Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. A. M. Vershik, Polymorphisms, Markov processes, and quasi-similarity, Discrete Contin. Dyn. Syst. 13 (2005), no. 5, 1305–1324. MR 2166671, 10.3934/dcds.2005.13.1305
  • 2. A. M. Veršik, Multivalued mappings with invariant measure (polymorphisms) and Markov operators, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 72 (1977), 26–61, 223 (Russian, with English summary). Problems of the theory of probability distributions, IV. MR 0476998
  • 3. A. M. Vershik, Superstability of hyperbolic automorphisms, and unitary dilations of Markov operators, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 3 (1987), 28–33, 127 (Russian, with English summary). MR 928157
  • 4. A. M. Veršik and O. A. Ladyženskaja, The evolution of measures as determined by Navier-Stokes equations, and the solvability of the Cauchy problem for the statistical Hopf equation, Dokl. Akad. Nauk SSSR 226 (1976), no. 1, 26–29 (Russian). MR 0397198
  • 5. O. A. Ladyzhenskaya and A. M. Vershik, Sur l’évolution des mesures déterminées par les équations de Navier-Stokes et la résolution du problème de Cauchy pour l’équation statistique de E. Hopf, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 2, 209–230 (French). MR 0447844
  • 6. Murray Rosenblatt, Markov processes. Structure and asymptotic behavior, Springer-Verlag, New York-Heidelberg, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 184. MR 0329037
  • 7. Béla Sz.-Nagy and Ciprian Foiaș, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR 0275190
  • 8. V. N. Sudakov, Geometric problems of the theory of infinite-dimensional probability distributions, Trudy Mat. Inst. Steklov. 141 (1976), 191 (Russian). MR 0431359
  • 9. D. Revuz, Markov chains, 2nd ed., North-Holland Mathematical Library, vol. 11, North-Holland Publishing Co., Amsterdam, 1984. MR 758799
  • 10. S. A. Juzvinskiĭ, Metric automorphisms with a simple spectrum, Dokl. Akad. Nauk SSSR 172 (1967), 1036–1038 (Russian). MR 0227357
  • 11. Robert R. Phelps, Lectures on Choquet’s theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. MR 1835574
  • 12. A. M. Vershik, Dynamic theory of growth in groups: entropy, boundaries, examples, Uspekhi Mat. Nauk 55 (2000), no. 4(334), 59–128 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 4, 667–733. MR 1786730, 10.1070/rm2000v055n04ABEH000314
  • 13. Steven Arthur Kalikow, 𝑇,𝑇⁻¹ transformation is not loosely Bernoulli, Ann. of Math. (2) 115 (1982), no. 2, 393–409. MR 647812, 10.2307/1971397
  • 14. T. Downarowich and B. Frei, Measure-theoretic and topological entropy of operators on function space, Ergodic Theory Dynam. Systems 25 (2005), no. 2, 105.

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 47B38, 47D07, 28D99, 60J99

Retrieve articles in all journals with MSC (2000): 47B38, 47D07, 28D99, 60J99


Additional Information

A. M. Vershik
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: vershik@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-06-00928-9
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