, Volume 179, Issue 1, pp 7-47

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

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We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ t μ = L * μ for bounded Borel measures on ℝ d  × (0, T), where L is a second order elliptic operator, for example, \( Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) \) , and the equation is understood as the identity $$ \int \left( {{\partial_t}u + Lu} \right)d\mu = 0 $$ for all smooth functions u with compact support in ℝ d  × (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.

Dedicated to Professor N. V. Krylov on the occasion of his 70th birthday
Translated from Problems in Mathematical Analysis 61, October 2011, pp. 9–42.