Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels

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.

💡 Research Summary

This paper presents a rigorous mathematical investigation into the prediction error of stochastic dynamical systems characterized by memory effects, specifically focusing on Generalized Langevin Equations (GLEs). The fundamental challenge addressed is the impact of approximation errors in the memory kernel—a critical component in describing non-Markovian dynamics—on the accuracy of trajectory predictions. Since the exact memory kernel is often inaccessible in complex physical or biological systems, researchers must rely on estimated or approximated kernels, making the quantification of resulting trajectory discrepancies a vital necessity.

The authors formulate the GLEs as stochastic Volterra equations, providing a robust framework to analyze how the decay properties of the memory kernel influence the stability of the system. A significant contribution of this work is the establishment of a quantitative link between the estimation error of the kernel, measured in a weighted norm, and the divergence of the system’s trajectories. The study demonstrates that the decay rate of the trajectory discrepancy is intrinsically tied to the decay rate of the memory kernel itself, covering both exponential and subexponential decay classes.

The analytical approach is bifurcated into first-order and second-order models to address varying levels of dynamical complexity. For first-order models, the authors utilize resolvent estimates in weighted spaces to derive precise moment and perturbation bounds. This ensures that the error in the kernel does not lead to unbounded divergence in the system’s statistical properties. For second-order models, which involve more complex kinetic-like structures with confining potentials, the authors employ a sophisticated hypocoercive Lyapunov-type distance. This technique allows them to prove that the system maintains contraction and stability properties even under kernel perturbations, effectively showing that the structural integrity of the dynamics is preserved despite the approximation.

Furthermore, the framework is generalized to accommodate non-translation-invariant kernels and white-noise forcing, making the findings applicable to a wide array of non-homogeneous stochastic processes. The paper concludes with numerical simulations that validate the theoretical bounds, reinforcing the core message: improvements in the precision of kernel estimation directly translate into enhanced reliability in long-term trajectory prediction. This research provides essential theoretical foundations for researchers working in molecular dynamics, statistical mechanics, and any field where data-driven modeling of memory-dependent stochastic systems is required.