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

TL;DR

This paper derives exact formulas for Lyapunov exponents in finite-dimensional isotropic stochastic systems with finite correlation time, linking them to the large deviation rate function of the system's coefficients.

Contribution

It provides a novel exact characterization of Lyapunov exponents in isotropic stochastic systems using large deviation principles.

Findings

Exact expressions for Lyapunov exponents derived.

Lyapunov exponents determined by the rate function of diagonal elements.

Results applicable to systems with finite correlation time and isotropy.

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 ${\bf A}$.