On Sobolev Classes Containing Solutions to Fokker–Planck–Kolmogorov Equations

Resumo

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 W^1,1(ℝ^d). 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.