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Long-period pulsations in the Pc5-6 band (periods about 3-20 min) have been known to be a persistent feature of the ULF activity at dayside high latitudes. Magnetopause surface eigenmodes were suggested as a potential source of high-latitude long-period pulsations. A ground response to these modes is expected to be beneath the ionospheric projection of the open-closed field line boundary (OCB). However, the comparison of the multi-instrument data from Svalbard with the latitudinal structure of ULF pulsations recorded by magnetometers covering near-cusp latitudes has shown that the latitudinal maximum of the broadband pulsations maximizes somewhat deeper in the magnetosphere than the OCB optical proxy. Therefore, these pulsations cannot be associated with the ground image of the magnetopause surface modes. Similarly, a localized peak in the latitudinal distribution of narrowband morning/dayside Pc5 power is located deeper in the magnetosphere, ~3o southward of the OCB. Therefore, narrowband dayside Pc5 wave activity cannot be associated with oscillations of the last closed field lines. The obtained result imposes an important limitation on possible mechanisms of high-latitude ULF variations. It is likely that a high variability of the magnetopause region may suppress the excitation efficiency. We suggest that stochastic fluctuations of the magnetospheric plasma and background magnetic field can provide an additional mechanism of damping of Alfven field line oscillations. To quantify this hypothesis, we consider a driven Alfven field line resonator with stochastic fluctuations of the Alfven eigenfrequency. This problem has been considered both analytically and numerically. The results indicate the deterioration of resonant properties of a MHD resonator owing to background fluctuations.