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What does a generic Markov operator look like? (English. Russian original) Zbl 1173.47306

St. Petersbg. Math. J. 17, No. 5, 763-772 (2006); translation from Algebra Anal. 17, No. 5, 91-104 (2005).
Generic (i.e., forming an everywhere dense massive subset) classes of Markov operators in the space \(L^2(X,\mu)\) with a finite continuous measure are considered. In a canonical way, each Markov operator is associated with a multivalued measure-preserving transformation (i.e., a polymorphism), and also with a stationary Markov chain; therefore, one can also talk of generic polymorphisms and generic Markov chains. Not only had the generic nature of the properties discussed in the paper been unclear before this research, but not even the very existence of Markov operators that enjoy these properties in full or partly was known. The most important result is that the class of totally nondeterministic nonmixing operators is generic. A number of problems is posed; there is some hope that generic Markov operators will find applications in various fields, including statistical hydrodynamics.

MSC:

47B38 Linear operators on function spaces (general)
47A35 Ergodic theory of linear operators
37A25 Ergodicity, mixing, rates of mixing
37A30 Ergodic theorems, spectral theory, Markov operators
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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References:

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