The time evolution of dynamical systems with random initial conditions is considered, by deriving the nth order probability density of the stochastic process which describes the response of the system, and the entropy function related to the said distribution. A constructive theorem is proved, which enables the explicit calculation of the nth order probability density in terms of the statistics of the initial conditions. Some monotonicity properties of the entropy are derived, and the results are applied in two examples. The same analysis can be applied to the study of the probabilistic response of dynamical systems with constant random parameters and deterministic initial conditions.
