  The paper discusses analytical and numerical results for non-harmonic,
undamped, single-well, stochastic oscillators driven by additive noises. It
focuses on average kinetic, potential and total energies together with the
corresponding distributions under random drivings, involving Gaussian white,
Ornstein-Uhlenbeck and Markovian dichotomous noises. It demonstrates that
insensitivity of the average total energy to the single-well potential type,
$V(x) \propto x^{2n}$, under Gaussian white noise does not extend to other
noise types. Nevertheless, in the long-time limit ($t \to \infty$), the average
energies grow as power-law with exponents dependent on the steepness of the
potential $n$. Another special limit corresponds to $n\to\infty$, i.e. to the
infinite rectangular potential well, when the average total energy grows as a
power-law with the same exponent for all considered noise types.
