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1991-1992 | 56 | 3 | 233-242
Tytuł artykułu

Structure of mixing and category of complete mixing for stochastic operators

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
56
Numer
3
Strony
233-242
Opis fizyczny
Daty
wydano
1992
otrzymano
1990-08-01
poprawiono
1991-11-04
Twórcy
autor
  • Institute of Mathematics, Technical University of Wrocław Wybrzeże, Wyspiańskiego 27, 50-370 Wrocław, Poland
autor
  • Institute of Mathematics, Technical University of Wrocław Wybrzeże, Wyspiańskiego 27, 50-370 Wrocław, Poland
Bibliografia
  • [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] W. Bartoszek, On the residuality of mixing by convolution probabilities, ibid., to appear.
  • [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.
  • [4] S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand-Reinhold, New York 1969.
  • [5] S. R. Foguel, Singular Markov operators, Houston J. Math. 11 (1985), 485-489.
  • [6] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 2, Springer, Berlin 1970.
  • [7] A. Iwanik, Baire category of mixing for stochastic operators, in: Proc. Measure Theory Conference, Oberwolfach 1990, Rend. Circ. Mat. Palermo (2), to appear.
  • [8] B. Jamison and S. Orey, Markov chains recurrent in the sense of Harris, Z. Wahrsch. Verw. Gebiete 8 (1967), 41-48.
  • [9] T. Komorowski and J. Tyrcha, Asymptotic properties of some Markov operators, Bull. Polish Acad. Sci. Math. 37 (1989), 221-228.
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv56z3p233bwm
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