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

TL;DR

The paper tackles non-Markovian Langevin dynamics driven by a combination of fractional Gaussian noise and Gaussian white noise by deriving memory-dependent PDEEs for the associated PDFs. It builds a non-Markovian probability density evolution method using the fractional Wick It\ô Skorohod integral and rough path theory to obtain pathwise solutions, and develops a local discontinuous Galerkin scheme to solve these PDEEs with high accuracy. The authors derive linear and nonlinear cases, demonstrate the LDG method achieves superior precision compared with finite difference, path integral, and Monte Carlo methods, and provide extensive numerical validations, including exact solutions in special cases. This approach offers a robust framework for uncertainty quantification in non-Markovian stochastic systems and extends numerical capabilities beyond traditional Markovian methods.

Abstract

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.

Figures (7)

• Figure 1: PDF evolution and stationary PDF of the FPK equation (\ref{['prob-exac-1']}) where $\sigma=1$ and $a=b=1$ in (a) and (b); $a=-1, b=0$ in (c) and (d).
• Figure 2: PDF evolution (a) of the FPK equation (\ref{['ldg1133']}), the absolute errors (b) between the exact solutions and the LDG solutions, and PDF evolution surface of LDG solutions (c) and the exact solutions (d) with $a=0.02, b=0.3, x_0=2$.
• Figure 3: PDF evolution of the memory-dependent PDEE (\ref{['ldg1139']}): (a) the comparisons among MC and LDG solutions with $a=-0.5, b=0.25, c=0.25$; (b) the comparisons among exact and LDG solutions with $a=-0.5, b=0, c=0.5$.
• Figure 4: PDF evolution of the memory-dependent PDEE (\ref{['ldg1136']}) with $a=-0.25, b=0.25, c=0.25, d=0.8$.
• Figure 5: The comparisons between MC and LDG solutions (a) of the memory-dependent PDEE (\ref{['ldg11310']}) and PDF evolution surface of the LDG solutions (b) with $a=-1, b=c=d=0.5, H=0.8, x_0=0.4$.

Theorems & Definitions (26)

• Definition 2.1: ducan2000stochastic
• Lemma 2.2: ducan2000stochastic
• Remark 1
• Lemma 2.3: ducan2000stochastic
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 2.6
• proof