Estimates for Solutions to Fokker–Planck–Kolmogorov Equations with Integrable Drifts

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 (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.

Log-Sobolev-type inequalities for solutions to stationary Fokker–Planck–Kolmogorov equations

“…as well as its infinite-dimensional analog, which is a constructive sufficient condition for the uniform integrability of the densities of finite-dimensional projections of solutions to infinite-dimensional equations with respect to the corresponding Gaussian measures. It has recently been shown in [14] that these results on Sobolev differentiability of densities break down in the L 1 -setting. It can happen that |b| ∈ L 1 (µ), but the solution density ̺ does not belong to the Sobolev class W 1,1 (R d ), i.e., |∇̺| does not belong to L 1 (R d ), and similarly for the density f the condition |v| ∈ L 1 (γ) does not guarantee that the function |∇f | belongs to L 1 (γ).…”