Scaling behavior of a nonlinear oscillator with additive noise , white and colored

We study analytically and numerically the problem of a nonlinear mechanical oscillator with additive noise in the absence of damping. We show that the amplitude, the velocity and the energy of the oscillator grow algebraically with time. For Gaussian white noise, an analytical expression for the probability distribution function of the energy is obtained in the long-time limit. In the case of colored, Ornstein-Uhlenbeck noise, a self-consistent calculation leads to (different) anomalous diffusion exponents. Dimensional analysis yields the qualitative behavior of the prefactors (generalized diffusion constants) as a function of the correlation time. PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.) – 05.45.-a Nonlinear dynamics and nonlinear dynamical systems