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

Abstract

We prove 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 $f=d\mu /d\gamma$ the integral of $f|\log (f+1){|}^{\alpha }$ against $\gamma$ is estimated via $\Vert v{\Vert }_{L^1(\mu )}$ for all $\alpha <1/4$, which is a weakened $L^1$-analog of the logarithmic Sobolev inequality. This means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolutely continuous with respect to Gaussian measures.