  We consider the impact of stochastic perturbations on otherwise coherent
oscillations of classical pulsators. The resulting dynamics are modelled by a
driven damped harmonic oscillator subject to either an external or an internal
forcing and white noise velocity fluctuations. We characterize the phase and
relative amplitude variations using analytical and numerical tools. When the
forcing is internal the phase variation displays a random walk behaviour and a
red noise power spectrum with a ragged erratic appearance. We determine the
dependence of the root mean square phase and relative amplitude variations
($\sigma_{\Delta \varphi}$ and $\sigma_{\Delta A/A}$, respectively) on the
amplitude of the stochastic perturbations, the damping constant $\eta$, and the
total observation time $t_{\rm obs}$ for this case, under the assumption that
the relative amplitude variations remain small, showing that $\sigma_{\Delta
\varphi}$ increases with $t_{\rm obs}^{1/2}$ becoming much larger than
$\sigma_{\Delta A/A}$ for $t_{\rm obs} \gg \eta^{-1}$. In the case of an
external forcing the phase and relative amplitude variations remain of the same
order, independent of the observing time. In the case of an internal forcing,
we find that $\sigma_{\Delta \varphi}$ does not depend on $\eta$. Hence, the
damping time cannot be inferred from fitting the power of the signal, as done
for solar-like pulsators, but the amplitude of the stochastic perturbations may
be constrained from the observations. Our results imply that, given sufficient
time, the variation of the phase associated to the stochastic perturbations in
internally driven classical pulsators will become sufficiently large to be
probed observationally.
