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Ornstein-Uhlenbeck operators and semigroups. (English. Russian original) Zbl 06945050

Russ. Math. Surv. 73, No. 2, 191-260 (2018); translation from Usp. Mat. Nauk 73, No. 2, 3-74 (2018).
Summary: 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.

MSC:

47D03 Groups and semigroups of linear operators
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
60H07 Stochastic calculus of variations and the Malliavin calculus
46G12 Measures and integration on abstract linear spaces
47B38 Linear operators on function spaces (general)
60B11 Probability theory on linear topological spaces
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