Stationary states in single-well potentials under symmetric Levy noises

Abstract

We discuss the existence of stationary states for subharmonic potentials $V(x)\propto |x{|}^c$, $c<2$, 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 $|x|\to \infty$, 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.

9 pages, 8 figures.

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