发表机构
Huazhong University of Science and Technology; China University of Geosciences (Wuhan)（华中科技大学; 中国地质大学（武汉））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对Lévy噪声下广义Langevin方程，利用Markov嵌入和条件矩统计量估计记忆核与随机强迫，并给出误差界与数值验证。
AI 中文摘要
我们考虑一维广义Langevin方程中记忆与随机强迫的识别问题，该方程具有状态依赖的Brownian噪声和可选的加性对称$\alpha$-稳定跳跃。有限维Markov嵌入提供了Kramers--Moyal条件矩关系，将记忆漂移、Brownian扩散和跳跃贡献分离。基于观测速度历史条件下的增量系综，截断统计量估计稳定参数，修正的二阶矩确定Brownian振幅，一阶矩导出记忆核的Volterra方程，并通过RKHS正则化求解。我们建立了矩估计器的条件有限样本界，以及具有数据选择正则化参数的核恢复的条件$L^2$误差界。数值实验考察了不同力、记忆核、观测长度、测量噪声和近似历史条件下的Brownian和Brownian--Lévy模型。
英文摘要
We consider the identification of memory and stochastic forcing in a one-dimensional generalized Langevin equation with state-dependent Brownian noise and optional additive symmetric $α$-stable jumps. A finite-dimensional Markov embedding provides Kramers--Moyal conditional moment relations that separate memory drift, Brownian diffusion, and jump contributions. From increment ensembles conditioned on the observed velocity history, truncated statistics estimate the stable parameters, corrected second moments determine the Brownian amplitude, and first moments lead to a Volterra equation for the memory kernel, solved by RKHS regularization. We establish conditional finite-sample bounds for the moment estimators and a conditional $L^2$ error bound for kernel recovery with a data-selected regularization parameter. Numerical experiments examine Brownian and Brownian--Lévy models under different forces, memory kernels, observation lengths, measurement noise, and approximate history conditioning.
