Effects of parametric noise on a nonlinear oscillator

We study a model of a nonlinear oscillator with a random frequency and derive the asymptotic behavior of the probability distribution function when the noise is white. In the small damping limit, we show that the physical observables grow algebraically with time before the dissipative time scale is reached, and calculate the associated anomalous diffusion exponents. In the case of colored noise, with a nonzero but arbitrarily small correlation time, the characteristic exponents are modified. We determine their values thanks to a self-consistent Ansatz.