Email this article On Sobolev Classes Containing Solutions to Fokker–Planck–Kolmogorov Equations
- Authors: Bogachev V.I.1,2,3, Popova S.N.1, Shaposhnikov S.V.1,2,3
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Affiliations:
- Department of Mechanics and Mathematics
- National Research University Higher School of Economics
- St.-Tikhon’s Orthodox University
- Issue: Vol 98, No 2 (2018)
- Pages: 498-501
- Section: Mathematics
- URL: https://journal-vniispk.ru/1064-5624/article/view/225566
- DOI: https://doi.org/10.1134/S1064562418060273
- ID: 225566
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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.
About the authors
V. I. Bogachev
Department of Mechanics and Mathematics; National Research University Higher School of Economics; St.-Tikhon’s Orthodox University
Author for correspondence.
Email: vibogach@mail.ru
Russian Federation, Moscow, 119991; Moscow; Moscow
S. N. Popova
Department of Mechanics and Mathematics
Email: vibogach@mail.ru
Russian Federation, Moscow, 119991
S. V. Shaposhnikov
Department of Mechanics and Mathematics; National Research University Higher School of Economics; St.-Tikhon’s Orthodox University
Email: vibogach@mail.ru
Russian Federation, Moscow, 119991; Moscow; Moscow
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