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

Abstract

A standard approach to analysis of noise-induced effects in stochastic dynamics assumes a Gaussian character of the noise term describing interaction of the analyzed system with its complex surroundings. An additional assumption about the existence of timescale separation between the dynamics of the measured observable and the typical timescale of the noise allows external fluctuations to be modeled as temporally uncorrelated and therefore white. However, in many natural phenomena the assumptions concerning the abovementioned properties of "Gaussianity" and "whiteness" of the noise can be violated. In this context, in contrast to the spatiotemporal coupling characterizing general forms of non-Markovian or semi-Markovian Lévy walks, so called Lévy flights correspond to the class of Markov processes which still can be interpreted as white, but distributed according to a more general, infinitely divisible, stable and non-Gaussian law. Lévy noise-driven non-equilibrium systems are known to manifest interesting physical properties and have been addressed in various scenarios of physical transport exhibiting a superdiffusive behavior. Here we present a brief overview of our recent investigations aimed to understand features of stochastic dynamics under the influence of Lévy white noise perturbations. We find that the archetypal phenomena of noise-induced ordering are robust and can be detected also in systems driven by non-Gaussian, heavy-tailed fluctuations with infinite variance.

7 pages, 8 figures

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• Lévy noise effects on Josephson junctions
• Lévy-noise versus Gaussian-noise-induced Transitions in the Ghil-Sellers Energy Balance Model
• Multiplicative Levy noise in bistable systems
• Lévy on-off intermittency
• Bistable generalised Langevin dynamics driven by correlated noise possessing a long jump distribution: barrier crossing and stochastic resonance
• Role of long jumps in Lévy noise induced multimodality
• Metastability for telegraph processes in a double-well potential
• Kantorovich-Rubinstein Distance and Approximation for Non-local Fokker-Planck Equations
• Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes
• How to simulate Lévy flights in a steep potential: An explicit splitting numerical scheme