Linear oscillators driven by Gaussian colored noise: crossovers and probability distributions

We study the statistical properties of linear oscillators driven by both internal and external Gaussian colored noise. We obtain the variances of displacement and velocity and analyze their asymptotic behavior and crossovers. We also derive exact equations for the joint and marginal probability density functions of the system and find their solutions. In order to test our general analysis, we study the case of Ornstein-Uhlenbeck driving noise.