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arXiv · 1805.09467

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

V. I. Bogachev·A. V. Shaposhnikov·S. V. Shaposhnikov
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Abstract

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 means that stationary measures of diffusions whose drifts are integrable perturbations of $-x$ are absolutely continuous with respect to Gaussian measures.

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V. I. Bogachev, A. V. Shaposhnikov, S. V. Shaposhnikov. 2019-08-08. Log-Sobolev-type inequalities for solutions to stationary Fokker-Planck-Kolmogorov equations. https://arxiv.org/abs/1805.09467

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