The Kolmogorov problem on uniqueness of probability solutions of elliptic and parabolic equations

A survey of recent results related to uniqueness problems for Kolmogorov equations will be given. Several long-standing problems will be discussed and solutions for some of them will be presented. In particular, we will demonstrate that in the one-dimensional case always there holds uniqueness of probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation with an unbounded drift coefficient and the unit diffusion coefficient and in all other dimensions it fails. Moreover, we will give several examples when the Kolmogorov equation has infinitely many linearly independent probability solutions. Finally we will discuss close problem on distances between stationary measures and transition probabilities of diffusion process.