  We consider in this work a model conservative system subject to dissipation
and Gaussian-type stochastic perturbations. The original conservative system
possesses a continuous set of steady states, and is thus degenerate. We
characterize the long-time limit of our model system as the perturbation
parameter tends to zero. The degeneracy in our model system carries features
found in some partial differential equations related, for example, to
turbulence problems.
