Derivation of Fokker–Planck equations for stochastic systems under excitation of multiplicative non-Gaussian white noise

We show that the increments of generalized Wiener process, useful to describe non-Gaussian white noise sources, have the properties of infinitely divisible random processes. Using functional approach and the new correlation formula for non-Gaussian white noise we derive directly from Langevin equation, with such a random source, the Kolmogorov's equation for Markovian non-Gaussian process. From this equation we obtain the Fokker–Planck equation for nonlinear system driven by white Gaussian noise, the Kolmogorov–Feller equation for discontinuous Markovian processes, and the fractional Fokker–Planck equation for anomalous diffusion. The stationary probability distributions for some simple cases of anomalous diffusion are derived.

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Governing equations for probability densities of Marcus stochastic differential equations with Lévy noise

Stochastics and Dynamics ◽

10.1142/s0219493717500332 ◽

2016 ◽

Vol 17 (05) ◽

pp. 1750033 ◽

Cited By ~ 3

Author(s):

Xu Sun ◽

Xiaofan Li ◽

Yayun Zheng

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Probability Density ◽

Lévy Processes ◽

Planck Equation ◽

Levy Processes ◽

Stochastic Dynamical Systems ◽

Non Gaussian ◽

Fokker Planck Equations ◽

Fokker Planck

Marcus stochastic differential equations (SDEs) often are appropriate models for stochastic dynamical systems driven by non-Gaussian Lévy processes and have wide applications in engineering and physical sciences. The probability density of the solution to an SDE offers complete statistical information on the underlying stochastic process. Explicit formula for the Fokker–Planck equation, the governing equation for the probability density, is well-known when the SDE is driven by a Brownian motion. In this paper, we address the open question of finding the Fokker–Planck equations for Marcus SDEs in arbitrary dimensions driven by non-Gaussian Lévy processes. The equations are given in a simple form that facilitates theoretical analysis and numerical computation. Several examples are presented to illustrate how the theoretical results can be applied to obtain Fokker–Planck equations for Marcus SDEs driven by Lévy processes.

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Dynamic behavior of periodic potential system driven by cross‐correlated non‐Gaussian noise and Gaussian white noise

International Journal of Robust and Nonlinear Control ◽

10.1002/rnc.5805 ◽

2021 ◽

Author(s):

Yongfeng Guo ◽

Xiaojuan Lou ◽

Qiang Dong ◽

Linjie Wang

Keyword(s):

White Noise ◽

Dynamic Behavior ◽

Gaussian Noise ◽

Periodic Potential ◽

Gaussian White Noise ◽

Potential System ◽

Non Gaussian

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Learning-based Adaptive Estimation for AOA Target Tracking with Non-Gaussian White Noise

2019 IEEE International Conference on Robotics and Biomimetics (ROBIO) ◽

10.1109/robio49542.2019.8961815 ◽

2019 ◽

Author(s):

Sheng Xu ◽

Yongsheng Ou ◽

Xinyu Wu

Keyword(s):

White Noise ◽

Target Tracking ◽

Adaptive Estimation ◽

Gaussian White Noise ◽

Non Gaussian

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EMD OF GAUSSIAN WHITE NOISE: EFFECTS OF SIGNAL LENGTH AND SIFTING NUMBER ON THE STATISTICAL PROPERTIES OF INTRINSIC MODE FUNCTIONS

Advances in Adaptive Data Analysis ◽

10.1142/s1793536909000217 ◽

2009 ◽

Vol 01 (04) ◽

pp. 517-527 ◽

Cited By ~ 13

Author(s):

GASTÓN SCHLOTTHAUER ◽

MARÍA EUGENIA TORRES ◽

HUGO L. RUFINER ◽

PATRICK FLANDRIN

Keyword(s):

White Noise ◽

Probability Density ◽

Kernel Smoothing ◽

Gaussian White Noise ◽

Large Data ◽

Intrinsic Mode Functions ◽

Mode Decomposition ◽

Non Gaussian ◽

Data Length ◽

Number Of Iterations

This work presents a discussion on the probability density function of Intrinsic Mode Functions (IMFs) provided by the Empirical Mode Decomposition of Gaussian white noise, based on experimental simulations. The influence on the probability density functions of the data length and of the maximum allowed number of iterations is analyzed by means of kernel smoothing density estimations. The obtained results are confirmed by statistical normality tests indicating that the IMFs have non-Gaussian distributions. Our study also indicates that large data length and high number of iterations produce multimodal distributions in all modes.

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Filtering with a limiter

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953398000240 ◽

1998 ◽

Vol 11 (3) ◽

pp. 289-300 ◽

Cited By ~ 1

Author(s):

R. Liptser ◽

P. Muzhikanov

Keyword(s):

Weak Convergence ◽

Diffusion Process ◽

White Noise ◽

Discrete Time ◽

Diffusion Approximation ◽

Conditional Expectation ◽

Gaussian White Noise ◽

Sampling Step ◽

Non Gaussian ◽

Filtering Problem

We consider a filtering problem for a Gaussian diffusion process observed via discrete-time samples corrupted by a non-Gaussian white noise. Combining the Goggin's result [2] on weak convergence for conditional expectation with diffusion approximation when a sampling step goes to zero we construct an asymptotic optimal filter. Our filter uses centered observations passed through a limiter. Being asymptotically equivalent to a similar filter without centering, it yields a better filtering accuracy in a prelimit case.

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Weighted Measurement Fusion White Noise Deconvolution Filter with Correlated Noise for Multisensor Stochastic Systems

Mathematical Problems in Engineering ◽

10.1155/2012/257619 ◽

2012 ◽

Vol 2012 ◽

pp. 1-16 ◽

Cited By ~ 5

Author(s):

Xin Wang ◽

Shu-Li Sun ◽

Kai-Hui Ding ◽

Jing-Yan Xue

Keyword(s):

White Noise ◽

Stochastic Systems ◽

Gaussian White Noise ◽

Minimum Variance ◽

Seismic Exploration ◽

Global Optimality ◽

Correlated Noise ◽

Noise Input ◽

Time Invariant ◽

Correlated Noises

For the multisensor linear discrete time-invariant stochastic control systems with different measurement matrices and correlated noises, the centralized measurement fusion white noise estimators are presented by the linear minimum variance criterion under the condition that noise input matrix is full column rank. They have the expensive computing burden due to the high-dimension extended measurement matrix. To reduce the computing burden, the weighted measurement fusion white noise estimators are presented. It is proved that weighted measurement fusion white noise estimators have the same accuracy as the centralized measurement fusion white noise estimators, so it has global optimality. It can be applied to signal processing in oil seismic exploration. A simulation example for Bernoulli-Gaussian white noise deconvolution filter verifies the effectiveness.

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