Geodesic
Ornstein--Uhlenbeck operators and semigroups
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 73 (2018) no. 2, pp. 191-260

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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. 
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     title = {Ornstein--Uhlenbeck operators and semigroups},
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V. I. Bogachev. Ornstein--Uhlenbeck operators and semigroups. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 73 (2018) no. 2, pp. 191-260. http://geodesic.mathdoc.fr/item/RM_2018_73_2_a0/