Brownian motion of harmonic systems with fluctuating parameters

We examine the stability properties of equilibrium moments of all orders for the damped mechanical oscillator with a delta correlated fluctuating frequency. A Markovian master equation is derived startimg from a frequency fluctuation process with finite correlation time τc and the limit τc→0 is taken. To approach this limit systematically, the oscillator and frequency fluctuation parameters are expressed in terms of a dimensionless scaling parameter. We derive exact integer moment transport equations in the limit of vanishing correlation time. These equations, and hence the moments, depend only on the second cumulant of the frequency fluctuations and not on the higher cumulants. The conjecture of Bourret et al.1) that for given frequency fluctuations, however weak, all moments beyond a certain order diverge is proved. We therefore conclude that the equilibrium distribution of the oscillator displacement and momentum cannot be Gaussian. A simple algebraic relation is established between the order of the lowest unstable moments and the system parameters.