Estimates for Solutions to Fokker–Planck–Kolmogorov Equations with Integrable Drifts¹

Abstract—The result of this paper states that every probability measure $\mu$ satisfying the stationary Fokker–Planck–Kolmogorov equation obtained by a $\mu$-integrable perturbation ${v}$ of the drift term –x of the Ornstein–Uhlenbeck operator is absolutely continuous with respect to the corresponding Gaussian measure $\gamma$ and for the density $f = \frac{{d\mu }}{{d\gamma }}$ the integral of$f{\text{|}}log(f + 1){{{\text{|}}}^{\alpha }}$with respect to $\gamma$ is estimated via ${\text{||}}{v}{\text{|}}{{{\text{|}}}_{{{{L}^{1}}(\mu )}}}$ for all $\alpha < \frac{1}{4}$. 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.