Hilbert space Lyapunov exponent stability

Froyland, Gary; González-Tokman, Cecilia; Quas, Anthony

doi:10.1090/tran/7521

Hilbert space Lyapunov exponent stability

Authors: Gary Froyland, Cecilia González-Tokman and Anthony Quas
Journal: Trans. Amer. Math. Soc. 372 (2019), 2357-2388
MSC (2010): Primary 37H15
DOI: https://doi.org/10.1090/tran/7521
Published electronically: May 20, 2019
MathSciNet review: 3988579
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Abstract: 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.

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