Structure of mixing and category of complete mixing for stochastic operators

A. Iwanik; R. 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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A. Iwanik, and R. 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{A1992,
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 = {A. Iwanik, R. 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 - A. Iwanik
AU - R. 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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  16. [16] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York 1974. Zbl0296.47023

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