Abstract
We prove that a probability solution of the stationary Kolmogorov equation generated by a first order perturbation v of the Ornstein–Uhlenbeck operator L possesses a highly integrable density with respect to the Gaussian measure satisfying the non-perturbed equation provided that v is sufficiently integrable. More generally, a similar estimate is proved for solutions to inequalities connected with Markov semigroup generators under the curvature condition \(CD(\theta ,\infty )\). For perturbations from Lp an analog of the Log-Sobolev inequality is obtained. It is also proved in the Gaussian case that the gradient of the density is integrable to all powers. We obtain dimension-free bounds on the density and its gradient, which also covers the infinite-dimensional case.
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