Lévy 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.

Non-Gaussian noise effects in the dynamics of a short overdamped Josephson junction

“…The curve of Figure 4 corresponding to γ T N = 0.0 shows a nonmonotonic behavior with a minimum, that is the signature of the RA phenomenon [76][77][78][79][80][81][82][83][84][85]. The resonant activation effect is a robust enough phenomenon to be observed also in the presence of Lévy noise sources.…”

Nonlinear Relaxation Phenomena in Metastable Condensed Matter Systems

“…Almost all curves of Figure 3 clearly show the presence of resonant activation [14,17,80,81,91,116,118,[128][129][130][131][132][133][134][135], specifically stochastic resonance activation [17,78,80,117], a noise induced phenomenon whose signature is the appearance of a minimum in the curve of MST vs ω.…”

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

“…Lévy processes are used in the context of stochastic resonance, where systems perturbed by white Lévy noises are more informative than systems perturbed by white Gaussian noise, see Dybiec (2009), who shows that double stochastic resonance can be observed in a single-well potential perturbed by Lévy stable noises. Dybiec and Gudowska-Nowak (2009) consider an approach to the analysis of noise-induced effects in stochastic dynamics under the influence of Lévy white noise perturbations, describing interactions of the analyzed system with its complex surroundings. In illicit economic transactions, Perc (2007a) shows that Lévy flights facilitate defection.…”