Stationary states in single-well potentials under symmetric Lévy noises

We discuss the existence of stationary states for subharmonic potentials V(x)∝|x|c, c < 2, under the action of symmetric α-stable noises. We show analytically that the necessary condition for the existence of the steady state is c > 2 − α. Consequently, for harmonic (c = 2) and superharmonic potentials (c > 2) driven by any α-stable noise, steady states always exist. Stationary states are characterized by probability density functions P(x)∝x-(c+α-1) for |x|→∞ having a lighter tail than the noise distribution for superharmonic potentials (c > 2) and a heavier tail than the noise distribution for subharmonic ones. Monte Carlo simulations confirm the existence of such stationary states and the form of the tails of the corresponding probability densities