On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures
 V. I. Bogachev,
 M. Röckner,
 S. V. Shaposhnikov
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Get AccessWe 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.
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 Title
 On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures
 Journal

Journal of Mathematical Sciences
Volume 179, Issue 1 , pp 747
 Cover Date
 20111101
 DOI
 10.1007/s1095801105816
 Print ISSN
 10723374
 Online ISSN
 15738795
 Publisher
 Springer US
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 Authors

 V. I. Bogachev ^{(1)}
 M. Röckner ^{(2)}
 S. V. Shaposhnikov ^{(3)}
 Author Affiliations

 1. Moscow State University, 119991, Moscow, Russia
 2. Universität Bielefeld, Bielefeld, 33501, Germany
 3. Moscow State University, 119991, Moscow, Russia