We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by both long- and short-range-dependent noise. The limit process has both Gaussian and non-Gaussian components. The theorem holds for any \(L^2\) functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology, the rough topology for processes in \(C^{\frac{1}{2}+}\). This leads to a ‘rough creation’ / ‘rough homogenization’ theorem, by which we mean the weak convergence of a family of random smooth curves to a non-Markovian random process with non-differentiable sample paths. In particular, we obtain effective dynamics for the second-order problem and for the kinetic fractional Brownian motion model.

我们证明了分数 Ornstein-Uhlenbeck 过程的向量值泛函的泛函极限定理，为由长程和短程相关噪声驱动的慢/快系统的波动理论提供了基础。极限过程同时具有高斯和非高斯分量。该定理适用于任何\(L^2\)函数，而对于具有更强可积性的函数，收敛性显示在 Hölder 拓扑中，即\(C^{\frac{1}{2中的过程的粗略拓扑}+}\). 这导致了“粗略创建”/“粗略同质化”定理，我们指的是随机平滑曲线族与具有不可微分样本路径的非马尔可夫随机过程的弱收敛。特别是，我们获得了二阶问题和动力学分数布朗运动模型的有效动力学。