arXiv subjects

Lifang Feng

Publications and source records attributed to Lifang Feng.

4 recordsLinked to original sources

arXiv2025

The memory-dependent FPK equation for fractional Gaussian noise

This paper aims to explore non-Markovian dynamics of nonlinear dynamical systems subjected to fractional Gaussian noise (FGN) and Gaussian white noise (GWN). A novel memory-dependent Fokker-Planck-Kolmogorov (memFPK) equation is developed to characterize the probability structure in such non-Markovian systems. The main challenge in this research comes from the long-memory characteristics of FGN. These features make it impossible to model the FGN-excited nonlinear dynamical systems as finite dimensional GWN-driven Markovian augmented filtering systems, so the classical FPK equation is no longer applicable. To solve this problem, based on fractional Wick-It\^o-Skorohod integral theory, this study first derives the fractional It\^o formula. Then, a memory kernel function is constructed to reflect the long-memory characteristics from FGN. By using fractional It\^o formula and integration by parts, the memFPK equation is established. {Importantly, the proposed memFPK equation is not limited to specific forms of drift and diffusion terms, making it broadly applicable to a wide class of nonlinear dynamical systems subjected to FGN and GWN.} Due to the historical dependence of the memory kernel function, a Volterra adjustable decoupling approximation is used to reconstruct the memory kernel dependence term. This approximation method can effectively solve the memFPK equation, thereby obtaining probabilistic responses of nonlinear dynamical systems subjected to FGN and GWN excitations. Finally, some numerical examples verify the accuracy and effectiveness of the proposed method.

math.PR↗

arXiv2025

Memory-Dependent FPK Equations for Nonlinear SDOF Oscillators Under Fractional Gaussian Noise Excitation

This paper investigates the transient probabilistic responses of nonlinear single-degree-of-freedom oscillators subjected to external fractional Gaussian noise (FGN) excitation. Owing to the inherent long-range correlations and memory characteristics of FGN, the resulting response process exhibits non-Markovian properties, rendering the classical Fokker-Planck-Kolmogorov (FPK) equation method inapplicable in its direct form. To overcome this critical challenge, a memory-dependent FPK (memFPK) equation is formulated for two-dimensional nonlinear stochastic systems within the fractional Wick-It\^o-Skorohod integral framework. The derived memFPK equation incorporates mixed second-order derivative terms, as well as time-dependent and state-dependent diffusion coefficients, which inherently capture the long-range correlations and memory effects induced by FGN excitation. For the numerical solution of the memFPK equation, a discretized local mean treatment is developed to estimate the memory-dependent diffusion coefficients involving conditional expectations. The proposed approach integrates local statistical averaging and smoothing techniques to enhance the stability of coefficient estimation. Subsequently, the memFPK equation is numerically solved using a finite difference scheme. The accuracy and effectiveness of the proposed framework are validated through linear and nonlinear numerical examples. Comparative results demonstrate the excellent agreement with analytical solutions or Monte Carlo simulations in terms of transient joint probability density functions (PDFs), marginal PDFs, low-probability tail regions, and statistical moments. These findings confirm that the proposed memFPK equation method serves as a robust and effective tool for analyzing the transient non-Markovian probabilistic responses of nonlinear SDOF systems under FGN excitation.

math.PR↗

arXiv2024

Non-Markovian dynamics: the memory-dependent probability density evolution equations

This paper aims to investigate the non-Markovian dynamics. The governing equations are derived for the probability density functions (PDFs) of non-Markovian stochastic responses to Langevin equation excited by combined fractional Gaussian noise (FGN) and Gaussian white noise (GWN). The main difficulty here is that the Langevin equation excited by FGN cannot be augmented by a filter excited by GWN, leading to the inapplicability of It\^o stochastic calculus theory. Thus, in the present work, based on the fractional Wick It\^o Skorohod integral and rough path theory, a new non-Markovian probability density evolution method is established to derive theoretically the memory-dependent probability density evolution equation (PDEEs) for the PDFs of non-Markovian stochastic responses to Langevin equation excited by combined FGN and GWN, which is a breakthrough to stochastic dynamics. Then, we extend an efficient algorithm, the local discontinuous Galerkin method, to numerically solve the memory-dependent PDEEs. Remarkably, this proposed method attains a higher accuracy compared to the prevalent methods such as finite difference, path integral (PI) and Monte Carlo methods, and boasts a broader applicability than the PI method, which fails to solve the memory-dependent PDEEs. Finally, several numerical examples are illustrated to verify the proposed scheme.

math.PR↗

arXiv2023

Averaging principle for McKean-Vlasov SDEs driven by multiplicative fractional noise with highly oscillatory drift coefficient

In this paper, we study averaging principle for a class of McKean-Vlasov stochastic differential equations (SDEs) that contain multiplicative fractional noise with Hurst parameter $H > $ 1/2 and highly oscillatory drift coefficient. Here the integral corresponding to fractional Brownian motion is the generalized Riemann-Stieltjes integral. Using Khasminskii's time discretization techniques, we prove that the solution of the original system strongly converges to the solution of averaging system as the times scale $ \epsilon $ gose to zero in the supremum- and H\"older-topologies which are sharpen existing ones in the classical Mckean-Vlasov SDEs framework.

math.PR↗