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). Few results on this topic are available, such as theoretical works of Il’in and Has’minskiı̆ and a recent more practical contribution of Mamontov and Willander. This is in a disproportion to an 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’minskiı̆ 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. © 2004 Elsevier Ltd. All rights reserved. Key words—Kolmogorov-forward/Fokker–Planck equation, trajectories of an imaginary particle, nonstationary invariant probability density The present work deals with a special topic in theory of diffusion stochastic processes (DSPs) (e.g., [1]–[3]) where not much is known until now. This is nonstationary invariant probability distributions of the processes. Few results on the topic are available, such as theoretical works of A. M. Il’in and R. Z. Has’minskiı̆ [4], [2] and a recent more practical contribution in [3, Chapter 3]. This is in a disproportion to an importance of nonstationary invariant DSPs. Indeed, they comprise the DSP generalizations (e.g., [3, pp. 54-55]; see also Definition 1 below) of the nonstationary steady states described with solutions of determinate ordinary differential equations (ODEs) or partial differential equations uniformly bounded in time on the entire axis (e.g., [5]–[7]). This points out a potentially wide application to mathematics and the natural sciences. Moreover, invariant DSPs arise in connection with such type of stability of systems of Itô’s stochastic differential equations as stability in distribution (e.g., see [4], [2] for the nonstationary case and also [8], [9] for the stationary case). In spite of the fact that this stability is the least restrictive one, it was studied to a lesser extent than other types of stochastic stability. The well-known textbooks (such as [1] or [2]) do not include much on nonstationary invariant DSPs. In particular, the nontransient analytical recipes to determine an invariant probability density (i.e. the recipes which, unlike [4, Theorem 5 and Condition A on p. 258], do not involve the transition distribution) are available only if the density is stationary and the so-called detailed-balance