Stationary states in single-well potentials under symmetric Levy noises

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