On uniqueness problems related to the Fokker-Planck-Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂tμ = L*μ for bounded Borel measures on ℝd×[0, T), where L is a second order elliptic operator, for example, Lu = Δxu+(b, ∇xu), and the equation is understood as the identity ∫ (∂tu + Lu) dμ = 0 for all smooth functions u with compact support in ℝd × (0, T). Our study are motivated by equations of such a type, namely, the Fokker-Planck-Kolmogorov equations for transition probabilities of diffusion processes. Solutions are considered in the class of probability measures and in the class of signed measures with integrable densities. We present some recent positive results, give counterexamples, and formulate open problems. Bibliography: 34 titles. © 2011 Springer Science+Business Media, Inc.