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Doklady Mathematics

, Volume 98, Issue 2, pp 498–501 | Cite as

On Sobolev Classes Containing Solutions to Fokker–Planck–Kolmogorov Equations

  • V. I. Bogachev
  • S. N. Popova
  • S. V. Shaposhnikov
Mathematics

Abstract

The main result of this paper answers negatively a long-standing question and shows that a density of a probability measure satisfying the Fokker–Planck–Kolmogorov equation with a drift integrable with respect to this density can fail to belong to the Sobolev class W1,1(ℝd). There is also a version of this result for densities with respect to Gaussian measures. On the other hand, we prove that the solution density belongs to certain fractional Sobolev classes.

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • V. I. Bogachev
    • 1
    • 2
    • 3
  • S. N. Popova
    • 1
  • S. V. Shaposhnikov
    • 1
    • 2
    • 3
  1. 1.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.St.-Tikhon’s Orthodox UniversityMoscowRussia

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