# 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

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topAnzelm 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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- [2] W. Bartoszek, On the residuality of mixing by convolution probabilities, ibid., to appear.
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- [10] U. Krengel and L. Sucheston, On mixing in infinite measure spaces, Z. Wahrsch. Verw. Gebiete 13 (1969), 150-164.
- [11] M. Lin, Mixing for Markov operators, ibid. 19 (1971), 231-242.
- [12] M. Lin, Convergence of the iterates of a Markov operator, ibid. 29 (1974), 153-163.
- [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] J. Rosenblatt, Ergodic and mixing random walks on locally compact groups, Math. Ann. 257 (1981), 31-42.
- [15] U. Sachdeva, On category of mixing in infinite measure spaces, Math. Systems Theory 5 (1978), 319-330.
- [16] H. H. Schaefer, Banach Lattices and Positive Operators, Springer, New York 1974.

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