The memory-dependent FPK equation for fractional Gaussian noise

TL;DR

This paper introduces a novel memory-dependent Fokker-Planck-Kolmogorov equation to model non-Markovian nonlinear systems driven by fractional Gaussian noise, enabling probabilistic analysis of systems with long-memory effects.

Contribution

It develops a broadly applicable memFPK equation using fractional Itô calculus and a Volterra approximation to handle long-memory FGN effects in nonlinear dynamical systems.

Findings

The memFPK equation accurately models systems with FGN.

Numerical examples confirm the method's effectiveness.

The approach is applicable to a wide class of nonlinear systems.

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systems as finite dimensional GWN-driven Markovian augmented filtering systems, so the classical FPK equation is no longer applicable.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