Abstract
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.
Highlights
The functional limit theorem we study here lays the foundation for the fluctuation problem for a slow/fast system with the fast variable given by a family of non-strong mixing stochastic processes, this will be discussed in [19], see [18] for the preliminary
A pivot theorem for obtaining effective dynamics for the slow moving particles in a fast turbulent environment are scaling limit theorems for the convergence of the following functionals
There exists no vector valued functional limit theorem with joint convergence, when the scaling limit of the components are mixed, in this article we provide a complete description for the joint convergence of {X k,ε} for Gk ∈ L2(μ)
Summary
The functional limit theorem we study here lays the foundation for the fluctuation problem for a slow/fast system with the fast variable given by a family of non-strong mixing stochastic processes, this will be discussed in [19], see [18] for the preliminary. They obtained uniform convergences in the continuous topology for a restrictive class of functions G (assuming sufficiently fast decay of the coefficients in the Wiener chaos expansion) This was extended in [30] to vector valued X ε, when each component of X ε falls in the Brownian case, with convergence understood in the sense of finite dimensional distributions. For Gk satisfying a stronger integrability condition, we can show weak convergence in the Cα([0, T ], Rd )-topology and for each fixed time in L2 for the low Hermite rank case, which already have interesting applications They are the basis for the convergence in a suitable rough topology, which due to the change of the nature of the problem will appear in [18] where rough path theory is used to study slow/fast systems, leading to ‘rough creation’ / ‘rough homogenization’ in which the effective limit is not necessarily a Markov process. Fε → U and Gε → V weakly imply that (Fε, Gε) → (U , V ) jointly, where U and V are taken to be independent random variables
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