Structure of mixing and category of complete mixing for stochastic operators

Anzelm Iwanik; Ryszard Rębowski

Annales Polonici Mathematici (1992)

  • Volume: 56, Issue: 3, page 233-242
  • ISSN: 0066-2216

Abstract

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Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak compactness. Finally, it is shown that most stochastic operators are completely mixing and that the same holds for convolution stochastic operators on l.c.a. groups.

How to cite

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Anzelm Iwanik, and Ryszard Rębowski. "Structure of mixing and category of complete mixing for stochastic operators." Annales Polonici Mathematici 56.3 (1992): 233-242. <http://eudml.org/doc/262386>.

@article{AnzelmIwanik1992,
abstract = {Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak compactness. Finally, it is shown that most stochastic operators are completely mixing and that the same holds for convolution stochastic operators on l.c.a. groups.},
author = {Anzelm Iwanik, Ryszard Rębowski},
journal = {Annales Polonici Mathematici},
keywords = {Markov operator; conservative deterministic factor; sweeping; stochastic operator on a -finite standard measure space; subinvariant measure; Frobenius-Perron operator; invertible measure preserving transformation; mixing transformation; weak compactness; completely mixing},
language = {eng},
number = {3},
pages = {233-242},
title = {Structure of mixing and category of complete mixing for stochastic operators},
url = {http://eudml.org/doc/262386},
volume = {56},
year = {1992},
}

TY - JOUR
AU - Anzelm Iwanik
AU - Ryszard Rębowski
TI - Structure of mixing and category of complete mixing for stochastic operators
JO - Annales Polonici Mathematici
PY - 1992
VL - 56
IS - 3
SP - 233
EP - 242
AB - Let T be a stochastic operator on a σ-finite standard measure space with an equivalent σ-finite infinite subinvariant measure λ. Then T possesses a natural "conservative deterministic factor" Φ which is the Frobenius-Perron operator of an invertible measure preserving transformation φ. Moreover, T is mixing ("sweeping") iff φ is a mixing transformation. Some stronger versions of mixing are also discussed. In particular, a notion of *L¹-s.o.t. mixing is introduced and characterized in terms of weak compactness. Finally, it is shown that most stochastic operators are completely mixing and that the same holds for convolution stochastic operators on l.c.a. groups.
LA - eng
KW - Markov operator; conservative deterministic factor; sweeping; stochastic operator on a -finite standard measure space; subinvariant measure; Frobenius-Perron operator; invertible measure preserving transformation; mixing transformation; weak compactness; completely mixing
UR - http://eudml.org/doc/262386
ER -

References

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  1. [1] J. Aaronson, M. Lin and B. Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products, Israel J. Math. 33 (1979), 198-224. 
  2. [2] W. Bartoszek, On the residuality of mixing by convolution probabilities, ibid., to appear. 
  3. [3] J. R. Choksi and V. S. Prasad, Approximation and Baire category theorems in ergodic theory, in: Proc. Sherbrooke Workshop Measure Theory (1982), Lecture Notes in Math. 1033, Springer, 1983, 94-113. 
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  6. [6] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 2, Springer, Berlin 1970. 
  7. [7] A. Iwanik, Baire category of mixing for stochastic operators, in: Proc. Measure Theory Conference, Oberwolfach 1990, Rend. Circ. Mat. Palermo (2), to appear. 
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  9. [9] T. Komorowski and J. Tyrcha, Asymptotic properties of some Markov operators, Bull. Polish Acad. Sci. Math. 37 (1989), 221-228. 
  10. [10] U. Krengel and L. Sucheston, On mixing in infinite measure spaces, Z. Wahrsch. Verw. Gebiete 13 (1969), 150-164. 
  11. [11] M. Lin, Mixing for Markov operators, ibid. 19 (1971), 231-242. 
  12. [12] M. Lin, Convergence of the iterates of a Markov operator, ibid. 29 (1974), 153-163. 
  13. [13] D. Ornstein and L. Sucheston, An operator theorem on L₁ convergence to zero with applications to Markov kernels, Ann. Math. Statist. 41 (1970), 1631-1639. 
  14. [14] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), 31-42. 
  15. [15] U. Sachdeva, On category of mixing in infinite measure spaces, Math. Systems Theory 5 (1978), 319-330. 
  16. [16] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York 1974. 

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