Dynamical inference for transitions in stochastic systems with<i>α</i>-stable Lévy noise

Gao, Ting; Duan, Jinqiao; Kan, Xingye; Cheng, Zhuan

doi:10.1088/1751-8113/49/29/294002

Cited by 12 publications

(3 citation statements)

References 32 publications

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“…Whereas, in the Gaussian case, transitions between competing attractors occur as a result of the very unlikely combination of many steps all going in the right direction, in the Lévy case, transitions result from individual, very large and very rare jumps. Recently, Duan and collaborators have made fundamental progress in achieving a variational formulation of the Lévy noise-perturbed dynamical systems (Hu and Duan, 2020) and in developing corresponding methods for data assimilation (Gao et al, 2016) and data analysis (Lu and Duan, 2020). In terms of applications, Lévy noise is becoming a more and more a popular concept and tool for studying and interpreting complex systems (Grigoriu and Samorodnitsky, 2003;Penland and Sardeshmukh, 2012;Zheng et al, 2016;Wu et al, 2017;Serdukova et al, 2017;Cai et al, 2017;Singla and Parthasarathy, 2020;Gottwald, 2021).…”

Section: Transitions Between Competing Metastable Statesmentioning

confidence: 99%

Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Lucarini

Serdukova

Margazoglou

2022

Nonlin. Processes Geophys.

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Abstract. We study the impact of applying stochastic forcing to the Ghil–Sellers energy balance climate model in the form of a fluctuating solar irradiance. Through numerical simulations, we explore the noise-induced transitions between the competing warm and snowball climate states. We consider multiplicative stochastic forcing driven by Gaussian and α-stable Lévy – α∈(0,2) – noise laws, examine the statistics of transition times, and estimate the most probable transition paths. While the Gaussian noise case – used here as a reference – has been carefully studied in a plethora of investigations on metastable systems, much less is known about the Lévy case, both in terms of mathematical theory and heuristics, especially in the case of high- and infinite-dimensional systems. In the weak noise limit, the expected residence time in each metastable state scales in a fundamentally different way in the Gaussian vs. Lévy noise case with respect to the intensity of the noise. In the former case, the classical Kramers-like exponential law is recovered. In the latter case, power laws are found, with the exponent equal to −α, in apparent agreement with rigorous results obtained for additive noise in a related – yet different – reaction–diffusion equation and in simpler models. This can be better understood by treating the Lévy noise as a compound Poisson process. The transition paths are studied in a projection of the state space, and remarkable differences are observed between the two different types of noise. The snowball-to-warm and the warm-to-snowball most probable transition paths cross at the single unstable edge state on the basin boundary. In the case of Lévy noise, the most probable transition paths in the two directions are wholly separated, as transitions apparently take place via the closest basin boundary region to the outgoing attractor. This property can be better elucidated by considering singular perturbations to the solar irradiance.

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Section: Transitions Between Competing Metastable Statesmentioning

confidence: 99%

Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Lucarini

Serdukova

Margazoglou

2022

Nonlin. Processes Geophys.

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“…Starting at every initial point, we may thus compute its maximal likely trajectory for the evolution of concentration for the transcription factor activator (and thus we occasionally call it the maximal likely evolution trajectory). The maximal likely trajectories [48,49] are also called 'paths of mode' in climate dynamics and data assimilation [50,51]. They may have jumps, as the solution sample paths of the stochastic system (3) have jumps due to Lévy motion.…”

Section: Maximal Likely Trajectorymentioning

confidence: 99%

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

Cheng

Wang

et al. 2019

Physica A: Statistical Mechanics and its Applications

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We study the most probable transition pathways and maximal likely trajectories in a genetic regulation model of the transcription factor activator's concentration evolution, with Gaussian noise and non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account, respectively. We compute the most probable transition pathways by the Onsager-Machlup least action principle, and calculate the maximal likely trajectories by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We have observed the rare most probable transition pathways in the case of Gaussian noise, for certain noise intensity, evolution time scale and system parameters. We have especially studied the maximal likely trajectories starting at the low concentration metastable state, and examined whether they evolve to or near the high concentration metastable state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period); (ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including: (a) A smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise; (b) The symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription.These findings provide insights for further experimental research, in order to achieve or to avoid specific gene transcriptions, with possible relevance for medical advances.

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“…Meanwhile, we derived the Fokker-Planck equations for Marcus SDEs driven by Lévy processes [20]. We also used a nonlocal Zakai equation to examine the most probable path for systems with α-stable Lévy systems and continuous-time observations [21]. Furthermore, we devised most probable phase portraits to capture certain aspects of stochastic dynamics [22], and applied to examine qualitative changes or bifurcation of equilibrium states under Lévy noise [23].…”

Section: Introductionmentioning

confidence: 99%

The maximum likelihood climate change for global warming under the influence of greenhouse effect and Levy noise

Zheng¹

2020

Preprint

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An abrupt climatic transition could be triggered by a single extreme event, an α-stable non-Gaussian Lévy noise is regarded as a type of noise to generate such extreme events. In contrast with the classic Gaussian noise, a comprehensive approach of the most probable transition path for systems under αstable Lévy noise is still lacking. We develop here a probabilistic framework, based on the nonlocal Fokker-Planck equation, to investigate the maximum likelihood climate change for an energy balance system under the influence of greenhouse effect and Lévy fluctuations. We find that a period of the cold climate state can be interrupted by a sharp shift to the warmer one due to larger noise jumps, and the climate change for warming 1.5 o C under an enhanced greenhouse effect generates a step-like growth process. These results provide important insights into the underlying mechanisms of abrupt climate transitions triggered by a Lévy process. arXiv:1909.08693v1 [cond-mat.stat-mech]

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Dynamical inference for transitions in stochastic systems withα-stable Lévy noise

Cited by 12 publications

References 32 publications

Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Lévy noise versus Gaussian-noise-induced transitions in the Ghil–Sellers energy balance model

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

The maximum likelihood climate change for global warming under the influence of greenhouse effect and Levy noise

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