Regularity of Solutions to Kolmogorov Equations with Perturbed Drifts

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