2025/10/23 by Feng, Lifang, Pei, Bin, Xu, Yong
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2510.20124
This paper investigates the probabilistic responses of nonlinear single-degree-of-freedom oscillators under fractional Gaussian noise (FGN) excitation. Unlike Gaussian white noise, FGN exhibits persistent correlations and memory effects, making conventional Fokker-Planck-Kolmogorov (FPK) equation methods inapplicable. To address this, we develop memory-dependent FPK (memFPK) equations based on fractional Wick-Ito-Skorohod calculus, capable of capturing the joint probability of system responses. A discretized local mean method (DLMM) is proposed to estimate the memory-dependent diffusion coefficient, and a finite difference scheme solves the memFPK equation numerically. Validation through linear and nonlinear examples shows excellent agreement with analytical or Monte Carlo solutions. This framework provides a practical tool for analyzing non-Markovian stochastic dynamics, with potential extensions to multidimensional and parametric FGN problems.