Random random matrix products

The paper deals with compositions of independent random bundle maps whose distributions form a stationary process which leads to study of Markov processes in random environments. A particular case of this situation is a product of independent random matrices with stationarity changing distributions. We obtain results concerning invariant filtrations for such systems, positivity and simplicity of the largest Lyapunov exponent, as well as central limit theorem type results. An application to random harmonic functions and measures is also considered. Continuous time versions of these results, which yield applications to linear stochastic differential equations in random environments, are also discussed.