Energetics of the single-well undamped stochastic oscillators

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 ($t\to \mathrm{\infty }$), the average energies grow as power-law with exponents dependent on the steepness of the potential $n$. Another special limit corresponds to $n\to \mathrm{\infty }$, 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.