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

  • V. I. Bogachev
  • M. Röckner
  • S. V. Shaposhnikov
Article
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.

Keywords

Probability Measure Cauchy Problem Lebesgue Measure Parabolic Equation Lyapunov Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • V. I. Bogachev
    • 1
  • M. Röckner
    • 2
  • S. V. Shaposhnikov
    • 3
  1. 1.Moscow State UniversityMoscowRussia
  2. 2.Universität BielefeldBielefeldGermany
  3. 3.Moscow State UniversityMoscowRussia

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