Brownian motion of harmonic oscillator with stochastic frequency

Abstract

The aim of this paper is to find the effect of stochastic frequency fluctuations on the brownian motion of a harmonic oscillator driven by white noise. The equilibrium values of the mean square displacement, of the autocorrelation function and of the spectral energy density are calculated explicity when the frequency is a two-valued Markov process. The dominant effect of the stochastic perturbation is a decrease of the effective damping coefficient which may lead to energetic instability. A threshold is found above which the spectral energy density exhibits two separated resonances.