Long-term properties of finite-correlation time isotropic stochastic systems

Abstract

We consider finite-dimensional systems of linear stochastic differential equations ${\partial_t}{x_k}\left( t \right) = {A_{kp}}\left( t \right){x_p}\left( t \right)$, ${\bf A}(t)$ being a stationary continuous statistically isotropic stochastic process with values in real $d \times d$ matrices. We suppose also that the laws of ${\bf A}(t)$ satisfy the large deviation principle. For these systems, we find exact expressions for the Lyapunov and generalized Lyapunov exponents and show that they are determined in a precise way only by the rate function of the diagonal elements of ${\bf A}$.