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Functional correlation bounds and deterministic homogenisation of fast-slow systems
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Fleming, Nicholas (2023) Functional correlation bounds and deterministic homogenisation of fast-slow systems. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3957267
Abstract
The main topic of this thesis is the problem of proving homogenisation (convergence to a stochastic differential equation) for fast-slow systems with deterministic fast direction. Kelly & Melbourne used rough path theory to show that this problem reduces to verifying certain statistical properties for the fast dynamics.
It is natural to consider the case where the fast dynamics is given by a nonuniformly hyperbolic diffeomorphism in the sense of L.-S. Young. Indeed, this class covers a wide variety of examples such as dispersing billiards, intermittent maps and H´enon attractors. Moreover, our understanding of the statistical properties of this class is generally very complete. However, until now it was not possible to verify one of the statistical properties required to apply rough path theory for certain nonuniformly maps with slow decay of correlations.
Our first result is that this property is satisfied by nonuniformly hyperbolic maps in the sense of Young under optimal assumptions on the rate of decay of correlations. The proof splits in two steps. First we prove that the map satisfies a condition introduced by Lepp¨anen which we call the Functional Correlation Bound. Then we use a weak dependence argument to show that the required property follows from the Functional Correlation Bound.
Our second main result is that the Functional Correlation Bound is in fact a sufficient condition for homogenisation. Since the Functional Correlation Bound is an elementary condition that is easy to write down, this could be useful for nondynamicists interested in applying homogenisation results. More generally, we give elementary and explicit sufficient conditions for homogenisation in the case where the fast dynamics is given by a family of dynamical systems.
Finally, we consider the problem of proving rates of convergence in the multidimensional weak invariance principle.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Homogenization (Differential equations), Stochastic differential equations, Diffeomorphisms, Differential equations, Hyperbolic, Transformations (Mathematics), Mappings (Mathematics) | ||||
Official Date: | May 2023 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Melbourne, Ian | ||||
Sponsors: | University of Warwick | ||||
Format of File: | |||||
Extent: | v, 89 pages | ||||
Language: | eng |
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