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

Abstract

The result of this paper states that every probability measure satisfying the stationary Fokker–Planck–Kolmogorov equation obtained by a -integrable perturbation of the drift term–x of the Ornstein–Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure γ and \(f = \frac{{d\mu }}{{d\gamma }}\) for the density the integral of

\(f\left| {\log } \right|{\left( {f + 1} \right)^\alpha }\)

with respect to γ is estimated via \({\left\| v \right\|_{{L^1}\left( \mu \right)}}\) for all α < \(\frac{1}{4}\). This shows that stationary measures of infinite-dimensional diffusions whose drifts are integrable perturbations of–are absolutely continuous with respect to Gaussian measures. A generalization is obtained for equations on Riemannian manifolds.