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Stationary states in single-well potentials under symmetric Levy noises

We discuss the existence of stationary states for subharmonic potentials $V(x) \propto |x|^c$, $c<2$, under action of symmetric $α$-stable noises. We show analytically that the necessary condition for the existence of the steady state is $c>2-α$. These states are characterized by heavy-tailed probability density functions which decay as $P(x) \propto x^{-(c+α-1)}$ for $|x| \to \infty$, i.e. stationary states posses a heavier tail than the corresponding $α$-stable law. Monte Carlo simulations confirm the existence of such stationary states and the form of the tails of corresponding probability densities.