Rendiconti Lincei - Matematica e Applicazioni


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Volume 30, Issue 1, 2019, pp. 205–221
DOI: 10.4171/RLM/843

Published online: 2019-04-01

On Sobolev regularity of solutions to Fokker–Planck–Kolmogorov equations with drifts in $L^1$

Vladimir I. Bogachev[1], Svetlana N. Popova[2] and Stanislav V. Shaposhnikov[3]

(1) Moscow State University and National Research University Higher School of Economics, Moscow, Russian Federation
(2) Moscow Institute of Physics and Technology, Russian Federation
(3) Moscow State University and National Research University Higher School of Economics, Moscow, Russian Federation

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.

Keywords: Fokker–Planck–Kolmogorov equations, $L^1$-estimate, Sobolev class

Bogachev Vladimir, Popova Svetlana, Shaposhnikov Stanislav: On Sobolev regularity of solutions to Fokker–Planck–Kolmogorov equations with drifts in $L^1$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), 205-221. doi: 10.4171/RLM/843