  We present an analytical study of a nonlinear oscillator subject to an
additive Ornstein-Uhlenbeck noise. Known results are mainly perturbative and
are restricted to the large dissipation limit (obtained by neglecting the
inertial term) or to a quasi-white noise (i.e., a noise with vanishingly small
correlation time). Here, in contrast, we study the small dissipation case (we
retain the inertial term) and consider a noise with finite correlation time.
Our analysis is non perturbative and based on a recursive adiabatic elimination
scheme: a reduced effective Langevin dynamics for the slow action variable is
obtained after averaging out the fast angular variable. In the conservative
case, we show that the physical observables grow algebraically with time and
calculate the associated anomalous scaling exponents and generalized diffusion
constants. In the case of small dissipation, we derive an analytic expression
of the stationary Probability Distribution Function (P.D.F.) which differs from
the canonical Boltzmann-Gibbs distribution. Our results are in excellent
agreement with numerical simulations.
