Lyapunov spectrum of ergodic stationary systems perturbed by white noise

Abstract

In the present paper it will be proved that the Lyapunov spectrum of an ergodic stationary homogenous linear differential system perturbed by Gaussian white noise dx = A(t)x dt + dW(t) (∈Rⁿ) consists of only one element, namely the number λ = max(λ_n) where λ_n is the top exponent of the unperturbed system dx = A{t)xdt. The proof is based on Oseledec-Millionshikov's famous multiplicative ergodic theorem and the Floquet type representation of Wihstutz for the fundamental matrix of the ergodic stationary system dx = A{t)xdt.

Keywords:

• Lyapunov exponent
• ergodic stationarysystems
• multiplicative ergodic theorem