Influence of the noise spectrum on the anomalous diffusion in a stochastic system

We use an effective Markovian description to study the long-time behavior of a nonlinear second-order Langevin equation with a Gaussian noise. When dissipation is neglected, the energy of the system grows as with time a power law with an anomalous scaling exponent that depends both on the confining potential and on the high-frequency distribution of the noise. The asymptotic expression of the probability distribution function in phase space is calculated analytically. The results are extended to the case where small dissipative effects are taken into account.