Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels
This work addresses error analysis for stochastic systems with memory, which is incremental as it builds on existing GLE frameworks to provide theoretical bounds.
The paper tackles prediction error in stochastic dynamical systems with memory using generalized Langevin equations, establishing that trajectory discrepancies decay based on memory kernel decay and are bounded by kernel estimation error, with numerical validation of these theoretical results.
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.