Etienne Pardoux Université d'Aix-Marseille We consider a toy example of a 3D dynamical system with two invariant quantities, which we perturb with both noise and dissipation, so that the ODE becomes a hypoelliptic diffusion, which has unique invariant probability measure. We precise exactly which invariant measure of the dynamical system is the unique limit of the invariant measures of the stochastic systems, as both noise and dissipation vanish. We thus prove much stronger results than are currently available concerning (S)PDEs. This is joint work with Jonathan Mattingly (Duke Univ.)