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

We prove that every probability measure μ satisfying the stationary Fokker–Planck–Kolmogorov equation obtained by a μ -integrable perturbation v of the drift term - x of the Ornstein–Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure γ and for the density f = d μ / d γ the integral of f | log ( f + 1 ) | α against γ is estimated via ‖ v ‖ L 1 ( μ ) for all α < 1 / 4 , which is a weakened L 1 -analog of the logarithmic Sobolev inequality. This yields that stationary measures of infinite-dimensional diffusions whose drifts are integrable perturbations of - x are absolutely continuous with respect to Gaussian measures. A generalization is obtained for equations on Riemannian manifolds.