Regularity of solutions to Kolmogorov equations with perturbed drifts

TL;DR

This paper establishes the regularity and integrability properties of solutions to perturbed Kolmogorov equations, including density bounds and gradient integrability, under various conditions and in infinite-dimensional settings.

Contribution

It provides new integrability and regularity estimates for solutions to perturbed Kolmogorov equations, extending results to infinite dimensions and establishing dimension-free bounds.

Findings

Solutions have highly integrable densities with respect to Gaussian measures.

Gradient of the density is integrable to all powers.

Dimension-free bounds on densities and gradients are obtained.

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 $L^p$ 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.