Energetics of single-well undamped stochastic oscillators

This paper discusses analytical and numerical results for nonharmonic, 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)∝x^{2n}, under Gaussian white noise does not extend to other noise types. Nevertheless, in the long-time limit (t→∞), the average energies grow as power law with exponents dependent on the steepness of the potential n. Another special limit corresponds to n→∞, 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.