Uniqueness problems for the Fokker-Planck-Kolmogorov equations

This talk concerns several elementary formulated uniqueness problems connected with the stationary Kolmogorov equation and the Cauchy problem for the evolution Fokker – Planck – Kolmogorov equation. Some of these problems, posed by Kolmogorov himself in the 1930s or naturally connected with Kolmogorov's questions, have been solved relatively recently, but others remain open. We shall mainly discuss the case of the unit diffusion coefficient and a smooth drift b, where the stationary equation with respect to a measure µ has the form ∆µ − div(bµ) = 0 or the same form for the solution density, and the parabolic equation for the density p(x,t) is written as ∂_t p = ∆p − div(bp). Such equations can be considered in the class of probability measures as well as in the class of bounded (possibly, signed) measures, which leads to different interesting problems. The uniqueness problems we discuss are also related to stationary measures of diffusions, Chapman – Kolmogorov equations and semigroups generated by second order elliptic operators. For understanding the essence of our main problems it is enough to have acquaintance with two semesters of calculus.