Fokker–Planck–Kolmogorov equations: the main achievements and open problems

The study of mathematical aspects of Fokker–Planck–Kolmogorov equations began with Kolmogorov’s papers at the beginning of the 1930s, although in the physics literature such equations had appeared earlier, in particular, in the papers of Fokker, Planck, Smoluchowski, Chapman, Bachelier, and other known scholars. The most important feature of Fokker–Planck–Kolmogorov equations, which distinguishes them from the classical direct and divergence form elliptic and parabolic equations and explains their frequently encountered name "double divergence form equations", is the action of partial derivatives not directly on solutions, but on the products of solutions with the equation coefficients. The talk is devoted to several key problems in the theory of Fokker–Planck–Kolmogorov equations: the existence of solutions and their properties, uniqueness problems, connections with diffusion processes. Some of these problems were posed by Kolmogorov himself, but, in spite of significant progress over the past decades, are not yet solved completely.