论文标题
分数Ornstein-Uhlenbeck过程的功能极限定理
Functional limit theorems for the fractional Ornstein-Uhlenbeck process
论文作者
论文摘要
我们证明了分数Ornstein-uhlenbeck过程的矢量值函数的功能极限定理，为慢速/快速系统的波动理论提供了基础。我们的主要贡献是与高斯和非高斯组成部分的关节收敛限制。这对于任何$ l^2 $函数都是有效的，而对于具有更强的集成性属性的功能，融合被证明在Hölder拓扑中保留。作为应用程序，我们证明了一个“粗糙创造”结果，即随机平滑曲线家族与具有粗糙样品路径的非马克维亚随机过程的弱收敛性。这包括二阶问题和动力学分数布朗运动模型。
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 such a noise. Our main contribution is on the joint convergence to a limit with both Gaussian and non-Gaussian components. This is valid for any $L^2$ functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology. As an application we prove a `rough creation' result, i.e. the weak convergence of a family of random smooth curves to a non-Markovian random process with rough sample paths. This includes the second order problem and the kinetic fractional Brownian motion model.
