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 $$L^2$$ 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 $$C^{\frac{1}{2}+}$$ 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.
We review recent developments of slow/fast stochastic differential equations, and also present a new result on Diffusion Homogenisation Theory with fractional and non-strong-mixing noise and providing new examples. The emphasise of the review will be on the recently developed effective dynamic theory for two scale random systems with fractional noise: Stochastic Averaging and 'Rough Diffusion Homogenisation Theory'. We also study the geometric models with perturbations to symmetries.
We prove an enhanced limit theorem for additive functionals of a multi-dimensional Volterra process ( y t ) t ⩾ 0 in the rough path topology. As an application, we establish weak convergence as ɛ → 0 of the solution of the random ordinary differential equation (ODE) d d t x t ε = 1 ε f ( x t ε , y t ε ) and show that its limit solves a rough differential equation driven by a Gaussian field with a drift coming from the Lévy area correction of the limiting rough driver. Furthermore, we prove that the stochastic flows of the random ODE converge to those of the Kunita type Itô SDE dx t = G(x t , dt), where G(x, t) is a semi-martingale with spatial parameters.
We aim to obtain a homogenization theorem for a passive tracer interacting with a fractional, possibly non-Gaussian, noise. To do so, we analyze limit theorems for normalized functionals of Hermite–Volterra processes and extend existing results to cover power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short- and long-range dependent components. Applying this theorem, we obtain a homogenization result for a slow/fast system driven by such Hermite noises.
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