  The paper presents analytical and numerical results on energetics of
non-harmonic, undamped, single-well, stochastic oscillators driven by additive
Gaussian white noises. Absence of damping and the action of noise are
responsible for lack of stationary states in such systems. We explore
properties of average kinetic, potential and total energies along with the
generalized equipartition relations. It is demonstrated that in the
friction-less dynamics nonequilibrium stationary states can be produced by
stochastic resetting. For an appropriate resetting protocol average energies
become bounded. If the resetting protocol is not characterized by finite
variance of renewal time intervals stochastic resetting can only slow down the
growth of average energies but does not bound them. Under special conditions
regarding frequency of resets, ratios of average energies follow the
generalized equipartition relations.
