Classical harmonic oscillator with multiplicative noise

The first two moments of an underdamped harmonic oscillator were found for additive and multiplicative noises. A detailed analysis is performed of the case of a random damping which supplements the case of a random frequency which has received the most study. In both cases the second moments are unstable for the sufficient large noise strength. The difference appears in the stability conditions of the first moments which can diverge already for white noise for the random damping and only for color noise for the random frequency. The difference between these two types of multiplicative noises shows up also in the response to an external periodic force, namely white noise influences the first moment only in the case of a random damping. However, the second moments in both cases show the non-monotonic dependence on the noise strength and correlation rate (stochastic resonance in linear systems).