Stationary states for underdamped anharmonic oscillators driven by Cauchy noise

Using numerical methods, we have studied stationary states in the underdamped anharmonic stochastic oscillators driven by Cauchy noise. The shape of stationary states depends on both the potential type and the damping. If the damping is strong enough, for potential wells which in the overdamped regime produce multimodal stationary states, stationary states in the underdamped regime can be multimodal with the same number of modes like in the overdamped regime. For the parabolic potential, the stationary density is always unimodal, and it is given by the two dimensional α-stable density. For the mixture of quartic and parabolic single-well potentials, the stationary density can be bimodal. Nevertheless, the parabolic addition, which is strong enough, can destroy the bimodality of the stationary state.