arXiv subjects

Bin Pei

Publications and source records attributed to Bin Pei.

18 recordsLinked to original sources

arXiv2026

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-Itô-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↗

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-Itô-Skorohod integral theory, this study first derives the fractional Itô formula. Then, a memory kernel function is constructed to reflect the long-memory characteristics from FGN. By using fractional Itô 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

Averaging principle for slow-fast systems of PDEs with rough drivers

This paper investigates a class of slow--fast systems of rough partial differential equations defined over a monotone family of interpolation Hilbert spaces. By employing the controlled rough path framework tailored to a monotone family of interpolation spaces, together with a time discretization argument, we demonstrate that the slow component strongly converges to the solution of the averaged system in the supremum norm as the time-scale parameter $\varepsilon$ tends to $0$.

math.PR↗

arXiv2025

Large deviation principle for slow-fast systems with infinite-dimensional mixed fractional Brownian motion

This work is concerned with the large deviation principle for a family of slow-fast systems perturbed by infinite-dimensional mixed fractional Brownian motion with Hurst parameter $H\in(\frac12,1)$. We adopt the weak convergence method which is based on the variational representation formula for infinite-dimensional mixed fractional Brownian motion. To obtain the weak convergence of the controlled systems, we apply the Khasminskii's averaging principle and the time discretization technique. In addition, we drop the boundedness assumption of the drift coefficients of the slow components and the diffusion coefficients of the fast components.Based on the proof of the large deviation principle, we also establish the moderate deviation principle for the slow-fast systems.

math.PR↗

arXiv2025

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 Itô stochastic calculus theory. Thus, in the present work, based on the fractional Wick Itô 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↗

arXiv2024

Averaging principle for semilinear slow-fast rough partial differential equations

In this paper, we investigate the averaging principle for a class of semilinear slow-fast partial differential equations driven by finite-dimensional rough multiplicative noise. Specifically, the slow component is driven by a general random $γ$-Hölder rough path for some $γ\in (1/3,1/2)$, while the fast component is driven by a Brownian rough path. Using controlled rough path theory and the classical Khasminskii's time discretization scheme, we demonstrate that the slow component converges strongly to the solution of the corresponding averaged equation under the Hölder topology.

math.PR↗

arXiv2024

Convergence of martingale solutions to the hybrid slow-fast system

This paper is devoted to studying the weak convergence for a slow-fast system with jumps modulated by Markovian switching regimes with the martingale method. However, due to the coexistence of fast component and Markovian switching regimes, the martingale method and perturbed test functions can not be applied directly. In this situation, a combination of perturbed test functions and the time discretization is applied efficiently. And the choice of appropriate perturbed test functions, which are related to the averaged coefficients, plays a decisive role. Our results also cover the case of slow-fast system without Markovian switching regimes. Finally, some examples are presented,and numerical simulations are carried out to observe a good agreement.

math.DS↗

arXiv2023

Stochastic averaging for non-Lipschitz multi-valued stochastic differential equations driven by G-Brownian motion

In this paper, we prove the validity of an averaging principle for multi-valued stochastic differential equations (MSDEs) driven by G-Brownian motion with non-Lipschitz coefficients. The convergence theorem between the solution of the averaged MSDEs and original one was obtained in the sense of p-th moments and also in capicity. Finally, one example is presented to illustrate our theory.

math.PR↗

arXiv2023

Almost Sure Averaging for Fast-slow Stochastic Differential Equations via Controlled Rough Path

This paper establishes the averaging method to a coupled system consisting of two stochastic differential equations which has a slow component driven by fractional Brownian motion (FBM) with less regularity $1/3< H \leq 1/2$ and a fast dynamics under additive FBM with Hurst-index $1/3< \hat H \leq 1/2$. We prove that the solution of the slow component converges almost surely to the solution of the corresponding averaged equation using the approach of time discretization and controlled rough path. To do this, we employ the random dynamical system (RDS) to obtain a stationary solution by an exponentially attracting random fixed point of the RDS generated by the non-Markovian fast component.

