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 $\|v\|_{L^1(\mu)}$ 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.