Path integral solutions of the governing equation of SDEs excited by Lévy white noise
Introduction
Non-Gaussian Lévy noise has been observed in turbulent fluid flows [1], plasmas [2], [3], heartbeat dynamics [4], neural networks [5], climate dynamics [6], and population dynamics problems [7]. Hence, it is reasonable to describe some actual systems by means of stochastic differential equations (SDEs) driven by Lévy white noise. A large number of investigations focused on the dynamical analysis of SDEs subject to α-stable Lévy white noise, including stochastic resonance [8], [9], phase transition [10], escape probability [11], [12], mean first passage time [13], [14] etc, in which the probability density functions (PDFs) hold the key to analyze stochastic dynamics. The evolution of the PDFs of the solutions of SDEs excited by α-stable Lévy white noise is governed by space fractional Fokker-Planck-Kolmogorov (FPK) equations which contain a spatial fractional derivation term [15], [16], [17], [18], [19]. The existence and uniqueness of the weak solution of this kind of nonlocal FPK equation was investigated in the cited article [20].
Much effort has been made on solving space fractional FPK equations. Due to the existence of the space fractional derivation term, exact solutions of this kind of equation can be obtained only for some special cases with quite restricted conditions [21], [22]. Thus, numerical techniques were developed to solve this kind of equation. For instance, Meerschaert and Tadjeran [23], [24] proposed a finite difference method to solve the one-dimensional fractional advection dispersion equation with variable coefficients on a finite domain. They also used the finite difference approximation to solve two-sided space fractional partial differential equations. Liu et al. [25] transformed the space fractional FPK equation into an ordinary differential equation, which can be solved using backward difference formulas. Gao and Duan [26] also proposed a fast and accurate numerical algorithm for this kind of FPK equation. Further, the development of the path integral (PI) method in recent years provides a new perspective even to solve space fractional FPK equations [27], [28], [29].
The PI method has been successfully used to solve FPK equations and SDEs driven by Gaussian white noise excitations. Initially, the PI method was proposed for solving the FPK equation and the corresponding SDE. Wehner and Wolfer first developed the PI method as a numerical tool to solve the nonlinear FPK equations [30]. Subsequently, the PI approach was improved to obtain a higher accuracy and efficiency by an interpolation method, and was extended to other systems. Naess and Johnsen combined the PI technique with an appropriate numerical scheme to analyze the response statistics of compliant offshore structure [31]. Yu et al. improved the PI method based on a Gauss-Legendre scheme [32]. Naess and Moe developed the PI approach combined with a splines interpolation method [33]. Xie et al. studied the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations by means of the PI method [34]. These studies have shown that the PI method in the Gaussian case is applicable and efficient. Due to the increasing interest about the use of Lévy noise in modeling real systems, the extension to non-Gaussian noise of the PI method also attracts much attention.
Recently, the PI method was extended to SDEs driven by non-Gaussian noise [28], [29]. For the Lévy case, Bucher et al. generalized the PI approach to investigate the first passage problem in nonlinear SDE under Lévy white noise [27]. Gulian et al. studied the high dimensional fractional Schrödinger equation using the path integral formulation [35]. To the author's knowledge, the PI method has not been used to the solution of the space fractional FPK equation, and this is our aim in this paper. In fact, the PI method is applicable not only to the SDEs excited by Gaussian noise but also to its corresponding governing equations.
In the present paper, the PI method is used to solve the one-dimensional space fractional FPK equation. This paper is organized as follows. In section 2, the scalar SDE excited by α-stable Lévy white noise and the corresponding one-dimensional space fractional FPK equation are presented. In section 3, the short time solution of the one-dimensional space fractional FPK equation is derived as the transition PDF in the Chapman-Kolmogorov-Smoluchowski (CKS) equation, which is the core of the PI technique. Then, a discrete representation of the PI solution is presented to facilitate the numerical calculation and the accuracy of the PI solution is analyzed. In section 4, three examples with different external function are presented to illustrate the validity and effectiveness of the PI method. We show that the PI solution agrees well with the exact solution or the Monte Carlo simulation result, which indicates that the PI method is suitable for solving the one-dimensional space fractional FPK equation. In section 5, our conclusions are presented.
Section snippets
Statement of problem
We start with the following scalar SDE where is a function of x, is the initial value of at time . Here, is α-stable Lévy white noise, which is the formal derivative of a symmetric α-stable Lévy motion . Moreover, is a zero-shift and zero-skewness Lévy motion, which can be described by the characteristic function where is the mathematical expectation operator. In the above, and
Methodology
In this section, the short time solution of the one-dimensional space fractional FPK equation is derived as the transition PDF in the PI method. Then the PI methodology and its numerical scheme are presented, and the accuracy of the PI solution is analyzed by means of the Fourier transformation.
Simulation results
To illustrate the validity and effectiveness of the PI method to solve the one-dimensional space fractional FPK equation, three kinds of examples are presented in this section. Next, the function in the space fractional FPK equation (6) is selected as zero, linear and nonlinear function, respectively.
Conclusions
For the space fractional FPK equation, corresponding to the SDE excited by α-stable Lévy white noise, the exact solution is known only for rare cases and usually numerical methods are used. In view of the recent development of the PI method in the non-Gaussian noise case, we extended the PI method to solve a class of one-dimensional space fractional FPK equations in this paper. The transition PDF is derived theoretically and used in the CKS equation to obtain the PI solution. Then the accuracy
Acknowledgements
This work was supported by the NSF of China (11572234), Shaanxi Province Project for Distinguished Young Scholars (2018JC-010), the Fundamental Research Funds for the Central Universities and Seed Foundation of Innovation and Creation for Graduate Students of NPU (Z2017189). Y.X. thanks to the Alexander von Humboldt Foundation.
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