Strong and weak stability of some Markov operators

Rudnicki, Ryszard

Colloquium Mathematicae, Tome 84/85 (2000), p. 255-263 / Harvested from The Polish Digital Mathematics Library

Résumé

An integral Markov operator $P$ appearing in biomathematics is investigated. This operator acts on the space of probabilistic Borel measures. Let $μ$ and $ν$ be probabilistic Borel measures. Sufficient conditions for weak and strong convergence of the sequence $(P^{n} μ - P^{n} ν)$ to $0$ are given.

Publié le : 2000-01-01

Zbl 0992.47016

EUDML-ID : urn:eudml:doc:210804

@article{bwmeta1.element.bwnjournal-article-cmv84i1p255bwm,
     author = {Ryszard Rudnicki},
     title = {Strong and weak stability of some Markov operators},
     journal = {Colloquium Mathematicae},
     volume = {84/85},
     year = {2000},
     pages = {255-263},
     zbl = {0992.47016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p255bwm}
}

Rudnicki, Ryszard. Strong and weak stability of some Markov operators. Colloquium Mathematicae, Tome 84/85 (2000) pp. 255-263. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p255bwm/

Bibliographie

[000] [1] M. F. Barnsley, Fractals Everywhere, Acad. Press, New York, 1988.

[001] [2] K. Baron and A. Lasota, Asymptotic properties of Markov operators defined by Volterra type integrals, Ann. Polon. Math. 58 (1993), 161-175. | Zbl 0839.47021

[002] [3] C. J. K. Batty, Z. Brzeźniak and D. A. Greenfield, A quantitative asymptotic theorem for contraction semigroups with countable unitary spectrum, Studia Math. 121 (1996), 167-183. | Zbl 0862.47020

[003] [4] S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Reinhold, New York, 1969.

[004] [5] S. R. Foguel, Harris operators, Israel J. Math. 33 (1979), 281-309.

[005] [6] H. Gacki and A. Lasota, Markov operators defined by Volterra type integrals with advanced argument, Ann. Polon. Math. 51 (1990), 155-166. | Zbl 0721.34094

[006] [7] A. Iwanik, Baire category of mixing for stochastic operators, Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992), 201-217. | Zbl 0764.60059

[007] [8] T. Komorowski and J. Tyrcha, Asymptotic properties of some Markov operators, Bull. Polish Acad. Sci. Math. 37 (1989), 221-228. | Zbl 0767.47012

[008] [9] A. Lasota and M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Appl. Math. Sci. 97, Springer, New York, 1994.

[009] [10] A. Lasota and M. C. Mackey, Global asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 43-62. | Zbl 0529.92011

[010] [11] A. Lasota, M. C. Mackey and J. Tyrcha, The statistical dynamics of recurrent biological events, ibid. 30 (1992), 775-800. | Zbl 0763.92001

[011] [12] M. Lin, Mixing for Markov operators, Z. Wahrsch. Verw. Gebiete 19 (1971), 231-242. | Zbl 0212.49301

[012] [13] K. Łoskot and R. Rudnicki, Sweeping of some integral operators, Bull. Polish Acad. Sci. Math. 37 (1989), 229-235. | Zbl 0767.47013

[013] [14] J. van Neerven, The Asymptotic Behaviour of a Semigroup of Linear Operators, Birkhäuser, Basel, 1996. | Zbl 0905.47001

[014] [15] E. Nummelin, General Irreducible Markov Chains and Non-Negative Operators, Cambridge Tracts in Math. 83, Cambridge Univ. Press, Cambridge, 1984. | Zbl 0551.60066

[015] [16] R. Rudnicki, Stability in $L^{1}$ of some integral operators, Integral Equations Operator Theory 24 (1996), 320-327. | Zbl 0843.47021

[016] [17] R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262. | Zbl 0838.47040

[017] [18] J. Tyrcha, Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle, J. Math. Biol. 26 (1988), 465-475. | Zbl 0716.92017

[018] [19] J. J. Tyson, Mini review: Size control of cell division, J. Theoret. Biol. 120 (1987), 381-391.

[019] [20] J. J. Tyson and K. B. Hannsgen, Global asymptotic stability of the size distribution in probabilistic models of the cell cycle, J. Math. Biol. 22 (1985), 61-68. | Zbl 0558.92012

[020] [21] J. J. Tyson and K. B. Hannsgen, Cell growth and division: A deterministic/probabilistic model of the cell cycle, ibid. 23 (1986), 231-246. | Zbl 0582.92020