Hilbert space Lyapunov exponent stability

We study cocycles of compact operators acting on a separable Hilbert space and investigate the stability of the Lyapunov exponents and Oseledets spaces when the operators are subjected to additive Gaussian noise. We show that as the noise is shrunk to 0, the Lyapunov exponents of the perturbed cocyc...

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Veröffentlicht in:Transactions of the American Mathematical Society 2019-08, Vol.372 (4), p.2357-2388
Hauptverfasser: Gary Froyland, Cecilia González-Tokman, Anthony Quas
Format: Artikel
Sprache:eng
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Zusammenfassung:We study cocycles of compact operators acting on a separable Hilbert space and investigate the stability of the Lyapunov exponents and Oseledets spaces when the operators are subjected to additive Gaussian noise. We show that as the noise is shrunk to 0, the Lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle, and the Oseledets spaces converge in probability to those of the unperturbed cocycle. This is, to our knowledge, the first result of this type with cocycles taking values in operators on infinite-dimensional spaces. The infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite-dimensional case.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/7521