Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

• Published in: Theoretical and applied mechanics letters Vol. 13; no. 3; p. 100436
• Main Authors: Luo, Yi, Lyu, Meng-Ze, Chen, Jian-Bing, Spanos, Pol D.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.05.2023; Elsevier
• Subjects: Analytical intrinsic drift coefficient; Dimension reduction; Globally-evolving-based generalized density evolution equation (GE-GDEE); Linear fractional differential system; Non-Markovian system; Analytical intrinsic drift coefficient; Dimension reduction; Linear fractional differential system; Non-Markovian system; Globally-evolving-based generalized density evolution equation (GE-GDEE)
• ISSN: 2095-0349
• DOI: 10.1016/j.taml.2023.100436

•An exact low-dimensional partial differential equation with analytical coefficients governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.•The remarkable demonstration on the existence and eligibility of the globally-evolving-based generalized density evolution equation (GE-GDEE) is provided for the case that the original high-dimensional stochastic differential system itself is non-Markovian.•The insights for the physical-mechanism-informed determination of the intrinsic drift coefficient of GE-GDEE is discussed. Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.