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 = dµ/dγ 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 infinitedimensional 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.
MSC: primary 35J15; Secondary 35B65Keywords: Ornstein-Uhlenbeck operator, stationary Fokker-Planck-Kolmogorov equation, integrable drift, logarithmic Sobolev inequality, Gaussian measureThe latter bound admits an infinite-dimensional version. To this end we write the drift b in the form b(x) = −x + v(x).If v = 0, then the only solution in the class of probability measures is the standard Gaussian measure γ with density (2π) −d/2 exp(−|x| 2 /2). Hence it is natural to express µ 1