Suppression of oscillations by Levy noise

Abstract

We find analytical solution of pair of stochastic equations with arbitrary forces and multiplicative Lévy noises in a steady-state nonequilibrium case. This solution shows that Lévy flights suppress always a quasi-periodical motion related to the limit cycle. We prove that difference between stochastic systems driven by Lévy and Gaussian noises is that the Lévy variation $ΔL\sim(Δt)^{1/α}$ with the exponent $α<2$ is much less than the Gaussian one $ΔW\sim(Δt)^{1/2}$ in the $Δt\to 0$ limit. Moreover, this difference is shown to remove the problem of the calculus choice because related addition to the physical force is of order $(Δt)^{2/α}\llΔt$.