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Uspekhi Mat. Nauk, 2018, Volume 73, Issue 2(440), Pages 3–74 (Mi umn9812)  

This article is cited in 17 scientific papers (total in 18 papers)

Ornstein–Uhlenbeck operators and semigroups

V. I. Bogachevab

a Lomonosov Moscow State University
b National Research University Higher School of Economics

Abstract: This survey gives an account of the state of the art of the theory of Ornstein–Uhlenbeck operators and semigroups. The domains of definition and the spectra of such operators are considered, along with related Sobolev classes with respect to Gaussian measures. Considerable attention is given to various functional inequalities involving such operators and semigroups. Generalized Mehler semigroups are briefly discussed. Major recent achievements are presented and remaining open problems are indicated.
Bibliography: 214 titles.

Keywords: Ornstein–Uhlenbeck operator, Ornstein–Uhlenbeck semigroup, Gaussian measure, Chebyshev–Hermite polynomial, Mehler formula.

Funding Agency Grant Number
Russian Science Foundation 17-11-01058
This research was conducted at Lomonosov Moscow State University with the support of the Russian Science Foundation (grant no. 17-11-01058).


DOI: https://doi.org/10.4213/rm9812

Full text: PDF file (1058 kB)
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English version:
Russian Mathematical Surveys, 2018, 73:2, 191–260

Bibliographic databases:

UDC: 519.2
MSC: Primary 47D03, 47A05, 60H07; Secondary 46G12, 47B38, 60B11
Received: 01.12.2017
Revised: 05.01.2018

Citation: V. I. Bogachev, “Ornstein–Uhlenbeck operators and semigroups”, Uspekhi Mat. Nauk, 73:2(440) (2018), 3–74; Russian Math. Surveys, 73:2 (2018), 191–260

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Garofalo N., Tralli G., “Hardy-Littlewood-Sobolev Inequalities For a Class of Non-Symmetric and Non-Doubling Hypoelliptic Semigroups”, Math. Ann.  crossref  mathscinet  isi
    2. V. I. Bogachev, A. V. Shaposhnikov, S. V. Shaposhnikov, “Estimates for solutions to Fokker-Planck-Kolmogorov equations with integrable drifts”, Dokl. Math., 98:3 (2018), 559–563  crossref  zmath  isi  scopus
    3. E. D. Kosov, “Besov classes on finite and infinite dimensional spaces”, Sb. Math., 210:5 (2019), 663–692  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    4. V. I. Bogachev, A. F. Miftakhov, S. V. Shaposhnikov, “Differential properties of semigroups and estimates of distances between stationary distributions of diffusions”, Dokl. Math., 99:2 (2019), 175–180  crossref  zmath  isi  scopus
    5. V. I. Bogachev, A. V. Shaposhnikov, S. V. Shaposhnikov, “Log-Sobolev-type inequalities for solutions to stationary Fokker-Planck-Kolmogorov equations”, Calc. Var. Partial Differential Equations, 58:5 (2019), 176, 16 pp.  crossref  mathscinet  zmath  isi  scopus
    6. Vladimir I. Bogachev, Egor D. Kosov, Svetlana N. Popova, “A new approach to Nikolskii–Besov classes”, Mosc. Math. J., 19:4 (2019), 619–654  mathnet  crossref
    7. I V. Bogachev, “Distributions of polynomials in many variables and Nikolskii-Besov spaces”, Real Anal. Exch., 44:1 (2019), 49–64  crossref  mathscinet  zmath  isi
    8. V. I. Bogachev, “Non-uniform Kozlov–Treschev averagings in the ergodic theorem”, Russian Math. Surveys, 75:3 (2020), 393–425  mathnet  crossref  crossref  mathscinet  isi  elib
    9. P. Patie, A. Vaidyanathan, “A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators”, Kinet. Relat. Mod., 13:3 (2020), 479–506  crossref  mathscinet  zmath  isi
    10. G. Tralli, “Some global Sobolev inequalities related to Kolmogorov-type operators”, Bruno Pini Math. Anal. Semin., 11:1 (2020), 143–156  isi
    11. U. C. Ji, M. R. Lee, P. Ch. Ma, “Generalized mehler semigroup on white noise functionals and white noise evolution equations”, Mathematics, 8:6 (2020), 1025  crossref  isi
    12. A. Lunardi, G. Metafune, D. Pallara, “The ornstein-uhlenbeck semigroup in finite dimension”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 378:2185 (2020), 20200217  crossref  mathscinet  isi
    13. A. Lunardi, D. Pallara, “Ornstein-uhlenbeck semigroups in infinite dimension”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 378:2185 (2020), 20190620  crossref  mathscinet  isi
    14. Miclo L., Patie P., “On Interweaving Relations”, J. Funct. Anal., 280:3 (2021), 108816  crossref  mathscinet  zmath  isi
    15. Shaposhnikov A., “A Note on Lusin-Type Approximation of Sobolev Functions on Gaussian Spaces”, J. Funct. Anal., 280:6 (2021), 108917  crossref  mathscinet  zmath  isi
    16. V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, “On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation”, Sb. Math., 212:6 (2021), 745–781  mathnet  crossref  crossref  isi  elib
    17. V. I. Bogachev, “Mnogochleny Chebysheva–Ermita i raspredeleniya mnogochlenov ot gaussovskikh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 66:4 (2021), 693–717  mathnet  crossref
    18. P. A. Borodin, I. A. Ibragimov, B. S. Kashin, V. V. Kozlov, A. V. Kolesnikov, S. V. Konyagin, E. D. Kosov, O. G. Smolyanov, N. A. Tolmachev, D. V. Treschev, A. V. Shaposhnikov, S. V. Shaposhnikov, A. N. Shiryaev, A. A. Shkalikov, “Vladimir Igorevich Bogachev (k shestidesyatiletiyu so dnya rozhdeniya)”, UMN, 76:6(462) (2021), 201–208  mathnet  crossref
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