Functional limit theorems for the fractional Ornstein-Uhlenbeck process
File(s)
Author(s)
Gehringer, Johann
Li, Xue-Mei
Type
Journal Article
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
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.
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.
Date Issued
2022-03-01
Date Acceptance
2020-09-28
Citation
Journal of Theoretical Probability, 2022, 35, pp.426-456
ISSN
0894-9840
Publisher
Springer
Start Page
426
End Page
456
Journal / Book Title
Journal of Theoretical Probability
Volume
35
Copyright Statement
© The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
License URL
Sponsor
Engineering and Physical Sciences Research Council
Engineering & Physical Science Research Council (EPSRC)
Grant Number
EP/V026100/1
EP/V026100/1
Subjects
passive tracer
fractional noise
multi-scale
mixed functional central and non-central limit theorems
rough creation
rough homogenization
rough topology
Publication Status
Published
Date Publish Online
2020-10-22