Brownian motion of harmonic systems with fluctuating parameters

Exact first and second order statistics of a damped mechanical oscillator with a fluctuating frequency and driven by a fluctuating force are obtained. All the cumulants of the fluctuating frequency are assumed to be delta correlated, while those of the driving force are of arbitrary form. The oscillator results depend only on the first and second order statistics of the driving force and on the coefficients of the first and second cumulants of the frequency fluctuations. Thus neither the higher order statistics of the driving force nor the coefficients of the higher cumulants of the frequency fluctuations need to be specified. When the fluctuating quantities are both delta correlated, the dynamics of the correction functions are shown to be independent of the frequency fluctuations.