Nonergodic Brownian oscillator: low-frequency response

Abstract An undisturbed Brownian oscillator may not reach thermal equilibrium with the bath due to formation of a localized normal mode. The latter emerge when spectrum has finite upper bound ω 0 and natural frequency exceeds critical value ω c , which depends on specific form spectrum. We consider response without mode external periodic force Ω lower than . results complement those obtained earlier for high-frequency at $\Omega\geqslant \omega_0$?> Ω ⩾ 0 require different mathematical approach. signature property is resonance coincides $\omega_*$?> ∗ In low-frequency domain $\Omega\lt\omega_0$?> Ω < condition $\Omega = \omega_*$?> = cannot be met (since $\omega_*\gt\omega_0$?> > ). Yet, in limits $\omega\to\omega_\textrm → $\Omega\to\omega_0^-$?> − shows peculiar quasi-resonance an amplitude increasing time sublinearly.