Suppression of oscillations by Levy noise

Abstract

We find analytical solution of pair of stochastic equations with arbitrary forces and multiplicative L\'evy noises in a steady-state nonequilibrium case. This solution shows that L\'evy flights suppress always a quasi-periodical motion related to the limit cycle. We prove that difference between stochastic systems driven by L\'evy and Gaussian noises is that the L\'evy variation $\Delta L\sim(\Delta t)^{1/\alpha}$ with the exponent $\alpha<2$ is much less than the Gaussian one $\Delta W\sim(\Delta t)^{1/2}$ in the $\Delta t\to 0$ limit. Moreover, this difference is shown to remove the problem of the calculus choice because related addition to the physical force is of order $(\Delta t)^{2/\alpha}\ll\Delta t$.