研究带记忆项的随机系统预测误差，给出误差与记忆核估计精度的关系。
Error Analysis of Generalized Langevin Equations with Approximated Memory Kernels
- 通过耦合噪声与伏尔泰拉比较定理分析误差
- 误差随记忆核衰减速率下降，且受核估计误差约束
- 适用于非平移不变核和白噪声，对模型改进有指导意义
我们研究具有记忆的随机动力系统中的预测误差，聚焦于作为随机伏尔泰拉方程的广义朗之万方程（GLE）。在强凸势能条件下，轨迹偏差的衰减速率由记忆核的衰减特性决定，并在加权范数下被核估计误差定量控制。分析结合了同步噪声耦合与伏尔泰拉比较定理，涵盖亚指数与指数型核类。对于一阶模型，利用加权空间中的再生核估计导出矩与扰动界；对于具有束缚势的二阶模型，通过拟柯西型李雅普诺夫距离证明在核扰动下的压缩性与稳定性。该框架可处理非平移不变核与白噪声驱动，明确揭示更优的核估计可提升轨迹预测性能。数值实验验证了理论结果。
原文摘要 · Abstract (English)
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.
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