Non-Boltzmann Equilibrium Probability Densities for Non-Linear L\'{e}vy Oscillator

We study, both analytically and by numerical modeling the equilibrium probability density function for an non-linear L'{e}vy oscillator with the L'{e}vy index \alpha, 1 \leq \alpha \leq 2, and the potential energy x^4. In particular, we show that the equilibrium PDF is bimodal and has power law asymptotics with the exponent -(\alpha +3).