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We construct asymptotic expansions for the exponential growth rate (Lyapunov exponent) and rotation number of the random oscillator when the noise is large, small, rapidly varying or slowly varying. We then apply our results to problems in the stability of the random oscillator, the spectrum of the one-dimensional random Schrödinger operator and wave propagation in a one-dimensional random medium.
Arnold et al. (Sun,) studied this question.