Nonstationary invariant distributions and the hydrodynamics-style generalization of the Kolmogorov-forward/Fokker-Planck equation

The work deals with nonstationary invariant probability distributions of diffusion stochastic processes (DSPs). A few results on this topic are available, such as theoretical works of Il'in and Has'minskiĭ and a recent more practical contribution of Mamontov and Willander. This is in disproportion to the importance of nonstationary invariant DSPs which have a potentially wide application to the natural sciences and mathematics, in particular, stability in distribution, the least restrictive type of stochastic stability. The nontransient analytical recipes to determine an invariant probability density are available only if the density is stationary and the so-called detailed-balance condition holds. If the invariant density is nonstationary, the recipes are unknown. This is one of the fundamental problems still unsolved in theory of DSPs. The present work proposes a solution of the problem and illustrates the solution with the new results on the Il'in-Has'minskiĭ example. The work also discusses the developed recipe in connection with stability in distribution and the uniform boundedness in time, and suggests a few directions for future research in mathematics and biology.