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A standard approach to analysis of noise-induced effects in stochasticdynamics assumes a Gaussian character of the noise term describing interactionof the analyzed system with its complex surroundings. An additional assumptionabout the existence of timescale separation between the dynamics of themeasured observable and the typical timescale of the noise allows externalfluctuations to be modeled as temporally uncorrelated and therefore white.However, in many natural phenomena the assumptions concerning theabovementioned properties of "Gaussianity" and "whiteness" of the noise can beviolated. In this context, in contrast to the spatiotemporal couplingcharacterizing general forms of non-Markovian or semi-Markovian L\'evy walks,so called L\'evy flights correspond to the class of Markov processes whichstill can be interpreted as white, but distributed according to a more general,infinitely divisible, stable and non-Gaussian law. L\'evy noise-drivennon-equilibrium systems are known to manifest interesting physical propertiesand have been addressed in various scenarios of physical transport exhibiting asuperdiffusive behavior. Here we present a brief overview of our recentinvestigations aimed to understand features of stochastic dynamics under theinfluence of L\'evy white noise perturbations. We find that the archetypalphenomena of noise-induced ordering are robust and can be detected also insystems driven by non-Gaussian, heavy-tailed fluctuations with infinitevariance.

Levy stable noise induced transitions: stochastic resonance, resonant activation and dynamic hysteresis