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 $\omega_0$ and the oscillator natural frequency exceeds a critical value $\omega_c$, 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 $\Omega$ lower than $\omega_0$. The results complement those obtained earlier for the high-frequency response at $\Omega\ge \omega_0$ and require a different mathematical approach. The signature property of the high-frequency response is resonance when the external force frequency $\Omega$ coincides with the frequency of the localized mode $\omega_*$. In the low-frequency domain $\Omega<\omega_0$ the condition of resonance $\Omega=\omega_*$ cannot be met (since $\omega_*>\omega_0$). Yet, in the limits $\omega\to\omega_c$ and $\Omega\to\omega_0^-$, the oscillator shows a peculiar quasi-resonance response with an amplitude increasing with time sublinearly.