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Mathematics > Probability

arXiv:1803.04568 (math)
[Submitted on 12 Mar 2018]

Title:On $L^1$-estimates for probability solutions to Fokker-Planck-Kolmogorov equations

Authors:Vladimir I. Bogachev, Svetlana N. Popova, Stanislav V. Shaposhnikov
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Abstract:We prove two new results connected with elliptic Fokker-Planck-Kolmogorov equations with drifts integrable with respect to solutions. The first result 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~$W^{1,1}(\mathbb{R}^d)$. There is also a version of this result for densities with respect to Gaussian measures. The second new result gives some positive information about properties of such solutions: the solution density is proved to belong to certain fractional Sobolev classes.
Subjects: Probability (math.PR)
MSC classes: 35J15
Cite as: arXiv:1803.04568 [math.PR]
  (or arXiv:1803.04568v1 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.1803.04568

Submission history

From: Vladimir Bogachev I. [view email]
[v1] Mon, 12 Mar 2018 23:09:20 UTC (18 KB)
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