Multiplicative Levy noise in bistable systems

Abstract

Stochastic motion in a bistable, periodically modulated potential is discussed. The system is stimulated by a white noise increments of which have a symmetric stable L\'evy distribution. The noise is multiplicative: its intensity depends on the process variable like |x|^{-\theta}. The Stratonovich and It\^o interpretations of the stochastic integral are taken into account. The mean first passage time is calculated as a function of \theta for different values of the stability index \alpha and size of the barrier. Dependence of the output amplitude on the noise intensity reveals a pattern typical for the stochastic resonance. Properties of the resonance as a function of \alpha, \theta\ and size of the barrier are discussed. Both height and position of the peak strongly depends on \theta\ and on a specific interpretation of the stochastic integral.