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The memory-dependent FPK equation for fractional Gaussian noise

Lifang Feng, Bin Pei, Yong Xu

TL;DR

The paper tackles non-Markovian stochastic dynamics driven by fractional Gaussian noise and Gaussian white noise, where classical Fokker-Planck-Kolmogorov theory fails due to long-memory effects. It develops a memory-dependent FPK equation (memFPK) using fractional Ito calculus and FWIS integration, introducing a Malliavin-based memory kernel that encodes historical state information via a function Psi. By applying a Lamperti transform and a Volterra adjustable decoupling approximation (VADA) to handle memory terms, the memFPK provides drift and diffusion coefficients that capture memory in a general nonlinear setting. Numerical results across four nonlinear systems (OU, Duffing, Verhulst, and a two-DOF Hamiltonian system) demonstrate that the memFPK predictions closely match analytical or Monte Carlo references, validating the approach and its potential for uncertainty quantification in non-Markovian dynamics.

Abstract

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systems as finite dimensional GWN-driven Markovian augmented filtering systems, so the classical FPK equation is no longer applicable.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.

The memory-dependent FPK equation for fractional Gaussian noise

TL;DR

The paper tackles non-Markovian stochastic dynamics driven by fractional Gaussian noise and Gaussian white noise, where classical Fokker-Planck-Kolmogorov theory fails due to long-memory effects. It develops a memory-dependent FPK equation (memFPK) using fractional Ito calculus and FWIS integration, introducing a Malliavin-based memory kernel that encodes historical state information via a function Psi. By applying a Lamperti transform and a Volterra adjustable decoupling approximation (VADA) to handle memory terms, the memFPK provides drift and diffusion coefficients that capture memory in a general nonlinear setting. Numerical results across four nonlinear systems (OU, Duffing, Verhulst, and a two-DOF Hamiltonian system) demonstrate that the memFPK predictions closely match analytical or Monte Carlo references, validating the approach and its potential for uncertainty quantification in non-Markovian dynamics.

Abstract

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.

Paper Structure

This paper contains 14 sections, 3 theorems, 73 equations, 18 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem setting and main result
  3. Derivation for the memFPK equation
  4. Overview of FPK equation based on Itô stochastic calculus theory
  5. Step-by-step procedure of the proposed method
  6. Numerical results
  7. Ornstein-Uhlenbeck process
  8. Duffing oscillator
  9. Verhulst model
  10. Two-DOF quasi-non-integrable Hamiltonian system
  11. Discussions and conclusions
  12. Some useful Lemmas
  13. Derivation of Eq. (\ref{['Malfbmx-1']})
  14. The memory kernel dependence term under approximation

Key Result

Theorem 1

Considering Eq. (mixSDE) with the initial condition $X_0=x_0$ which is a constant or random variable independent of FGN and GWN with known distribution. Suppose that the function $f(\cdot)$ is differentiable, while the functions $g(\cdot)$ and $h(\cdot)$ are twice-differentiable. Additionally, assum where with where $\mathrm{d}W_t$ denotes the Itô integral with respect to Brownian motion (BM) $W

Figures (18)

  • Figure 1: Flowchart of the proposed method.
  • Figure 2: Comparison of the PDF surface via memFPK equation.
  • Figure 3: Comparison of the PDF surface via memFPK equation.
  • Figure 4: Comparison of the transient PDFs via memFPK equation.
  • Figure 5: Comparison of the transient CDFs via memFPK equation.
  • ...and 13 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Remark 2
  • Lemma A.3
  • Lemma A.4
  • Remark C.5