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St. Petersburg Mathematical Journal

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What does a generic Markov operator look like?

Author: A. M. Vershik
Translated by: N. Tsilevich
Original publication: Algebra i Analiz, tom 17 (2005), nomer 5.
Journal: St. Petersburg Math. J. 17 (2006), 763-772
MSC (2000): Primary 47B38, 47D07, 28D99, 60J99
Published electronically: July 20, 2006
MathSciNet review: 2241424
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Abstract: 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 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.

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Additional Information

A. M. Vershik
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Markov operator, polymorphism, bistochastic measure
Received by editor(s): March 18, 2005
Published electronically: July 20, 2006
Additional Notes: Partially supported by RFBR (project no. 05-01-00899) and by INTAS (project no. 03-51-5018)
Dedicated: To the memory of O. A. Ladyzhenskaya
Article copyright: © Copyright 2006 American Mathematical Society