nonlinearity.
