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.

翻译：我们分析了具有记忆效应的随机动力系统的预测误差，重点关注以随机Volterra方程形式表述的广义朗之万方程。我们证明，在强凸势条件下，轨迹偏差的衰减速率由记忆核的衰减速率决定，并在加权范数下被核的估计误差定量界定。我们的分析将同步噪声耦合与Volterra比较定理相结合，涵盖了亚指数和指数两类核函数。对于一阶模型，我们利用加权空间中的预解估计推导了矩界和扰动界。对于具有约束势的二阶模型，我们通过亚强制李雅普诺夫型距离证明了核扰动下的收缩性和稳定性。该框架适用于非平移不变核和白噪声激励，明确将改进的核估计与提升的轨迹预测联系起来。数值算例验证了这些理论结果。