Harmonic oscillator with non-Gaussian frequency fluctuations

The moment and probability characteristics of harmonic oscillator with white non-Gaussian frequency fluctuations are investigated. Using a functional approach we derive the integro-differential Kolmogorov equation for the joint probability density function of oscillator coordinate and velocity. Since it is difficult to find a solution of this equation in the steady state the set of equations for joint moments and the hypothesis of time-reversal symmetry which is valid for zero friction are applied. For the case of small friction we obtain the approximate probability distributions of oscillator coordinate and velocity which transform into exact stable Cauchy distributions in the limit of zero friction.