  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.
