Abstract
In this paper, the probability density functions (PDFs) of scalar stochastic differential equations (SDEs) subject to α-stable Lévy white noise are investigated. The path integral (PI) method is extended to solve one-dimensional space fractional Fokker-Planck-Kolmogorov (FPK) equations, which are the governing equations corresponded to scalar SDEs excited by α-stable Lévy white noise. First, we derive a short time solution of the one-dimensional space fractional FPK equation, which is used in the Chapman-Kolmogorov-Smoluchowski (CKS) equation to obtain the PI solution. Then, the accuracy of the PI solution is analyzed theoretically in terms of its characteristic function. Our results demonstrate that the PI method has a higher accuracy than the first order finite difference method for one step iteration in time. Finally, several illustrative examples are carried out in detail to verify the feasibility and effectiveness of the PI method for solving one-dimensional space fractional FPK equations. We find that the PI solution agrees well with the exact solution or the Monte Carlo one.
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