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

Vladimir I. Bogachev; Svetlana N. Popova; Stanislav V. Shaposhnikov

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arXiv · 1803.04568

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

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.

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Vladimir I. Bogachev, Svetlana N. Popova, Stanislav V. Shaposhnikov. 2018-03-12. On $L^1$-estimates for probability solutions to Fokker-Planck-Kolmogorov equations. https://arxiv.org/abs/1803.04568

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