Nonergodic Brownian oscillator: low-frequency response

An undisturbed Brownian oscillator may not reach thermal equilibrium with the thermal bath due to the formation of a localized normal mode. The latter may emerge when the spectrum of the thermal bath has a finite upper bound.0 and the oscillator natural frequency exceeds a critical value.c, which depends on the specific form of the bath spectrum. We consider the response of the oscillator with and without a localized mode to the external periodic force with frequency O lower than.0. The results complement those obtained earlier for the high-frequency response at O..0 and require a different mathematical approach. The signature property of the high-frequency response is resonance when the external force frequency O coincides with the frequency of the localized mode.*. In the low-frequency domain O <.0 the condition of resonance O =.* cannot be met (since.w(*) >w(0)). Yet, in the limits...c and O..- 0, the oscillator shows a peculiar quasi-resonance response with an amplitude increasing with time sublinearly.