Extremal Character of the Lyapunov Exponent of the Stochastic Harmonic Oscillator

Abstract

We give a formula for the quadratic Lyapunov exponent of the harmonic oscillator in the presence of a finite-state Markov noise process. In case the noise process is reversible, the quadratic Lyapunov exponent is strictly less than that for the corresponding white-noise process obtained from the central limit theorem. An example is presented of a nonreversible Markov noise process for which this inequality is reversed.