We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.

中文翻译：

---

Koopman 算子基于核的近似的误差界限

我们考虑再生核希尔伯特空间（RKHS）上随机微分方程的库普曼算子的数据驱动近似。如果数据是从长期遍历模拟中收集的，我们的重点是估计误差。我们推导了以希尔伯特-施密特范数测量的核互协方差算子方差的精确表达式，以及有限数据估计误差的概率界限。此外，我们使用有限 Mercer 级数展开推导了 RKHS 中可观测量预测误差的界限。此外，假设 RKHS 具有库普曼不变性，我们提供了完全近似误差的界限。使用 Ornstein-Uhlenbeck 过程的数值实验说明了我们的结果。