Fokker–Planck Equations and Maximal Dissipativity for Kolmogorov Operators for SPDE Driven by Lévy Noise

2012 ◽  
Vol 38 (2) ◽  
pp. 381-396 ◽  
Author(s):  
Yueling Li ◽  
Xiaobin Sun ◽  
Yingchao Xie
Keyword(s):  
Lévy Noise ◽  
Kolmogorov Operators ◽  
Fokker Planck Equations ◽  
Levy Noise ◽  
Fokker Planck

scholarly journals An Enhanced Intrinsic Time-Scale Decomposition Method Based on Adaptive Lévy Noise and Its Application in Bearing Fault Diagnosis

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 617
Author(s):  
Jianpeng Ma ◽  
Shi Zhuo ◽  
Chengwei Li ◽  
Liwei Zhan ◽  
Guangzhu Zhang
Keyword(s):  
Fault Diagnosis ◽  
Time Scale ◽  
Lévy Noise ◽  
Model Parameters ◽  
Rolling Bearings ◽  
Intrinsic Time ◽  
Time Scale Decomposition ◽  
Weak Fault ◽  
Levy Noise ◽  
The Impact

When early failures in rolling bearings occur, we need to be able to extract weak fault characteristic frequencies under the influence of strong noise and then perform fault diagnosis. Therefore, a new method is proposed: complete ensemble intrinsic time-scale decomposition with adaptive Lévy noise (CEITDALN). This method solves the problem of the traditional complete ensemble intrinsic time-scale decomposition with adaptive noise (CEITDAN) method not being able to filter nonwhite noise in measured vibration signal noise. Therefore, in the method proposed in this paper, a noise model in the form of parameter-adjusted noise is used to replace traditional white noise. We used an optimization algorithm to adaptively adjust the model parameters, reducing the impact of nonwhite noise on the feature frequency extraction. The experimental results for the simulation and vibration signals of rolling bearings showed that the CEITDALN method could extract weak fault features more effectively than traditional methods.

scholarly journals Quantifying the parameter dependent basin of the unsafe regime of asymmetric Lévy-noise-induced critical transitions

2020 ◽  
Vol 42 (1) ◽  
pp. 65-84
Author(s):  
Jinzhong Ma ◽  
Yong Xu ◽  
Yongge Li ◽  
Ruilan Tian ◽  
Shaojuan Ma ◽  
...  
Keyword(s):  
Lévy Noise ◽  
Stable Model ◽  
Efficient Tool ◽  
Escape Probability ◽  
Critical Transitions ◽  
Current State ◽  
Undesirable State ◽  
Parameter Dependent ◽  
Levy Noise ◽  
The Given

AbstractIn real systems, the unpredictable jump changes of the random environment can induce the critical transitions (CTs) between two non-adjacent states, which are more catastrophic. Taking an asymmetric Lévy-noise-induced tri-stable model with desirable, sub-desirable, and undesirable states as a prototype class of real systems, a prediction of the noise-induced CTs from the desirable state directly to the undesirable one is carried out. We first calculate the region that the current state of the given model is absorbed into the undesirable state based on the escape probability, which is named as the absorbed region. Then, a new concept of the parameter dependent basin of the unsafe regime (PDBUR) under the asymmetric Lévy noise is introduced. It is an efficient tool for approximately quantifying the ranges of the parameters, where the noise-induced CTs from the desirable state directly to the undesirable one may occur. More importantly, it may provide theoretical guidance for us to adopt some measures to avert a noise-induced catastrophic CT.

scholarly journals Fokker–Planck representations of non-Markov Langevin equations: application to delayed systems

2019 ◽  
Vol 377 (2153) ◽  
pp. 20180131 ◽  
Author(s):  
Luca Giuggioli ◽  
Zohar Neu
Keyword(s):  
Langevin Equation ◽  
Complex Dynamics ◽  
Probability Distributions ◽  
Delay Systems ◽  
Langevin Equations ◽  
Delayed Systems ◽  
Temporal Features ◽  
Fokker Planck Equations ◽  
Random Fluctuations ◽  
Fokker Planck

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker–Planck equations for the n -time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n  = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker–Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

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