Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels

TL;DR

This paper provides a comprehensive error analysis for generalized Langevin equations with approximated memory kernels, linking kernel estimation accuracy to trajectory prediction quality in stochastic systems with memory.

Contribution

It introduces a unified analytical framework for error bounds in GLEs with various kernel classes, including non-translation-invariant kernels, and demonstrates the impact of kernel estimation on prediction accuracy.

Findings

Trajectory errors decay with kernel decay rate.

Kernel estimation errors directly affect prediction accuracy.

Numerical examples confirm theoretical bounds.

Abstract

We analyze prediction error in stochastic dynamical systems with memory, focusing on generalized Langevin equations (GLEs) formulated as stochastic Volterra equations. We establish that, under a strongly convex potential, trajectory discrepancies decay at a rate determined by the decay of the memory kernel and are quantitatively bounded by the estimation error of the kernel in a weighted norm. Our analysis integrates synchronized noise coupling with a Volterra comparison theorem, encompassing both subexponential and exponential kernel classes. For first-order models, we derive moment and perturbation bounds using resolvent estimates in weighted spaces. For second-order models with confining potentials, we prove contraction and stability under kernel perturbations using a hypocoercive Lyapunov-type distance. This framework accommodates non-translation-invariant kernels and white-noise forcing, explicitly linking improved kernel estimation to enhanced trajectory prediction. Numerical examples validate these theoretical findings.