Path integral solutions of the governing equation of SDEs excited by Lévy white noise

Xu, Yong; Zan, Wanrong; Jia, Wantao; Kurths, Jürgen

doi:10.1016/j.jcp.2019.05.023

Cited by 45 publications

(14 citation statements)

References 31 publications

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“…Several authors [5][6][7][8] have dealt with different research interests for classical SDEs. Then, they extended their studies to the fractional case (FSDEs with constant order α) and investigated many results like existence, uniqueness, and stability for various classes of FSDEs (see [9][10][11][12][13][14][15][16]).…”

Section: Introductionmentioning

confidence: 99%

On the Existence and Uniqueness of Solutions for Multidimensional Fractional Stochastic Differential Equations with Variable Order

Moualkia

2021

Mathematics

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Fractional stochastic differential equations are still in their infancy. Based on some existing results, the main difficulties here are how to deal with those equations if the fractional order is varying with time and how to confirm the existence of their solutions in this case. This paper is about the existence and uniqueness of solutions to the fractional stochastic differential equations with variable order. We prove the existence by using the Picard iterations and propose new sufficient conditions for the uniqueness.

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Section: Introductionmentioning

confidence: 99%

On the Existence and Uniqueness of Solutions for Multidimensional Fractional Stochastic Differential Equations with Variable Order

Moualkia

2021

Mathematics

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“…Its exact stationary solution can only be obtained for few conceptual models under very strict conditions, and it is difficult to find its exact stationary solution in general [10]. Therefore, several numerical techniques have been developed to achieve solving of the FP equations [11][12][13][14][15][16][17][18][19][20][21].…”

Section: Introductionmentioning

confidence: 99%

“…In general, the PDFs of stochastic dynamics can be obtained from two main categories: One is directly to solve the FP equation through several techniques like finite element method [11,12], finite difference method [13,14], path integral method [15][16][17], variation method [18,19] and Galerkin method [20,21]. These approachs focus on discretizing the computational domain into a set of grid points and solving approximate solutions on the grid, but the solution is only calculated at the grid point, and the evaluation of any other point requires an interpolation or other reconstruction techniques.…”

Section: Introductionmentioning

confidence: 99%

Solving Fokker-Planck equation using deep learning

Zhang

et al. 2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed algorithm does not require any interpolation and coordinate transformation, which is different from the traditional numercial methods. The main novelty of this paper is that penalty factors are introduced to overcome the local optimization for the deep learning approach, and the corresponding setting rules are given. Meanwhile, we consider a normalization condition as a supervision condition to effectively avoid that the trial solution is zero. Several numerical examples are presented to illustrate performances of the proposed algorithm, including one-and twodimensional systems. All the results suggest that the deep learning is quite feasible and effective to calculate the FP equation. Further, influences of the number of hidden layers, the penalty factors, and the optimization algorithm are discussed in detail. These results indicate that the performances of the machine learning technique can be improved through constructing the neural networks appropriately.

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“…Therefore, a kind of the non-Gaussian noise needs to be considered to model unpredictable jumps of random environment. Lévy flights, a stochastic process characterized by the occurrence of extremely long jumps, can solve this problem [18][19][20][21] . The length of these jumps is distributed as Lévy stable statistics, which exhibits heavy tails and makes the moments divergent.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2020

Appl. Math. Mech.-Engl. Ed.

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In 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.

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Path integral solutions of the governing equation of SDEs excited by Lévy white noise

Cited by 45 publications

References 31 publications

On the Existence and Uniqueness of Solutions for Multidimensional Fractional Stochastic Differential Equations with Variable Order

On the Existence and Uniqueness of Solutions for Multidimensional Fractional Stochastic Differential Equations with Variable Order

Solving Fokker-Planck equation using deep learning

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

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