Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

Johann Gehringer () and Xue-Mei Li ()
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Johann Gehringer: Imperial College
Xue-Mei Li: Imperial College

Journal of Theoretical Probability, 2022, vol. 35, issue 1, 426-456

Abstract: 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 $$L^2$$ 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.

Keywords: Passive tracer; Fractional noise; Multi-scale; Mixed functional central and non-central limit theorems; Rough creation; Rough homogenization; Rough topology; 34F05; 60F05; 60F17; 60G18; 60G22; 60H05; 60H07; 60H10 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-020-01044-7

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