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

TL;DR

This paper establishes a weakened logarithmic Sobolev inequality for stationary measures of certain diffusions, showing their absolute continuity with respect to Gaussian measures under integrable perturbations of the drift.

Contribution

It proves a new inequality relating stationary measures' densities to perturbations of the Ornstein-Uhlenbeck drift, extending the understanding of measure regularity.

Findings

Stationary measures are absolutely continuous w.r.t. Gaussian measures.

The integral of the density times a logarithmic factor is bounded by the perturbation norm.

The result generalizes classical inequalities to a broader class of diffusions.

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.