Stochastic dynamics driven by combined Lévy–Gaussian noise: fractional Fokker–Planck–Kolmogorov equation and solution

Zan, Wanrong; Xu, Yong; Kurths, Jürgen; Chechkin, Aleksei V.; Metzler, Ralf

doi:10.1088/1751-8121/aba654

“…Many directions remain yet to be explored, including the behaviour of the system under truncated Levy noise [69], combined Lévy-Gaussian noise [108], finite-velocity Lévy walks [109], different nonlinearities [110] and higher dimensions [66,111,112]. Other problems of interest concern noise with memory, of which few studies exist to date, such as [113,114].…”

Section: Discussionmentioning

confidence: 99%

1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency

Kan¹,

Pétrélis²

2023

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On–off intermittency occurs in nonequilibrium physical systems close to bifurcation points, and is characterised by an aperiodic switching between a large-amplitude ‘on’ state and a small-amplitude ‘off’ state. Lévy on–off intermittency is a recently introduced generalisation of on–off intermittency to multiplicative Lévy noise, which depends on a stability parameter α and a skewness parameter β. Here, we derive two novel results on Lévy on–off intermittency by leveraging known exact results on the first-passage time statistics of Lévy flights. First, we compute anomalous critical exponents explicitly as a function of arbitrary Lévy noise parameters ( α , β ) for the first time, by a heuristic method, extending previous results. The predictions are verified using numerical solutions of the fractional Fokker–Planck equation. Second, we derive the power spectrum S(f) of Lévy on–off intermittency, and show that it displays a power law S ( f ) ∝ f κ at low frequencies f, where κ ∈ ( − 1 , 0 ) depends on the noise parameters α , β . An explicit expression for κ is obtained in terms of ( α , β ) . The predictions are verified using long time series realisations of Lévy on–off intermittency. Our findings help shed light on instabilities subject to non-equilibrium, power-law-distributed fluctuations, emphasizing that their properties can differ starkly from the case of Gaussian fluctuations.

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“…Such correlations are associated with the viscoelasticity of the environment and are observed, e.g., in tracer diffusion in complex liquids [50] and in the crowded cytoplasm of biological cells and membranes [28,32,35,42,[51][52][53]. Superdiffusion motion, e.g., comes about due to positive correlations in FBM [43] or in spatiotemporally coupled Lévy walks [41,[54][55][56][57][58][59][60][61]. Examples for such statistics are found, i.a., in correlated vesicle motion in cells [37] or for the motion of molecular motors in cells [38,62].…”

Section: Introductionmentioning

confidence: 97%

Characterising stochastic motion in heterogeneous media driven by coloured non-Gaussian noise

Mutothya

¹

,

Xu

²

,

Li

³

et al. 2021

J. Phys. A: Math. Theor.

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We study the stochastic motion of a test particle in a heterogeneous medium in terms of a position dependent diffusion coefficient mimicking measured deterministic diffusivity gradients in biological cells or the inherent heterogeneity of geophysical systems. Compared to previous studies we here investigate the effect of the interplay of anomalous diffusion effected by position dependent diffusion coefficients and coloured non-Gaussian noise. The latter is chosen to be distributed according to Tsallis’ q-distribution, representing a popular example for a non-extensive statistic. We obtain the ensemble and time averaged mean squared displacements for this generalised process and establish its non-ergodic properties as well as analyse the non-Gaussian nature of the associated displacement distribution. We consider both non-stratified and stratified environments.

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“…(62), (56) remain unknown leading to only a non-rigorous estimate of the scaling exponents of the different moments with µ. Furthermore, the behavior of the system under truncated Lévy noise [42,63,64,101,102], combined Lévy-Gaussian noise [103], a finite-velocity Lévy walk [104], different nonlinearities [105], higher dimensions [93,106,107] and its time statistics [4-10, 23, 108] would also be interesting to understand. Finally, since Lévy statistics are found in many physical systems, we permit ourselves speculate that the anomalous critical exponents predicted here for instabilities in the presence of power-law noise may be observable experimentally.…”

Section: Discussionmentioning

confidence: 99%

Lévy on-off intermittency

van Kan,

Alexakis,

Brachet

2021

Preprint

0

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We present a new form of intermittency, Lévy on-off intermittency, which arises from multiplicative α-stable white noise close to an instability threshold. We study this problem in the linear and nonlinear regimes, both theoretically and numerically, for the case of a pitchfork bifurcation with fluctuating growth rate. We compute the stationary distribution analytically and numerically from the associated fractional Fokker-Planck equation in the Stratonovich interpretation. We characterize the system in the parameter space (α, β) of the noise, with stability parameter α ∈ (0, 2) and skewness parameter β ∈ [−1, 1]. Five regimes are identified in this parameter space, in addition to the well-studied Gaussian case α = 2. Three regimes are located at 1 < α < 2, where the noise has finite mean but infinite variance. They are differentiated by β and all display a critical transition at the deterministic instability threshold, with on-off intermittency close to onset. Critical exponents are computed from the stationary distribution. Each regime is characterised by a specific form of the density and specific critical exponents, which differ starkly from the Gaussian case. A finite or infinite number of integer-order moments may converge, depending on parameters. Two more regimes are found at 0 < α ≤ 1. There, the mean of the noise diverges, and no critical transition occurs. In one case the origin is always unstable, independently of the distance µ from the deterministic threshold. In the other case, the origin is conversely always stable, independently of µ. We thus demonstrate that an instability subject to non-equilibrium, power-law-distributed fluctuations can display substantially different properties than for Gaussian thermal fluctuations, in terms of statistics and critical behavior.

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Stochastic dynamics driven by combined Lévy–Gaussian noise: fractional Fokker–Planck–Kolmogorov equation and solution

Cited by 21 publications

References 79 publications

1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency

1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency

Characterising stochastic motion in heterogeneous media driven by coloured non-Gaussian noise

Lévy on-off intermittency

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