Simple exact expressions are derived for all the Lyapunov exponents of certain N-dimensional stochastic linear dynamical systems. In the case of the product of independent random matrices, each of which has independent Gaussian entries with mean zero and variance 1/N, the exponents have an exponential distribution as N→∞. In the case of the time-ordered product integral of exp[N-1/2dW], where the entries of the N×N matrix W(t) are independent standard Wiener processes, the exponents are equally spaced for fixed N and thus have a uniform distribution as N→∞.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics