Lecture Notes in Mathematics Volume 1186, 1986, pp 85-125
Date: 17 Sep 2006

Lyapunov exponents of linear stochastic systems

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The asymptotic behavior of linear stochastic systems in Rd of the form ẋ = A(ξ(t))x, x(o) = xo ε Rd, ξ(t) stationary stochastic process, is investigated by means of geometric nonlinear control theory. Concerning the rotational behavior it is proved that the projection onto the unit sphere has a unique invariant probability. Concerning the stability it is proved that the solution x(t;xo) has an exponential growth rate $$\lambda = \mathop {\lim }\limits_{t \to \infty } \tfrac{1}{t} \log |x(t;x_O )|$$ which is independent of chance and of xo and equal to the biggest Lyapunov exponent from the multiplicative ergodic theorem.