Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamichysteresis

Abstract

A standard approach to analysis of noise-induced effects in stochastic dynamics assumes aGaussian character of the noise term describing interaction of the analyzed system with itscomplex surroundings. An additional assumption about the existence of timescaleseparation between the dynamics of the measured observable and the typical timescale ofthe noise allows external fluctuations to be modeled as temporally uncorrelated andtherefore white. However, in many natural phenomena the assumptions concerning theabove mentioned properties of ‘Gaussianity’ and ‘whiteness’ of the noise can beviolated. In this context, in contrast to the spatiotemporal coupling characterizinggeneral forms of non-Markovian or semi-Markovian Lévy walks, so called Lévy flightscorrespond to the class of Markov processes which can still be interpreted aswhite, but distributed according to a more general, infinitely divisible, stable andnon-Gaussian law. Lévy noise-driven non-equilibrium systems are known to manifestinteresting physical properties and have been addressed in various scenarios ofphysical transport exhibiting a superdiffusive behavior. Here we present a briefoverview of our recent investigations aimed at understanding features of stochasticdynamics under the influence of Lévy white noise perturbations. We find that thearchetypal phenomena of noise-induced ordering are robust and can be detected also insystems driven by memoryless, non-Gaussian, heavy-tailed fluctuations with infinitevariance.