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

We consider finite-dimensional systems of linear stochastic differential equations ${\partial }_t{x}_k\left(t\right)=A_{kp}\left(t\right)x_p\left(t\right)$, $\mathbf{A}\left(t\right)$ being a stationary continuous statistically isotropic stochastic process with values in real $d\times d$ matrices. We suppose that the laws of $\mathbf{A}\left(t\right)$ 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 $\mathbf{A}$.