Modeling evolution sample distributions of random quantities by the equation of Liuville

The difference approximation of the one-dimensional Liouville equation for the sample distribution density of the non-stationary time series estimated by the histogram is considered. We prove a necessary and sufficient condition that the change in the sample density of the distribution over a certain period of time can be modeled as the evolution of the density according to the Liouville equation. This condition is a strong positivity of the initial density distribution in the inner class intervals. The determination of the corresponding velocity algorithm is constructed and its mechanical-statistical meaning is shown as a semigroup equivalent in Chernoff sense to the average semigroup, generating the evolution of the distribution function.