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

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.