On Sobolev Classes Containing Solutions to Fokker–Planck–Kolmogorov Equations
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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 W1,1(ℝd). 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.
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