Statistical averages in dynamical systems

General formulas are deduced for calculating the correlation of a random process with a functional over it. These formulas yield closed equations for the probability densities of dynamical systems with random parameters. Parametric resonance in a system whose eigenfrequency is a random function of the time is considered as an example. An equation of general form is obtained for the joint probability density of the coordinate and velocity, this going over into the Einstein–Fokker equation if the frequency is a Gaussian $\delta$-correlated function of the time and into the Kolmogorov–Feller equation if the frequency is a generalized Poisson $\delta$-correlated random process.