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Regularity of Solutions to Kolmogorov Equations with Perturbed Drifts

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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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Acknowledgements

We are very grateful to the anonymous referee for thorough reading and important corrections. This research is supported by the Russian Science Foundation Grant 17-11-01058 at Lomonosov Moscow State University (the results in Section ?? and Section ??). The second author is a winner of the “Young Russian Mathematics” contest and thanks its sponsors and jury. The work of A.V. Shaposhnikov (the results in Section ??) was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2019-1614).

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Authors and Affiliations

  1. Department of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991, Russia

    Vladimir I. Bogachev & Egor D. Kosov

  2. National Research University, Higher School of Economics, Usacheva 6, 119048, Moscow, Russia

    Vladimir I. Bogachev & Egor D. Kosov

  3. Steklov Mathematical Institute of Russian Academy of Science, 8 Gubkina Street, 119991, Moscow, Russia

    Alexander V. Shaposhnikov

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  1. Vladimir I. Bogachev
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  2. Egor D. Kosov
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  3. Alexander V. Shaposhnikov
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Correspondence to Vladimir I. Bogachev.

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Bogachev, V.I., Kosov, E.D. & Shaposhnikov, A.V. Regularity of Solutions to Kolmogorov Equations with Perturbed Drifts. Potential Anal 58, 681–702 (2023). https://doi.org/10.1007/s11118-021-09954-9

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