  This paper concerns singular perturbation problems where the dynamics of the
fast variable evolve in the whole space according to an operator whose
infinitesimal generator is formed by a Grushin type second order part and a
Ornstein-Uhlenbeck first order part. We prove that the dynamics of the fast
variables admits an invariant measure and that the associated ergodic problem
has a viscosity solution which is also regular and with logarithmic growth at
infinity. These properties play a crucial role in the main theorem which
establishes that the value functions of the starting perturbation problems
converge to the solution of an effective problem whose operator and initial
datum are given in terms of the associated invariant measure.