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 $ ε$ gose to zero in the supremum- and Hölder-topologies which are sharpen existing ones in the classical Mckean-Vlasov SDEs framework.

math.PR↗

arXiv2023

Almost Sure Averaging for Evolution Equations driven by fractional Brownian motions

We apply the averaging method to a coupled system consisting of two evolution equations which has a slow component driven by fractional Brownian motion (FBM) with the Hurst parameter $H_1> \frac12$ and a fast component driven by additive FBM with the Hurst parameter $ H_2\in(1-H_1,1)$. The main purpose is to show that the slow component of such a couple system can be described by a stochastic evolution equation with averaged coefficients. Our first result provides a pathwise mild solution for the system of mixed stochastic evolution equations. Our main result deals with an averaging procedure which proves that the slow component converges almost surely to the solution of the corresponding averaged equation using the approach of time discretization. To do this we generate a stationary solution by a exponentially attracting random fixed point of the random dynamical system generated by the fast component.

math.PR↗

arXiv2023

Averaging principle for fast-slow system driven by mixed fractional Brownian rough path

This paper is devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index $H ~(1/3 < H\leq 1/2)$. Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the $L^1$-sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new.

math.PR↗

arXiv2022

Precise Laplace approximation for mixed rough differential equation

This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path with as . Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we imbed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale.

math.PR↗

arXiv2021

Pathwise unique solutions and stochastic averaging for mixed stochastic partial differential equations driven by fractional Brownian motion and Brownian motion

This paper is devoted to a system of stochastic partial differential equations (SPDEs) that have a slow component driven by fractional Brownian motion (fBm) with the Hurst parameter $H >1/2$ and a fast component driven by fast-varying diffusion. It improves previous work in two aspects: Firstly, using a stopping time technique and an approximation of the fBm, we prove an existence and uniqueness theorem for a class of mixed SPDEs driven by both fBm and Brownian motion; Secondly, an averaging principle in the mean square sense for SPDEs driven by fBm subject to an additional fast-varying diffusion process is established. To carry out these improvements, we combine the pathwise approach based on the generalized Stieltjes integration theory with the Itô stochastic calculus. Then, we obtain a desired limit process of the slow component which strongly relies on an invariant measure of the fast-varying diffusion process.

math.PR↗

arXiv2021

L^p(p>2)-strong convergence in stochastic averaging principle for two time-scales stochastic evolution equations driven by Lévy process

The main goal of the work is to study the stochastic averaging principle for two time-scales stochastic evolution equations driven by Lévy process. The solution of reduced equation with modified coefficient is derived to approximate the slow component of original equation under suitable condition. It is shown that the slow component can strongly converge to the solution of corresponding reduced equation in L^p(p>2)-strong convergence sense.Our key and novelty is how to cope with the changes caused by Lévy process and higher order moments.

math.DS↗

arXiv2021

Averaging Principles for Mixed Fast-Slow Systems Driven by Fractional Brownian Motion

We focus on fast-slow systems involving both fractional Brownian motion (fBm) and standard Brownian motion (Bm). The integral with respect to Bm is the standard Ito integral, and the integral with respect to fBm is the generalised Riemann-Stieltjes integral using the tools of fractional calculus. An averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a noise to be averaged out in the limit is established. It is shown that the slow process has a limit in the mean square sense, which is characterized by the solution of stochastic differential equations driven by fBm whose coefficients are averaged with respect to the stationary measure of the fast-varying diffusion. The implication is that one can ignore the complex original systems and concentrate on the averaged systems instead. This averaging principle paves the way for reduction of computational complexity.

math.DS↗

arXiv2020

Positivity of the density for rough differential equations

Due to recent developments of Malliavin calculus for rough differential equations, it is now known that, under natural assumptions, the law of a unique solution at a fixed time has a smooth density function. Therefore, it is quite natural to ask whether or when the density is strictly positive. In this paper we study this problem from the viewpoint of Aida-Kusuoka-Stroock's general theory.

math.PR↗

arXiv2019

Averaging principles for non-autonomous two-time-scale stochastic reaction-diffusion equations with polynomial growth

In this paper, we develop the averaging principle for a class of two-time-scale stochastic reaction-diffusion equations driven by Wiener processes and Poisson random measures. We assume that all coefficients of the equation have polynomial growth, and the drift term of the equation is non-Lipschitz. Hence, the classical formulation of the averaging principle under the Lipschitz condition is no longer available. To prove the validity of the averaging principle, the existence and uniqueness of the mild solution are proved firstly. Then, the existence of time-dependent evolution family of measures associated with the fast equation is studied, by which the averaged coefficient is obtained. Finally, the validity of the averaging principle is verified.

math.DS↗