Functional limit theorems for the fractional Ornstein-Uhlenbeck process

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 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